# What is MAD in Math? (Mean Absolute Deviation) Lesson

The **Mean Absolute Deviation (MAD)** is a measure of how much the values in a set of data vary from the mean value. Simply stated it is the average distance between each data point and the mean of the set of data.

We can calculate the **Mean Absolute Deviation** by first finding the mean of the dataset, then find the distance (absolute difference) between each data point and the mean, and lastly, calculate the average of the distances from the second step.

The 3 Steps are:

1) Find the mean (average) of the data set

2) Find the distance(absolute value of difference) of each point to the mean.

3) Find the average of all the distances from Step 2.

The following formula is used to calculate the **Mean Absolute Deviation** of grouped data.

Where n is the number of data points in the set and x _{i}represents each data point.

The MAD is often used as a measure of dispersion or variability in a dataset. It’s particularly useful when the dataset has outliers or extreme values because it considers the absolute differences rather than squared differences, which can give undue weight to outliers in other measures like the standard deviation.

The Mean Absolute Deviation provides a measure of how spread out the data points are from the mean of the dataset, and it’s calculated by finding the average of the absolute differences between each data point and the mean.

**Examples**

**Example 1:** Find the mean absolute deviation of the dataset: {5, 7, 8, 9, 10}.

**Solution:** Start by finding the mean of the dataset. To do this we will find the sum of the data set, and then divide that value by 5, since there are 5 numbers in the set of data.

5 + 7 + 8 + 9 + 10 = 39

39/5 = 7.8

Therefore, the mean of the dataset is 7.8.

Next we will find the distance (absolute difference) between each data point and the mean.

|5 – 7.8| = 2.8

|7 – 7.8| = 0.8

|8 – 7.8| = 0.2

|9 – 7.8| = 1.2

|10 – 7.8| = 2.2

Next we will find the average of the absolute differences.

2.8 + 0.8 + 0.2 + 1.2 + 2.2 = 7.2

7.2/5 = 1.44

Therefore, the Mean Absolute Deviation for this dataset is 1.44.

**Example 2: **Find the mean absolute deviation of the data set {-7, -2, 3, 8, 13, 18}

**Solution:** Start by finding the mean of the data set, by taking the sum and dividing by 6, since there are 6 terms in the set.

-7 + -2 + 3 + 8 + 13 + 18 = 33

33/6 = 5.5

The mean of the dataset is 5.5.

Next we need to find the absolute difference between each point and the mean.

|-7 – 5.5| = 12.5

|-2 – 5.5| = 7.5

|3 – 5.5| = 2.5

|8 – 5.5| = 2.5

|13 – 5.5| = 7.5

|18 – 5.5| = 12.5

Last we will find the average of the absolute differences from above.

12.5 + 7.5 + 2.5 + 2.5 + 7.5 + 12.5 = 45

45/6 = 7.5

Therefore, the mean absolute deviation is 7.5.

**Example 3:** If the mean of the following dataset {12, 10, 9, 8, 10, 11, x} is 10.6. Find the mean absolute deviation of the dataset.

**Solution:** Start by finding the value of x. In order to do this we will create an equation of finding the mean and set that equal to 10.6.

To solve this we will combine like terms in the numerator, and multiply both sides by 7, to cancel the fraction.

From there, we can subtract 60 from both sides to obtain the value of x.

x = 14.2

Now that we know the value of x and the mean, we can find the absolute difference of each data point to the mean.

|12 – 10.6| = 1.4

|10 – 10.6| = 0.6

|9 – 10.6| = 1.6

|8 – 10.6| = 2.6

|10 – 10.6| = 0.6

|11 – 10.6| = 0.4

|14.2 – 10.6| = 3.6

Lastly, we will find the average of the absolute differences.

1.4 + 0.6 + 1.6 + 2.6 + 0.6 + 0.4 + 3.6 = 10.8

10.8/7 ≈ 1.543

Therefore, the mean absolute deviation for this dataset is approximately 1.543.

**FAQs on What is MAD in Math (Mean Absolute Deviation)**

**1) What does the Mean Absolute Deviation (MAD) measure?**

The MAD measures the average distance between each data point in a dataset and the mean of that dataset. It provides a measure of the dispersion or variability of the data.

**2) How is the MAD calculated?**

To calculate the MAD:

- Find the mean (average) of the dataset.
- For each data point in the dataset, find the absolute difference between the data point and the mean.
- Calculate the average of these absolute differences.

**3) How is the MAD different from standard deviation?**

The MAD and standard deviation are both measures of dispersion, but they differ in how they treat deviations from the mean. The MAD uses absolute differences, while the standard deviation uses squared differences. As a result, the MAD is less sensitive to outliers and extreme values compared to the standard deviation.

**4) When should I use the MAD?**

The MAD is particularly useful when dealing with datasets containing outliers or extreme values because it provides a robust measure of variability that is not unduly influenced by these outliers. It’s also simpler to calculate and interpret compared to the standard deviation in some cases.

**5) What are the limitations of the MAD?**

While the MAD is robust to outliers, it may not capture the full extent of variability in the data, especially in datasets with heavy-tailed distributions. Additionally, because it only considers the absolute differences, it may not provide information about the direction of deviations from the mean.

**6) How can I interpret the MAD?**

A smaller MAD indicates that the data points are closer to the mean, suggesting less variability in the dataset. Conversely, a larger MAD suggests greater variability or dispersion in the data.