This Data Is From A Sample. Calculate The Mean, Standard Deviation, And Variance. Round Answers To Two Decimal Places.$\[ \begin{tabular}{|c|} \hline $x$ \\ \hline 19.7 \\ \hline 31.1 \\ \hline 46.8 \\ \hline 10.4 \\ \hline 18.7 \\ \hline 42.7

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Calculating Mean, Standard Deviation, and Variance from a Sample Data

In statistics, calculating the mean, standard deviation, and variance are essential steps in understanding the distribution of a dataset. These measures provide valuable insights into the central tendency and dispersion of the data. In this article, we will calculate the mean, standard deviation, and variance from a sample dataset using the given values.

Sample Dataset

The sample dataset consists of the following values:

x
19.7
31.1
46.8
10.4
18.7
42.7

Calculating the Mean

The mean is the average value of the dataset. To calculate the mean, we add up all the values and divide by the number of values.

Formula for Calculating the Mean:

xˉ=∑i=1nxin\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}

where xˉ\bar{x} is the mean, xix_i is the individual value, and nn is the number of values.

Calculating the Mean from the Sample Dataset:

x
19.7
31.1
46.8
10.4
18.7
42.7

To calculate the mean, we add up all the values:

∑i=16xi=19.7+31.1+46.8+10.4+18.7+42.7=169.4\sum_{i=1}^{6} x_i = 19.7 + 31.1 + 46.8 + 10.4 + 18.7 + 42.7 = 169.4

There are 6 values in the dataset, so we divide the sum by 6:

xˉ=169.46=28.23\bar{x} = \frac{169.4}{6} = 28.23

Calculating the Standard Deviation

The standard deviation measures the amount of variation or dispersion from the mean value. To calculate the standard deviation, we use the following formula:

Formula for Calculating the Standard Deviation:

s=∑i=1n(xi−xˉ)2n−1s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}

where ss is the standard deviation, xix_i is the individual value, xˉ\bar{x} is the mean, and nn is the number of values.

Calculating the Standard Deviation from the Sample Dataset:

To calculate the standard deviation, we first need to calculate the deviations from the mean:

x x - xˉ\bar{x} (x - xˉ\bar{x})^2
19.7 -9.23 85.19
31.1 2.87 8.25
46.8 18.57 345.19
10.4 -17.83 317.19
18.7 -9.53 90.29
42.7 14.47 209.29

Next, we calculate the sum of the squared deviations:

∑i=16(xi−xˉ)2=85.19+8.25+345.19+317.19+90.29+209.29=1055.4\sum_{i=1}^{6} (x_i - \bar{x})^2 = 85.19 + 8.25 + 345.19 + 317.19 + 90.29 + 209.29 = 1055.4

There are 6 values in the dataset, so we divide the sum by 5 (n-1):

s=1055.45=211.08=14.55s = \sqrt{\frac{1055.4}{5}} = \sqrt{211.08} = 14.55

Calculating the Variance

The variance measures the average of the squared deviations from the mean value. To calculate the variance, we use the following formula:

Formula for Calculating the Variance:

s2=∑i=1n(xi−xˉ)2n−1s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}

where s2s^2 is the variance, xix_i is the individual value, xˉ\bar{x} is the mean, and nn is the number of values.

Calculating the Variance from the Sample Dataset:

To calculate the variance, we divide the sum of the squared deviations by 5 (n-1):

s2=1055.45=211.08s^2 = \frac{1055.4}{5} = 211.08

In this article, we calculated the mean, standard deviation, and variance from a sample dataset using the given values. The mean is the average value of the dataset, which is 28.23. The standard deviation measures the amount of variation or dispersion from the mean value, which is 14.55. The variance measures the average of the squared deviations from the mean value, which is 211.08. These measures provide valuable insights into the central tendency and dispersion of the data.

  • "Statistics for Dummies" by Deborah J. Rumsey
  • "Mathematics for Dummies" by Mary Jane Sterling
    Frequently Asked Questions (FAQs) about Calculating Mean, Standard Deviation, and Variance

In our previous article, we calculated the mean, standard deviation, and variance from a sample dataset using the given values. In this article, we will answer some frequently asked questions (FAQs) about calculating these measures.

Q: What is the difference between the mean and the median?

A: The mean and the median are both measures of central tendency, but they are calculated differently. The mean is the average value of the dataset, while the median is the middle value of the dataset when it is arranged in order. The mean is sensitive to outliers, while the median is not.

Q: Why do we use the standard deviation instead of the variance?

A: The standard deviation is a more intuitive measure of dispersion than the variance. The standard deviation is the square root of the variance, and it is measured in the same units as the data. The variance is measured in squared units, which can make it difficult to interpret.

Q: Can I use the sample standard deviation as an estimate of the population standard deviation?

A: Yes, you can use the sample standard deviation as an estimate of the population standard deviation. However, you should be aware that the sample standard deviation is a biased estimator of the population standard deviation. To get an unbiased estimate, you can use the formula:

s2=∑i=1n(xi−xˉ)2n−1s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}

Q: How do I calculate the standard deviation when the data is not normally distributed?

A: When the data is not normally distributed, you can use the following methods to calculate the standard deviation:

  • Use the interquartile range (IQR) instead of the standard deviation. The IQR is the difference between the 75th percentile and the 25th percentile.
  • Use the median absolute deviation (MAD) instead of the standard deviation. The MAD is the median of the absolute deviations from the median.

Q: Can I use the sample variance as an estimate of the population variance?

A: Yes, you can use the sample variance as an estimate of the population variance. However, you should be aware that the sample variance is a biased estimator of the population variance. To get an unbiased estimate, you can use the formula:

s2=∑i=1n(xi−xˉ)2n−1s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}

Q: How do I calculate the variance when the data is not normally distributed?

A: When the data is not normally distributed, you can use the following methods to calculate the variance:

  • Use the IQR instead of the variance. The IQR is the difference between the 75th percentile and the 25th percentile.
  • Use the MAD instead of the variance. The MAD is the median of the absolute deviations from the median.

Q: Can I use the sample standard deviation and sample variance as estimates of the population standard deviation and population variance?

A: Yes, you can use the sample standard deviation and sample variance as estimates of the population standard deviation and population variance. However, you should be aware that the sample standard deviation and sample variance are biased estimators of the population standard deviation and population variance. To get unbiased estimates, you can use the formulas:

s2=∑i=1n(xi−xˉ)2n−1s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}

s=s2s = \sqrt{s^2}

In this article, we answered some frequently asked questions (FAQs) about calculating the mean, standard deviation, and variance. We discussed the differences between the mean and the median, the standard deviation and the variance, and how to calculate these measures when the data is not normally distributed. We also discussed how to use the sample standard deviation and sample variance as estimates of the population standard deviation and population variance.