Mean Absolute Deviation: Formula, Steps & Solved Examples

Mean absolute deviation (MAD) is a variable that indicates the average distance between their observation and the mean. Mean absolute deviation is necessary to calculate the mean data. Mean measures the average of the observation, while deviation refers to the variance of the previous data. Thus, mean absolute deviation refers to the average distance of each observation from the mean of given data information. When we combine mean and deviation, we can define deviation as the average distance for each observation from the mean of data.

Keyterms: Mean, Mean absolute deviation, Standard deviation, Calculations, Data point, Statistics

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Mean Absolute Deviation

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Mean absolute deviation (MAD) is a variable that indicates the average distance between their observation and the mean. Mean absolute deviation uses initial data units to make interpretations easier. Large numbers indicate that data points are spread out from the average of the data. 

In contrast, lower values are associated with closely related data points. The rate of total deviation is also known as the average deviation. This definition of deviation sounds similar to standard deviation (SD). Although they both measure variability, they have different calculations. 

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Mean Absolute Deviation Formula

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Mean Absolute Deviation formula is given by,

\(MAD = \frac{\sum |X - \mu|}{N}\)

Where, 

|X – µ| = Absolute deviation

X = Value of the data point

µ = Mean

N = Sample size

Statistics include total deviation. The deviation is the difference between a data point and the mean. The total deviation rate automatically removes any subtraction that occurs.

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Steps to Calculate Mean Absolute Deviation

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We can obtain the mean deviation from the mean by following the below methods.

1. Find the mean of the given observations.
2. Calculate the difference between each observation and the calculated mean.
3. Evaluate the mean of the differences obtained in the second step.

\(\frac{1}{n} \sum^n_{i=1} |x_i -m| = \frac{|x_1-m| + |x_2 -m|+...+|x_n -m|}{n}\)

Calculation of Mean Absolute Deviation

Suppose a deviation from the average value a is given as (x - a), where x is the perception of any data set. To find the average deviation, we need to find the sum of all deviations in a data set. Since the average inclination is between high values and minimum set data, we can see that some deviations will be positive and some will be negative. The sum of such deviations will give zero.

Also Read: Types of Probability

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Example of Mean Absolute Deviation

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Consider this example where students of a class scored the following marks in an exam - 55, 65.70, 70, 72, 85, 90, 93, 100. Here, we should find the mean absolute deviation.

Let us first find the mean of the given data,

Mean = (55+65+70+70+72+85+90+93+100)/ 10

Mean = 790/ 10 

Mean = 79

Now, let us calculate the deviation,

Now calculate the mean absolute deviation.

Mean Absolute Deviation = |- 24 - 14 - 9 - 9 - 7 + 6 + 11 + 11 + 14 + 21 = 0|/ 10

Mean Absolute Deviation = (24 + 14 + 9 + 9 + 7 + 6 + 11 + 11 + 14 + 21)/ 10

Mean Absolute Deviation = 126/ 10

Mean Absolute Deviation = 12.6

This gives us a clear picture of the deviation of observation. Therefore, we can say that each point in the data set measures a distance of 12.6 from the mean of 79.

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Things to Remember 

• Mean is often used to measure central tendency. It represents the amount of data collection provided and also, it is suitable for accurate and continuous data.
• The mean deviation indicates the scattering of various series objects from their average value. The values of the series extremes are not significantly affected.
• In mathematics, mean absolute deviation (MAD) is a strong measure of the variability of a consistent sample of quantitative data. 

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