What is Mean Absolute Deviation?
The average of the difference between observations in a set or data values and the mean is called mean absolute deviation (MAD). Hence, we can say that MAD is a measure of variability in the data set.
A larger value of MAD indicates that the data points are far away from the average and a lower deviation value indicates that the data points are closer to the average value.
Mean absolute deviation is also called mean deviation or average absolute deviation.
How to determine Mean Absolute Deviation?
Step 1: Determine the mean of the data set.
Step 2: Determine the distances between each data point and the mean. These are the absolute deviations.
Step 3: Add the absolute deviations obtained in step 2.
Step 4: Now, divide the sum obtained in step 3 by the number of data points.
Rapid Recall
Mean Absolute Deviation, MAD = \(\frac{\text{Sum of the distances between each data value and their mean}}{\text{Number of data values}}\)
Solved Examples
Example 1
Find and analyze the mean absolute deviation for the data set:
69, 51, 71, 77, 71, 80, 75, 63, 73
Solution:
The data values provided: 69, 51, 71, 77, 71, 80, 75, 63, 73
From the data, the number of samples = 9
Now,
Step 1: Find the mean.
Mean = \(\frac{69 ~+ ~51~ +~ 71~ +~ 77~ +~ 71~ +~ 80 ~+ ~75 ~+ ~63 ~+~ 73}{9} \)= 70
Step 2: We can use a dot plot to organize and find the distance between each data point and the mean. The distances are 1, 19, 1, 7, 1, 10, 5, 7, and 3.
Step 3: The sum of the distances is 1 + 19 + 1 + 7 + 1 + 10 + 5 + 7 + 3 = 54
Step 4: The mean absolute deviation is \(\frac{54}{9}\) = 6.
So, the data values differ from the mean by an average of 6 points.
Example 2
Olivia finds and interprets the mean absolute deviation of the data set: 35, 40, 38, 32, 42, and 41.
Is Olivia correct?
Solution:
To check whether Olivia is correct or not, let’s find the mean absolute deviation.
Given data values are 35, 40, 38, 32, 42, 41
From the data, the number of samples = 6
Now,
Step 1: Find the Mean = \(\frac{35 ~+~ 40~ +~ 38 ~+ ~32 ~+ ~42~ +~ 41}{6}\) = 38
Step 2: Now, find the distance between each data point and mean. The distances are 3, 2, 0, 6, 4, and 3.
Step 3: The sum of the distances is 3 + 2 + 0 + 6 + 4 + 3 = 18
Step 4: The mean absolute deviation is \(\frac{18}{6} \) = 3.
Hence, the data values are different from the mean by an average of 3.
As a result, we can conclude that Olivia is correct.
Example 3
The entry fees for various doodle workshops conducted round the year are: $20, $20, $16, $12, $15, $25, $11
Determine and analyze the mean absolute deviation for the given data.
Solution:
As mentioned in the question, the entry fees of the workshops are:
$20, $20, $16, $12, $15, $25, $11
Number of samples = 7
Step 1: Find the Mean = \(\frac{20~ +~ 20~ +~ 16 ~+ ~12 ~+ ~15 ~+ ~25~ +~ 11}{7}\) = 17
Step 2: Find the distance between each data point and the mean. The distances are 3, 3, 1, 5, 2, 8 and 6.
Step 3: The sum of the distances is 3 + 3 + 1 + 5 + 2 + 8 + 6 = 28
Step 4: The mean absolute deviation is \(\frac{28}{7} \) = 4.
Therefore, the data values are different from the mean by an average of 4.
Mean deviation is simple to calculate and comprehend. As a result, mean deviation is frequently used in daily life. For instance, when students take tests, teachers compute the mean of the results to determine whether the class average and hence performance is good or if it needs improvement.
The absolute deviation of a dataset is the average distance between each data point and the mean. It gives us a sense of the variability in a dataset.
Add all the data values to get the total or the sum. Divide the sum by the total number of values in the data set. The quotient thus obtained is the mean of the data set.
A dot plot helps in organizing the data and determining the distance of each data point from the mean. These distances are then added and the sum obtained is divided by the number of samples to find the mean absolute deviation.