# Measures of Variability

The rangemean absolute deviation, and interquartile range are all ways to measure the variability for a set of data.

• The range measures how spread out the data is. To find the range, subtract the smallest value from the largest value in a set of data.
• The mean absolute deviation (MAD) is the average distance of each data value from the mean.
• A small MAD means the values do not vary much from the mean.   A large MAD means that the values vary greatly from the mean.

To find the mean absolute deviation (MAD):

1. Find the mean.
2. Find the distance between each data value and the mean. That is, find the absolute value of the difference between each data value and the mean.
3. Find the average of those differences.

Here’s an example of finding MAD for the following data set.

7, 3, 5, 7, 8, 3, 9

1. Find the mean.
(7+3+5+7+8+3+9)/7 = 42/7 = 6
2. Find the distance (absolute value) between each data value and the mean
|7 – 6| = 1
|3 – 6| = 3
|5 – 6| = 1
|7 – 6| = 1
|8 – 6| = 2
|3 – 6| = 3
|9 – 6| = 3
3. Find the average of those absolute value differences.
(1+3+1+1+2+3+3)/7 = 14/7 = 2

We’ll get to interquartile range on the next day.

## Practice

Find the range and mean absolute deviation of each data set. Round to the nearest tenth.

1.   30, 21, 18, 19, 23, 24, 26, 32, 30, 22, 12, 15, 21

2.   88, 74, 98, 76, 68, 74, 89, 92

3.   6, 45, 87, 23, 51, 15, 32

4.   11, 7, 14, 2, 5, 13, 3, 6, 10, 3, 8, 4, 8, 4, 7

(source)