How to Find Measures of Variability? (+FREE Worksheet!)

A variance is a number that indicates how a series of data is distributed around the mean value. The sample variance of a set of \(n\) sample data is the number \(s^2\) defined by the formula \(s^2={\frac{∑(x- \bar x)^2}{n-1}}\).

The square root \(s\) of the sample variance is called the sample standard deviation of a set of \(n\) sample data \(s=\sqrt{\frac{∑(x- \bar x)^2}{n-1}}\). The standard deviation \((σ)\) is one of the scattering indices that shows how far the data is from the average value. If the standard deviation of a set of data is close to zero, it indicates that the data are close to the mean and have little scatter; While a large standard deviation indicates significant data scatter.

Find Measures of Variability Example 1:

Find the range and sample variance of the data set table:

Solution: for the data set the maximum is \(47\) and the minimum is \(8\), so the range is \(R=47-8=39\). For sample variance use defining formula \(s^2={\frac{∑(x- \bar x)^2}{n-1}}\). We first compute for each observation \(x\) its deviation \(x- \bar x\) from the sample mean. Since the mean of the data is \(27\), we obtain the numbers displayed in the second line of the supplied table

The Ultimate Algebra Bundle From Pre-Algebra to Algebra II

Download

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

Thus \(∑(x- \bar x)^2=(19)^2+(-19)^2+(5)^2+(-11)^2+(-2)^2+(20)^2+(-8)^2+(-5)^2+(4)^2=1,377\)

So the variance is \(s^2={\frac{∑(x- \bar x)^2}{n-1}}=\frac{1377}{8}=172.125\)

Find Measures of Variability Example 2:

Find the range and sample variance of the data set table:

Solution: for the data set the maximum is \(32\) and the minimum is \(18\), so the range is \(R=32-18=14\). For sample variance use defining formula \(s^2={\frac{∑(x- \bar x)^2}{n-1}}\). We first compute for each observation \(x\) its deviation \(x- \bar x\) from the sample mean. Since the mean of the data is \(25.4\), we obtain the numbers displayed in the second line of the supplied table

Thus \(∑(x- \bar x)^2=(0.6)^2+(-6.4)^2+(-7.4)^2+(3.6)^2+(6.6)^2+(-1.4)^2+(4.6)^2+(-0.4)^2=175.88\)

So the variance is \(s^2={\frac{∑(x- \bar x)^2}{n-1}}=\frac{175.88}{7}=25.13\)

Exercises for Finding Measures of Variability

Find the sample variance and standard deviation of each data set table.

1)

2)

1) Sample variance: 36.24

standard deviation: 6.02

The Ultimate Middle School Math Bundle Grades 6-8

Download

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

2) Sample variance: 67.14

standard deviation: 8.19

The Ultimate Elementary School Math Bundle Grade 3-5

Download

$114.99 Original price was: $114.99.$54.99Current price is: $54.99.