7. Statistics

Lesson

Measures of distribution tell us how far the scores in a data set are spread out.

We've already looked at one measure of spread, the range, which is the difference between the greatest and least score in a data set.

Now we are going to learn about a new measure of spread called the Mean absolute deviation (MAD). The MAD of a set of data is the average distance between each data value and the mean.

Let's use an example to help explain this.

**Find** the mean absolute deviation of $23,18,31,28,20$23,18,31,28,20.

**Think/Do:**

1. Find the mean.

$\frac{23+18+31+28+20}{5}$23+18+31+28+205$=$=$24$24

2. Find the difference between each individual score and the mean.

$23-24=-1$23−24=−1

$18-24=-6$18−24=−6

$31-24=7$31−24=7

$28-24=4$28−24=4

$20-24=-4$20−24=−4

Take the absolute value of each difference.

$\left|-1\right|=1$|−1|=1

$\left|-6\right|=6$|−6|=6

$\left|7\right|=7$|7|=7

$\left|4\right|=4$|4|=4

$\left|-4\right|=4$|−4|=4

3. Find the mean of these differences.

$\frac{1+6+7+4+4}{5}$1+6+7+4+45$=$=$4.4$4.4

Therefore, the mean absolute deviation is $4.4$4.4 units.

**Reflect:** This means that, on average, scores in this data set are $4.4$4.4 units above or below the mean.

The box below summarizes our steps.

Mean absolute deviation (MAD)

The mean absolute deviation (MAD) of a set of data is the average distance between each data value and the mean.

To calculate the mean absolute deviation of a set of data:

- Calculate the mean.
- Find the absolute value of the differences between each value in the set and the mean.
- Find the average of those values.

Calculate the mean absolute deviation of the values below, by answering each question.

$2$2, $8$8, $6$6, $3$3, $10$10, $15$15, $6$6 and $6$6.

State your answer to 2 decimal places if necessary.

First, calculate the mean of

$2$2, $8$8, $6$6, $3$3, $10$10, $15$15, $6$6 and $6$6.

Complete the table of values, finding the distance of each value from the mean.

Value Distance from $7$7 $2$2 $\editable{}$ $8$8 $\editable{}$ $6$6 $\editable{}$ $3$3 $\editable{}$ $10$10 $\editable{}$ $15$15 $\editable{}$ $6$6 $\editable{}$ $6$6 $\editable{}$ Using your values from the table above, calculate the mean of the differences.

Which of the following is true concerning the mean absolute deviation of a set of data?

It describes the average distance between each data value and the mean.

AIt describes the variation of the data values around the median.

BIt describes the absolute value of the mean.

CIt describes the variation of the data values around the mode.

D

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Summarize numerical data sets in relation to their context.

Find the quantitative measures of center (median and/or mean) for a numerical data set and recognize that this value summarizes the data set with a single number. Interpret mean as an equal or fair share. Find measures of variability (range and interquartile range) as well as informally describe the shape and the presence of clusters, gaps, peaks, and outliers in a distribution.