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7.05 Mean absolute deviation (MAD)

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.

Worked example

Question 1

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$2324=1

$18-24=-6$1824=6

$31-24=7$3124=7

$28-24=4$2824=4

$20-24=-4$2024=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:

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

 

Practice questions

Question 2

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.

  1. First, calculate the mean of

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

  2. 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 $\editable{}$
  3. Using your values from the table above, calculate the mean of the differences.

Question 3

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

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

    A

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

    B

    It describes the absolute value of the mean.

    C

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

    D

 

Outcomes

NC.6.SP.2

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.

NC.6.SP.3a

Determine the measure of center of a data set and understand that it is a single number that summarizes all the values of that data set.• Understand that a mean is a measure of center that represents a balance point or fair share of a data set and can be influenced by the presence of extreme values within the data set.• Understand the median as a measure of center that is the numerical middle of an ordered data set.

NC.6.SP.3

Understand that both a measure of center and a description of variability should be considered when describing a numerical data set.

NC.6.SP.5

Summarize numerical data sets in relation to their context.

NC.6.SP.5b

Analyze center and variability by:• Giving quantitative measures of center, describing variability, and any overall pattern, and noting any striking deviations.• Justifying the appropriate choice of measures of center using the shape of the data distribution.

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