In this article, you will learn how to Find Measures of Variability in a few simple steps.

## Step by step guide to finding Measures of Variability

Measures of Variability allow us to understand how far apart the data points are from the distribution center. Also, Measures of Variability give us descriptive statistics and summarize our data.

The three main measures of variability: Range, Variance, and Standard Deviation.

The range \((R)\) of a data set is the difference between its largest and smallest values \(R=xmax-xmin\). The range is one of the best scattering indices for low volume data, but at the same time, its drawback is that only two data are involved and the rest of the data is ignored and therefore does not fully determine the scatter.

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:

Data | 43 | 8 | 32 | 16 | 25 | 47 | 19 | 22 | 31 |

**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

\(x\) | 43 | 8 | 32 | 16 | 25 | 47 | 19 | 22 | 31 |

\(x- \bar x\) | 19 | -19 | 5 | -11 | -2 | 20 | -8 | -5 | 4 |

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:

Data | 26 | 19 | 18 | 29 | 32 | 24 | 30 | 25 |

**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

\(x\) | 26 | 19 | 18 | 29 | 32 | 24 | 30 | 25 |

\(x- \bar x\) | 0.6 | -6.4 | -7.4 | 3.6 | 6.6 | -1.4 | 4.6 | -0.4 |

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)

Data | 32 | 45 | 50 | 42 | 37 | 46 | 40 |

2)

Data | 14 | 9 | 22 | 31 | 11 | 19 | 29 | 25 |

1) Sample variance: 36.24

standard deviation: 6.02

2) Sample variance: 67.14

standard deviation: 8.19

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