We can use a measure of center and a measure of variation to describe the distribution of a data set. To determine which measures are the most appropriate to use, we can see the shape of the distribution.

If the data distribution is symmetric, we can use the mean to describe the center and the mean absolute deviation (MAD) to describe the variation.
If the data distribution is skewed, we can use the median to describe the center and the interquartile range (IQR) to describe the variation.

Consider the given data plot.

The left side of the graph is a mirror image of the right side of the graph, which means that the distribution is symmetric. Then, we can use the mean and the MAD to describe the center and the variation of the distribution. Before finding the measures, we will write the data as a list. Remember that each point is a data value.

8, 10, 10, 12, 12, 12, 14, 14, 14, 14, 16, 16, 16, 18, 18, 20

Now, let's find one measure at a time!

Mean

The mean of a data set is the sum of the data divided by the number of data values. Let's start by calculating the sum of the data.

8 + 10 + 10 + 12 + 12 + 12 + 14 + 14 + 14 + 14 + 16 + 16 + 16 + 18 + 18 + 20 = 224

There are

16

data values. Now that we have the sum of the data and the number of data values, we can calculate the mean.

\frac{2 2 4}{1 6}

CalcQuot

Calculate quotient

14

The mean is

14 .

Mean Absolute Deviation

Now, we will find the mean absolute deviations.

Mean Absolute Deviation
An average of how much data values differ from the mean.

To calculate the mean absolute deviation, we follow these four steps.

Find the mean of the data.
Find the distance between each data value and the mean.
Find the sum of the distances from Step $2 .$
Divide the sum of Step $3$ by the total number of data values.

We already know the mean of our data set, so we can move on to finding the distances. To do so, we will replace the dots in our graph with their distances from the mean.

Finally, to find the mean absolute deviation, we need to divide the sum of the found distances by the number of values in each data set.

\frac{6 + 4 + 4 + 2 + 2 + 2 + 2 + 2 + 2 + 4 + 4 + 6}{1 6}

AddTerms

Add terms

\frac{4 0}{1 6}

CalcQuot

Calculate quotient

2.5

The MAD is

2.5 .

Exercise

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