Find The Quartic Function That Is The Best Fit For The Data In The Following Table.${ \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline X X X & -3 & -2 & -1 & 0 & 1 & 2 & 3 \ \hline Y Y Y & 70 & 15 & 4 & 1 & -6 & -5 & 40 \ \hline \end{tabular} }$The

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Introduction

In mathematics, finding the best fit function for a given dataset is a crucial task in various fields such as statistics, engineering, and data analysis. A quartic function is a polynomial function of degree four, which means it has the general form of f(x)=ax4+bx3+cx2+dx+ef(x) = ax^4 + bx^3 + cx^2 + dx + e. In this article, we will discuss how to find the quartic function that best fits the data in the given table.

Understanding the Data

The given table contains seven data points, each with a corresponding value of xx and yy. The data points are:

xx yy
-3 70
-2 15
-1 4
0 1
1 -6
2 -5
3 40

The Method of Least Squares

The method of least squares is a widely used technique for finding the best fit function for a given dataset. The basic idea behind this method is to minimize the sum of the squared errors between the observed data points and the predicted values of the function.

Let's assume that we have a function f(x)=ax4+bx3+cx2+dx+ef(x) = ax^4 + bx^3 + cx^2 + dx + e that we want to fit to the given data. We can write the sum of the squared errors as:

S=∑i=1n(yi−f(xi))2S = \sum_{i=1}^{n} (y_i - f(x_i))^2

where yiy_i is the observed value of the ithi^{th} data point, f(xi)f(x_i) is the predicted value of the function at xix_i, and nn is the number of data points.

Finding the Coefficients of the Quartic Function

To find the coefficients of the quartic function, we need to minimize the sum of the squared errors SS. This can be done by taking the partial derivatives of SS with respect to each of the coefficients aa, bb, cc, dd, and ee, and setting them equal to zero.

Let's denote the partial derivatives of SS with respect to each of the coefficients as:

∂S∂a=0\frac{\partial S}{\partial a} = 0

∂S∂b=0\frac{\partial S}{\partial b} = 0

∂S∂c=0\frac{\partial S}{\partial c} = 0

∂S∂d=0\frac{\partial S}{\partial d} = 0

∂S∂e=0\frac{\partial S}{\partial e} = 0

Solving these equations simultaneously will give us the values of the coefficients aa, bb, cc, dd, and ee that minimize the sum of the squared errors.

Calculating the Coefficients

To calculate the coefficients, we need to use the given data points and the equations of the partial derivatives. Let's denote the given data points as:

xix_i yiy_i
-3 70
-2 15
-1 4
0 1
1 -6
2 -5
3 40

Using the equations of the partial derivatives, we can calculate the coefficients as follows:

a=∑i=1nyixi4−∑i=1nxi4∑i=1nyin∑i=1nxi8−(∑i=1nxi4)2na = \frac{\sum_{i=1}^{n} y_i x_i^4 - \frac{\sum_{i=1}^{n} x_i^4 \sum_{i=1}^{n} y_i}{n}}{\sum_{i=1}^{n} x_i^8 - \frac{(\sum_{i=1}^{n} x_i^4)^2}{n}}

b=∑i=1nyixi3−∑i=1nxi3∑i=1nyin∑i=1nxi6−(∑i=1nxi3)2nb = \frac{\sum_{i=1}^{n} y_i x_i^3 - \frac{\sum_{i=1}^{n} x_i^3 \sum_{i=1}^{n} y_i}{n}}{\sum_{i=1}^{n} x_i^6 - \frac{(\sum_{i=1}^{n} x_i^3)^2}{n}}

c=∑i=1nyixi2−∑i=1nxi2∑i=1nyin∑i=1nxi4−(∑i=1nxi2)2nc = \frac{\sum_{i=1}^{n} y_i x_i^2 - \frac{\sum_{i=1}^{n} x_i^2 \sum_{i=1}^{n} y_i}{n}}{\sum_{i=1}^{n} x_i^4 - \frac{(\sum_{i=1}^{n} x_i^2)^2}{n}}

d=∑i=1nyixi−∑i=1nxi∑i=1nyin∑i=1nxi2−(∑i=1nxi)2nd = \frac{\sum_{i=1}^{n} y_i x_i - \frac{\sum_{i=1}^{n} x_i \sum_{i=1}^{n} y_i}{n}}{\sum_{i=1}^{n} x_i^2 - \frac{(\sum_{i=1}^{n} x_i)^2}{n}}

e=∑i=1nyi−∑i=1n∑i=1nyin∑i=1n1−(∑i=1n1)2ne = \frac{\sum_{i=1}^{n} y_i - \frac{\sum_{i=1}^{n} \sum_{i=1}^{n} y_i}{n}}{\sum_{i=1}^{n} 1 - \frac{(\sum_{i=1}^{n} 1)^2}{n}}

Calculating the Coefficients Values

Using the given data points, we can calculate the values of the coefficients as follows:

a=70(−3)4+15(−2)4+4(−1)4+1(0)4−6(1)4−5(2)4+40(3)4−70(−3)4+15(−2)4+4(−1)4+1(0)4−6(1)4−5(2)4+40(3)47(−3)8+(−2)8+(−1)8+08+18+28+387−((−3)4+(−2)4+(−1)4+04+14+24+34)27a = \frac{70(-3)^4 + 15(-2)^4 + 4(-1)^4 + 1(0)^4 - 6(1)^4 - 5(2)^4 + 40(3)^4 - \frac{70(-3)^4 + 15(-2)^4 + 4(-1)^4 + 1(0)^4 - 6(1)^4 - 5(2)^4 + 40(3)^4}{7}}{\frac{(-3)^8 + (-2)^8 + (-1)^8 + 0^8 + 1^8 + 2^8 + 3^8}{7} - \frac{((-3)^4 + (-2)^4 + (-1)^4 + 0^4 + 1^4 + 2^4 + 3^4)^2}{7}}

b=70(−3)3+15(−2)3+4(−1)3+1(0)3−6(1)3−5(2)3+40(3)3−70(−3)3+15(−2)3+4(−1)3+1(0)3−6(1)3−5(2)3+40(3)37(−3)6+(−2)6+(−1)6+06+16+26+367−((−3)3+(−2)3+(−1)3+03+13+23+33)27b = \frac{70(-3)^3 + 15(-2)^3 + 4(-1)^3 + 1(0)^3 - 6(1)^3 - 5(2)^3 + 40(3)^3 - \frac{70(-3)^3 + 15(-2)^3 + 4(-1)^3 + 1(0)^3 - 6(1)^3 - 5(2)^3 + 40(3)^3}{7}}{\frac{(-3)^6 + (-2)^6 + (-1)^6 + 0^6 + 1^6 + 2^6 + 3^6}{7} - \frac{((-3)^3 + (-2)^3 + (-1)^3 + 0^3 + 1^3 + 2^3 + 3^3)^2}{7}}

c=70(−3)2+15(−2)2+4(−1)2+1(0)2−6(1)2−5(2)2+40(3)2−70(−3)2+15(−2)2+4(−1)2+1(0)2−6(1)2−5(2)2+40(3)27(−3)4+(−2)4+(−1)4+04+14+24+347−((−3)2+(−2)2+(−1)2+02+12+22+32)27c = \frac{70(-3)^2 + 15(-2)^2 + 4(-1)^2 + 1(0)^2 - 6(1)^2 - 5(2)^2 + 40(3)^2 - \frac{70(-3)^2 + 15(-2)^2 + 4(-1)^2 + 1(0)^2 - 6(1)^2 - 5(2)^2 + 40(3)^2}{7}}{\frac{(-3)^4 + (-2)^4 + (-1)^4 + 0^4 + 1^4 + 2^4 + 3^4}{7} - \frac{((-3)^2 + (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 + 3^2)^2}{7}}

d=70(−3)+15(−2)+4(−1)+1(0)−6(1)−5(2)+40(3)−70(−3)+15(−2)+4(−1)+1(0)−6(1)−5(2)+40(3)7(−3)2+(−2)2+(−1)2+02+12+22+327−((−3)+(−2)+(−1)+0+1+2+3)27d = \frac{70(-3) + 15(-2) + 4(-1) + 1(0) - 6(1) - 5(2) + 40(3) - \frac{70(-3) + 15(-2) + 4(-1) + 1(0) - 6(1) - 5(2) + 40(3)}{7}}{\frac{(-3)^2 + (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 + 3^2}{7} - \frac{((-3) + (-2) + (-1) + 0 + 1 + 2 + 3)^2}{7}}

e = \frac{70 + 15 + 4 + 1 - 6 - 5 + 40 - \frac{70 + 15 + 4 + 1 - 6 - 5 + 40}{7}}{\frac{1 + 1 + 1 + 1 + 1 + 1 + 1}{7} - \<br/> **Q&A: Finding the Best Fit Quartic Function for Given Data** ===========================================================

Q: What is the method of least squares?

A: The method of least squares is a widely used technique for finding the best fit function for a given dataset. The basic idea behind this method is to minimize the sum of the squared errors between the observed data points and the predicted values of the function.

Q: How do I calculate the coefficients of the quartic function using the method of least squares?

A: To calculate the coefficients of the quartic function, you need to use the given data points and the equations of the partial derivatives. The equations are:

a=∑i=1nyixi4−∑i=1nxi4∑i=1nyin∑i=1nxi8−(∑i=1nxi4)2n</span></p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>b</mi><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub><msubsup><mi>x</mi><mi>i</mi><mn>3</mn></msubsup><mo>−</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>3</mn></msubsup><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>6</mn></msubsup><mo>−</mo><mfrac><mrow><mostretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>3</mn></msubsup><msup><mostretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac></mrow></mfrac></mrow><annotationencoding="application/x−tex">b=∑i=1nyixi3−∑i=1nxi3∑i=1nyin∑i=1nxi6−(∑i=1nxi3)2n</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">b</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:3.2281em;vertical−align:−1.3342em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.8939em;"><spanstyle="top:−2.1697em;"><spanclass="pstrut"style="height:3.1589em;"></span><spanclass="mord"><spanclass="mop"><spanclass="mopop−symbolsmall−op"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8043em;"><spanstyle="top:−2.4003em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−3.2029em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2997em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7959em;"><spanstyle="top:−2.4231em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">i</span></span></span><spanstyle="top:−3.0448em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">6</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2769em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.0992em;"><spanstyle="top:−2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.5356em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopenmtight">(</span><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7047em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.8971em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8051em;"><spanstyle="top:−2.1777em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmathnormalmtight">i</span></span></span><spanstyle="top:−2.8448em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight">3</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3223em;"><span></span></span></span></span></span></span><spanclass="mclosemtight"><spanclass="mclosemtight">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7463em;"><spanstyle="top:−2.786em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.345em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span><spanstyle="top:−3.3889em;"><spanclass="pstrut"style="height:3.1589em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.8939em;"><spanclass="pstrut"style="height:3.1589em;"></span><spanclass="mord"><spanclass="mop"><spanclass="mopop−symbolsmall−op"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8043em;"><spanstyle="top:−2.4003em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−3.2029em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2997em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3117em;"><spanstyle="top:−2.55em;margin−left:−0.0359em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8141em;"><spanstyle="top:−2.4413em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">i</span></span></span><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">3</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2587em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.1589em;"><spanstyle="top:−2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.535em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.931em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8913em;"><spanstyle="top:−2.214em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmathnormalmtight">i</span></span></span><spanstyle="top:−2.931em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight">3</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.286em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.931em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmtight"><spanclass="mordmathnormalmtight"style="margin−right:0.03588em;">y</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3281em;"><spanstyle="top:−2.357em;margin−left:−0.0359em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.345em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:1.3342em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></span></p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>c</mi><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>4</mn></msubsup><mo>−</mo><mfrac><mrow><mostretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><msup><mostretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac></mrow></mfrac></mrow><annotationencoding="application/x−tex">c=∑i=1nyixi2−∑i=1nxi2∑i=1nyin∑i=1nxi4−(∑i=1nxi2)2n</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">c</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:3.2281em;vertical−align:−1.3342em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.8939em;"><spanstyle="top:−2.1697em;"><spanclass="pstrut"style="height:3.1589em;"></span><spanclass="mord"><spanclass="mop"><spanclass="mopop−symbolsmall−op"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8043em;"><spanstyle="top:−2.4003em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−3.2029em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2997em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7959em;"><spanstyle="top:−2.4231em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">i</span></span></span><spanstyle="top:−3.0448em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">4</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2769em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.0992em;"><spanstyle="top:−2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.5356em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopenmtight">(</span><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7047em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.8971em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8051em;"><spanstyle="top:−2.1777em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmathnormalmtight">i</span></span></span><spanstyle="top:−2.8448em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight">2</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3223em;"><span></span></span></span></span></span></span><spanclass="mclosemtight"><spanclass="mclosemtight">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7463em;"><spanstyle="top:−2.786em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.345em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span><spanstyle="top:−3.3889em;"><spanclass="pstrut"style="height:3.1589em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.8939em;"><spanclass="pstrut"style="height:3.1589em;"></span><spanclass="mord"><spanclass="mop"><spanclass="mopop−symbolsmall−op"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8043em;"><spanstyle="top:−2.4003em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−3.2029em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2997em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3117em;"><spanstyle="top:−2.55em;margin−left:−0.0359em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8141em;"><spanstyle="top:−2.4413em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">i</span></span></span><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2587em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.1589em;"><spanstyle="top:−2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.535em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.931em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8913em;"><spanstyle="top:−2.214em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmathnormalmtight">i</span></span></span><spanstyle="top:−2.931em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight">2</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.286em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.931em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmtight"><spanclass="mordmathnormalmtight"style="margin−right:0.03588em;">y</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3281em;"><spanstyle="top:−2.357em;margin−left:−0.0359em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.345em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:1.3342em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></span></p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>d</mi><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><mfrac><mrow><mostretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><msup><mostretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac></mrow></mfrac></mrow><annotationencoding="application/x−tex">d=∑i=1nyixi−∑i=1nxi∑i=1nyin∑i=1nxi2−(∑i=1nxi)2n</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">d</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:3.09em;vertical−align:−1.295em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.795em;"><spanstyle="top:−2.11em;"><spanclass="pstrut"style="height:3.06em;"></span><spanclass="mord"><spanclass="mop"><spanclass="mopop−symbolsmall−op"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8043em;"><spanstyle="top:−2.4003em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−3.2029em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2997em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7959em;"><spanstyle="top:−2.4231em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">i</span></span></span><spanstyle="top:−3.0448em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2769em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.06em;"><spanstyle="top:−2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.535em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopenmtight">(</span><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7047em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.8971em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3281em;"><spanstyle="top:−2.357em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.143em;"><span></span></span></span></span></span></span><spanclass="mclosemtight"><spanclass="mclosemtight">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7463em;"><spanstyle="top:−2.786em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.345em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span><spanstyle="top:−3.29em;"><spanclass="pstrut"style="height:3.06em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.795em;"><spanclass="pstrut"style="height:3.06em;"></span><spanclass="mord"><spanclass="mop"><spanclass="mopop−symbolsmall−op"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8043em;"><spanstyle="top:−2.4003em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−3.2029em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2997em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3117em;"><spanstyle="top:−2.55em;margin−left:−0.0359em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3117em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.06em;"><spanstyle="top:−2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.535em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.931em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmtight"><spanclass="mordmathnormalmtight">x</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3281em;"><spanstyle="top:−2.357em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.143em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.931em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmtight"><spanclass="mordmathnormalmtight"style="margin−right:0.03588em;">y</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3281em;"><spanstyle="top:−2.357em;margin−left:−0.0359em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.345em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:1.295em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></span></p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>e</mi><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mn>1</mn><mo>−</mo><mfrac><mrow><mostretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mn>1</mn><msup><mostretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac></mrow></mfrac></mrow><annotationencoding="application/x−tex">e=∑i=1nyi−∑i=1n∑i=1nyin∑i=1n1−(∑i=1n1)2n</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">e</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:3.09em;vertical−align:−1.295em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.795em;"><spanstyle="top:−2.11em;"><spanclass="pstrut"style="height:3.06em;"></span><spanclass="mord"><spanclass="mop"><spanclass="mopop−symbolsmall−op"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8043em;"><spanstyle="top:−2.4003em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−3.2029em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2997em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">1</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.06em;"><spanstyle="top:−2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.535em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopenmtight">(</span><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7047em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.8971em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmtight">1</span><spanclass="mclosemtight"><spanclass="mclosemtight">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7463em;"><spanstyle="top:−2.786em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.345em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span><spanstyle="top:−3.29em;"><spanclass="pstrut"style="height:3.06em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.795em;"><spanclass="pstrut"style="height:3.06em;"></span><spanclass="mord"><spanclass="mop"><spanclass="mopop−symbolsmall−op"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8043em;"><spanstyle="top:−2.4003em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−3.2029em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2997em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3117em;"><spanstyle="top:−2.55em;margin−left:−0.0359em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.06em;"><spanstyle="top:−2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.535em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.931em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mopmtight"><spanclass="mopop−symbolsmall−opmtight"style="position:relative;top:0em;">∑</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7385em;"><spanstyle="top:−2.1786em;margin−left:0em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">i</span><spanclass="mrelmtight">=</span><spanclass="mordmtight">1</span></span></span></span><spanstyle="top:−2.931em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmathnormalmtight">n</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.3214em;"><span></span></span></span></span></span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmtight"><spanclass="mordmathnormalmtight"style="margin−right:0.03588em;">y</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3281em;"><spanstyle="top:−2.357em;margin−left:−0.0359em;margin−right:0.0714em;"><spanclass="pstrut"style="height:2.5em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmathnormalmtight">i</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.345em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:1.295em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></span></p><h2><strong>Q:Whatisthesignificanceofthequarticfunctioninreal−worldapplications?</strong></h2><p>A:Thequarticfunctionhassignificantapplicationsinvariousfieldssuchasphysics,engineering,andeconomics.Forexample,itisusedtomodelthemotionofobjectsundertheinfluenceofgravity,todescribethebehaviorofelectricalcircuits,andtopredictthegrowthofpopulations.</p><h2><strong>Q:HowdoIdeterminethebestfitquarticfunctionforagivendataset?</strong></h2><p>A:Todeterminethebestfitquarticfunctionforagivendataset,youneedtousethemethodofleastsquarestocalculatethecoefficientsofthefunction.Youcanthenusethesecoefficientstoplotthefunctionandcompareitwiththegivendatapoints.</p><h2><strong>Q:Whatarethelimitationsofthemethodofleastsquares?</strong></h2><p>A:Themethodofleastsquareshasseverallimitations.Forexample,itassumesthattheerrorsinthedatapointsarenormallydistributed,whichmaynotalwaysbethecase.Additionally,itcanbesensitivetooutliersinthedata,whichcanaffecttheaccuracyoftheresults.</p><h2><strong>Q:HowdoIhandleoutliersinthedatawhenusingthemethodofleastsquares?</strong></h2><p>A:Tohandleoutliersinthedata,youcanusetechniquessuchasdatatransformation,datacleaning,androbustregression.Thesetechniquescanhelptoreducetheimpactofoutliersontheresultsandimprovetheaccuracyofthemodel.</p><h2><strong>Q:Whataresomecommonapplicationsofthequarticfunctioninmathematics?</strong></h2><p>A:Thequarticfunctionhasseveralapplicationsinmathematics,including:</p><ul><li>Modelingthemotionofobjectsundertheinfluenceofgravity</li><li>Describingthebehaviorofelectricalcircuits</li><li>Predictingthegrowthofpopulations</li><li>Solvingsystemsofequations</li><li>Findingtherootsofpolynomials</li></ul><h2><strong>Q:HowdoIusethequarticfunctiontosolvesystemsofequations?</strong></h2><p>A:Tousethequarticfunctiontosolvesystemsofequations,youneedtofirstwritethesystemofequationsintheformofaquarticequation.Youcanthenusethemethodofleastsquarestocalculatethecoefficientsofthequarticfunctionandsolvefortherootsoftheequation.</p>a = \frac{\sum_{i=1}^{n} y_i x_i^4 - \frac{\sum_{i=1}^{n} x_i^4 \sum_{i=1}^{n} y_i}{n}}{\sum_{i=1}^{n} x_i^8 - \frac{(\sum_{i=1}^{n} x_i^4)^2}{n}} </span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>b</mi><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub><msubsup><mi>x</mi><mi>i</mi><mn>3</mn></msubsup><mo>−</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>3</mn></msubsup><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>6</mn></msubsup><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>3</mn></msubsup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac></mrow></mfrac></mrow><annotation encoding="application/x-tex">b = \frac{\sum_{i=1}^{n} y_i x_i^3 - \frac{\sum_{i=1}^{n} x_i^3 \sum_{i=1}^{n} y_i}{n}}{\sum_{i=1}^{n} x_i^6 - \frac{(\sum_{i=1}^{n} x_i^3)^2}{n}} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.2281em;vertical-align:-1.3342em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8939em;"><span style="top:-2.1697em;"><span class="pstrut" style="height:3.1589em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.4231em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0992em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.5356em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7047em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.8971em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8051em;"><span style="top:-2.1777em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-2.8448em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3223em;"><span></span></span></span></span></span></span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.3889em;"><span class="pstrut" style="height:3.1589em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8939em;"><span class="pstrut" style="height:3.1589em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1589em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.535em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.214em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3342em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>c</mi><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>4</mn></msubsup><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac></mrow></mfrac></mrow><annotation encoding="application/x-tex">c = \frac{\sum_{i=1}^{n} y_i x_i^2 - \frac{\sum_{i=1}^{n} x_i^2 \sum_{i=1}^{n} y_i}{n}}{\sum_{i=1}^{n} x_i^4 - \frac{(\sum_{i=1}^{n} x_i^2)^2}{n}} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.2281em;vertical-align:-1.3342em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8939em;"><span style="top:-2.1697em;"><span class="pstrut" style="height:3.1589em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.4231em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0992em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.5356em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7047em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.8971em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8051em;"><span style="top:-2.1777em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-2.8448em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3223em;"><span></span></span></span></span></span></span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.3889em;"><span class="pstrut" style="height:3.1589em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8939em;"><span class="pstrut" style="height:3.1589em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1589em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.535em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.214em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3342em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>d</mi><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac></mrow></mfrac></mrow><annotation encoding="application/x-tex">d = \frac{\sum_{i=1}^{n} y_i x_i - \frac{\sum_{i=1}^{n} x_i \sum_{i=1}^{n} y_i}{n}}{\sum_{i=1}^{n} x_i^2 - \frac{(\sum_{i=1}^{n} x_i)^2}{n}} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.09em;vertical-align:-1.295em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.795em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.06em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.4231em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.06em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.535em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7047em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.8971em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.29em;"><span class="pstrut" style="height:3.06em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.795em;"><span class="pstrut" style="height:3.06em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.06em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.535em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.295em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>e</mi><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mrow><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mn>1</mn><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mi>n</mi></mfrac></mrow></mfrac></mrow><annotation encoding="application/x-tex">e = \frac{\sum_{i=1}^{n} y_i - \frac{\sum_{i=1}^{n} \sum_{i=1}^{n} y_i}{n}}{\sum_{i=1}^{n} 1 - \frac{(\sum_{i=1}^{n} 1)^2}{n}} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.09em;vertical-align:-1.295em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.795em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.06em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.06em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.535em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7047em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.8971em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight">1</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.29em;"><span class="pstrut" style="height:3.06em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.795em;"><span class="pstrut" style="height:3.06em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.06em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.535em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.295em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <h2><strong>Q: What is the significance of the quartic function in real-world applications?</strong></h2> <p>A: The quartic function has significant applications in various fields such as physics, engineering, and economics. For example, it is used to model the motion of objects under the influence of gravity, to describe the behavior of electrical circuits, and to predict the growth of populations.</p> <h2><strong>Q: How do I determine the best fit quartic function for a given dataset?</strong></h2> <p>A: To determine the best fit quartic function for a given dataset, you need to use the method of least squares to calculate the coefficients of the function. You can then use these coefficients to plot the function and compare it with the given data points.</p> <h2><strong>Q: What are the limitations of the method of least squares?</strong></h2> <p>A: The method of least squares has several limitations. For example, it assumes that the errors in the data points are normally distributed, which may not always be the case. Additionally, it can be sensitive to outliers in the data, which can affect the accuracy of the results.</p> <h2><strong>Q: How do I handle outliers in the data when using the method of least squares?</strong></h2> <p>A: To handle outliers in the data, you can use techniques such as data transformation, data cleaning, and robust regression. These techniques can help to reduce the impact of outliers on the results and improve the accuracy of the model.</p> <h2><strong>Q: What are some common applications of the quartic function in mathematics?</strong></h2> <p>A: The quartic function has several applications in mathematics, including:</p> <ul> <li>Modeling the motion of objects under the influence of gravity</li> <li>Describing the behavior of electrical circuits</li> <li>Predicting the growth of populations</li> <li>Solving systems of equations</li> <li>Finding the roots of polynomials</li> </ul> <h2><strong>Q: How do I use the quartic function to solve systems of equations?</strong></h2> <p>A: To use the quartic function to solve systems of equations, you need to first write the system of equations in the form of a quartic equation. You can then use the method of least squares to calculate the coefficients of the quartic function and solve for the roots of the equation.</p>