Measures of Variability in Data Set: Mean Absolute Deviation Formula
The Mean Absolute Deviation, or MAD, is the average of the distance between the observations and the mean. Mean Deviation and Average Absolute Deviation are the other names for Mean Absolute Deviation.
The Formula for the Mean absolute deviation
The mean absolute deviation is the sum of all absolute values of deviation from the central measure divided by the total number of observations.
Mean Absolute Division \(=\Sigma\frac{\text{Absolute values of Deviation from Central Measures}}{\text{Total Number of Observations}} \)
Mathematically, it can be represented as:
Mean Absolute Deviation \(=\frac{\Sigma\left | X-\mu \right |}{N}\)
Where,
- \(X \) is a data set.
- \(\mu \) is the mean.
- \( X-\mu \) is the absolute deviation of the data point.
- \( \Sigma\left | X-\mu \right | \) is the absolute value of the sum of all deviations
- \(N \) is the sample size of the data set.
How to calculate the mean Absolute deviation?
The steps to find the mean absolute deviation are as follows:
Step 1: Calculate the average of the dataset by adding the values and dividing by the total number of data points.
Step 2: Calculate the absolute deviation by subtracting each value in the data set from the mean. The sign will be ignored and only the magnitude will be considered.
Step 3: Now, add the values from the previous step and divide by the sample size.
Solved Examples
Example 1:
Determine the mean absolute deviation of the given data sets:
26, 46, 56, 45, 19, 22, 24
Solution:
Step 1: Calculate the mean of the given data.
Mean \( =~\frac{26+46+56+45+19+22+24}{7}~=~34 \)
The mean of the given data is 34.
Step 2: Calculate the absolute value of the differences between each value in the data item and the mean.
Data points | Absolute values of distance from mean |
|---|---|
26 | 8 |
46 | 12 |
56 | 22 |
45 | 11 |
19 | 15 |
22 | 12 |
24 | 10 |
Step 3: Calculate the mean of the distances between each value in the data item and the mean.
\( \frac{8+12+22+11+15+12+10}{7}~=~12.85 \)
The mean absolute deviation of the data provided is 12.85.
Example 2:
Determine the mean absolute deviation of the given data sets:
36, 56, 66, 55, 29, 22
Solution:
Step 1: Calculate the mean of the given data.
Mean \( =~\frac{36+56+66+65+29+22}{6}~=~44 \)
The mean of the given data is 44.
Step 2: Calculate the absolute value of the differences between each value in the data and the mean.
Data points | Absolute values of distance from mean |
|---|---|
36 | 8 |
56 | 12 |
66 | 22 |
65 | 11 |
29 | 15 |
22 | 22 |
Step 3: Calculate the mean of the absolute values of the differences between each value in the data item and the mean.
\( \frac{8+12+22+11+15+22}{6}~=~15 \)
The mean absolute deviation of the data provided is 15.
Example 3:
Erica likes to share photos of her cat on social media. Here is the total number of “likes” for the last six photos:
10, 15, 15, 17, 18, 21
Calculate the mean absolute deviation.
Solution:
Step 1: Find the mean.
The total number of “likes” is 96, and there are six pictures.
mean \(=~ \frac{96}{6}~=~16 \)
Step 2: Determine the distance between each data point and the mean.
Data points | Distance from mean |
|---|---|
10 | \(\left | 10-16 \right |=6 \) |
15 | \(\left | 15-16 \right |=1 \) |
15 | \(\left | 15-16 \right |=1 \) |
17 | \(\left | 17-16 \right |=1 \) |
18 | \(\left | 18-16 \right |=2 \) |
21 | \(\left | 21-16 \right |=5 \) |
Step 3: Add the distances together to get the total.
6 + 1 + 1 + 1 + 2 + 5 = 16
Step 4: Divide the sum by the number of data points.
Mean Absolute Deviation \(~=\frac{16}{6}\approx~2.67 \)
Example 4:
The table shows the maximum speeds of eight magic carpet rides at an amusement park. Find the mean absolute deviation of the set of data.
Maximum speeds of magic carpets (mph) |
|---|
58 |
72 |
88 |
66 |
40 |
80 |
60 |
48 |
Solution:
Step 1: Calculate the mean of the given data.
Mean \(~=\frac{58+72+88+66+40+80+60+48}{8}~=~64 \)
The mean of the given data is 64 mph.
Step 2: Calculate the absolute value of the differences between each value in the data item and the mean.
Data points | Absolute values of distance from mean |
|---|---|
58 | 6 |
88 | 24 |
40 | 24 |
60 | 4 |
72 | 8 |
66 | 2 |
80 | 16 |
48 | 16 |
Step 3: Calculate the mean of the absolute values of the differences between each value in the data item and the mean.
\(\frac{24+16+6+4+2+8+16+24}{8}~=~12.5 \)
The mean absolute deviation of the data provided is 12.5.
The formula for the mean absolute deviation is
\(=\Sigma\frac{\text{Absolute values of Deviation from Central Measures}}{\text{Total Number of Observations}} \)
The absolute deviations are all zero if the mean absolute deviation is zero. The mean absolute deviation determines the average deviation of the data points from the mean, and the distance is never negative. The mean absolute deviation can only average to zero if all of the absolute deviations are zero, but can never be negative.
The sum of all observations divided by the total number of observations in a data set is known as the mean in statistics.
To find the mean absolute deviation of the data, start by taking the mean of the data set. Find the absolute value of the difference between the value of each data point and the mean value. Divide the sum of all distances by the total number of data points. This value is the mean absolute deviation.
Mainly, there are three types of means in statistics: Arithmetic Mean, Geometric Mean, and Harmonic Mean.