Find The Variance Of The Data.Data: 87 , 94 , 103 , 84 , 112 , 90 87, 94, 103, 84, 112, 90 87 , 94 , 103 , 84 , 112 , 90 Mean: X ˉ = 95 \bar{x} = 95 X ˉ = 95 Variance $(\sigma^2) = [?]

Mar 13, 2025 by ADMIN 185 views

**Calculating Variance: A Step-by-Step Guide**

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Understanding Variance

Variance is a measure of the spread or dispersion of a set of data from its mean value. It is an essential concept in statistics and is used to quantify the amount of variation or dispersion of a set of data. In this article, we will learn how to calculate the variance of a given data set.

Given Data Set

The given data set is: $87, 94, 103, 84, 112, 90$

Calculating the Mean

The mean of a data set is calculated by adding up all the values and dividing by the number of values. In this case, the mean is given as $\bar{x} = 95$ .

Calculating the Variance

The variance is calculated using the following formula:

$\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$

where $x_i$ is each individual data point, $\bar{x}$ is the mean, and $n$ is the number of data points.

Step-by-Step Calculation

Let's calculate the variance step by step:

Calculate the deviations from the mean: We need to calculate the difference between each data point and the mean.

Data Point	Deviation from Mean
87	87 - 95 = -8
94	94 - 95 = -1
103	103 - 95 = 8
84	84 - 95 = -11
112	112 - 95 = 17
90	90 - 95 = -5

Calculate the squared deviations: We need to square each deviation.

Data Point	Squared Deviation
87	(-8)^2 = 64
94	(-1)^2 = 1
103	(8)^2 = 64
84	(-11)^2 = 121
112	(17)^2 = 289
90	(-5)^2 = 25

Calculate the sum of squared deviations: We need to add up all the squared deviations.

$\sum_{i=1}^{n} (x_i - \bar{x})^2 = 64 + 1 + 64 + 121 + 289 + 25 = 464$

Calculate the variance: We need to divide the sum of squared deviations by the number of data points.

$\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n} = \frac{464}{6} = 77.33$

Conclusion

In this article, we learned how to calculate the variance of a given data set. We used the formula for variance and calculated the squared deviations, sum of squared deviations, and finally the variance. The variance of the given data set is 77.33.

Importance of Variance

Variance is an essential concept in statistics and is used in various fields such as finance, economics, and engineering. It helps to understand the spread or dispersion of a set of data and is used to make informed decisions.

Real-World Applications

Variance is used in various real-world applications such as:

Finance: Variance is used to measure the risk of a portfolio of stocks or bonds.
Economics: Variance is used to measure the dispersion of economic data such as GDP or inflation rates.
Engineering: Variance is used to measure the dispersion of data in quality control and process improvement.

Limitations of Variance

While variance is a useful measure of dispersion, it has some limitations. For example:

Sensitivity to outliers: Variance is sensitive to outliers and can be affected by a single data point that is far away from the mean.
Assumes normal distribution: Variance assumes that the data follows a normal distribution, which may not always be the case.

Alternatives to Variance

There are some alternatives to variance such as:

Standard deviation: Standard deviation is a measure of dispersion that is similar to variance but is less sensitive to outliers.
Interquartile range: Interquartile range is a measure of dispersion that is less sensitive to outliers and is used in situations where the data is not normally distributed.

Conclusion

In conclusion, variance is an essential concept in statistics that is used to measure the spread or dispersion of a set of data. It has various real-world applications and is used in various fields such as finance, economics, and engineering. However, it has some limitations and there are some alternatives to variance such as standard deviation and interquartile range.

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What is Variance?

Q: What is the formula for Variance?

A: The formula for variance is:

$\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$

where $x_i$ is each individual data point, $\bar{x}$ is the mean, and $n$ is the number of data points.

Q: What is the difference between Variance and Standard Deviation?

A: Variance and standard deviation are both measures of dispersion, but they are related in a way that standard deviation is the square root of variance. In other words, standard deviation is a more intuitive measure of dispersion, while variance is a more mathematical measure.

Q: What is the importance of Variance in Real-World Applications?

A: Variance is used in various real-world applications such as finance, economics, and engineering. It helps to understand the spread or dispersion of a set of data and is used to make informed decisions.

Q: What are the limitations of Variance?

A: While variance is a useful measure of dispersion, it has some limitations. For example, it is sensitive to outliers and assumes that the data follows a normal distribution, which may not always be the case.

Q: What are some alternatives to Variance?

A: There are some alternatives to variance such as standard deviation, interquartile range, and median absolute deviation. These measures of dispersion are less sensitive to outliers and are used in situations where the data is not normally distributed.

Q: How do I calculate Variance in Excel?

A: To calculate variance in Excel, you can use the following formula:

=VAR(range)

where range is the range of cells that contains the data.

Q: How do I calculate Variance in Python?

A: To calculate variance in Python, you can use the following code:

import numpy as np

data = [1, 2, 3, 4, 5] variance = np.var(data)

Q: What is the difference between Population Variance and Sample Variance?

A: Population variance is used when the data is a sample of the entire population, while sample variance is used when the data is a sample of a larger population. Population variance is calculated using the formula:

$\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$

while sample variance is calculated using the formula:

$s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$

Q: How do I calculate Variance in a Time Series?

A: To calculate variance in a time series, you can use the following formula:

$\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$

where $x_i$ is each individual data point, $\bar{x}$ is the mean, and $n$ is the number of data points.

Q: What is the difference between Variance and Covariance?

A: Variance and covariance are both measures of dispersion, but they are related in a way that covariance measures the relationship between two variables, while variance measures the dispersion of a single variable.

Q: How do I calculate Variance in a Multivariate Distribution?

A: To calculate variance in a multivariate distribution, you can use the following formula:

$\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$

where $x_i$ is each individual data point, $\bar{x}$ is the mean, and $n$ is the number of data points.