Averages give us an idea of the concentration of the observations about the central part of the distribution. If we know the average alone we cannot form a complete idea about the distribution. They must be supported and supplemented by some other measures called measures of dispersion. The Median Absolute Deviation is one such measure of dispersion that gives us some idea of the degree of “spread” in our data.
How to calculate the Median Absolute Deviation:
- Arrange the given data in ascending or descending order.
- Calculate the median of the given data. If there are an odd number of data values, take the middlemost value. If there are an even number of data points the average of the two middlemost values is the required median.
- Calculate the difference of each data value from the median.
- Take the absolute value of each of the differences calculated above. By taking absolute value we disregard the sign of the differences and consider them to be positive number.
- Arrange the absolute differences in ascending or descending order and once again calculate the median of the absolute differences.
- Note that in the above method we take the median twice. We first calculate the median of the data and then calculate the median of the absolute differences.
Median Absolute Deviation Formula:
The median absolute deviation for a given set of data values can be calculated using the formula, \text{Median Absolute Deviation }= \text{Median}_{i}(|x_i - \text{Median}|).
Example 1:
Consider the following set of data arranged in ascending order: 12, 33, 36, 45, 47, 53, 55. Since there are n=7 data values we have an odd number of data points. Therefore we take the middlemost data value to be the median \text{Median }= 4^{th}\text{ term } = 45. We now calculate the differences of the data values from the median,
| x_i | x_i-\text{Median} = x_i - 45 | |x_i-\text{Median}| |
| 12 | -33 | 33 |
| 33 | -12 | 12 |
| 36 | -9 | 9 |
| 45 | 0 | 0 |
| 47 | 2 | 2 |
| 53 | 8 | 8 |
| 55 | 10 | 10 |
We now arrange the absolute differences in the last column in ascending order as follows, 0, 2, 8, 9, 10, 12, 13. The median of the above values is the middlemost term. Therefore, we have that Median Absolute Deviation for the given set of data values is equal to 9.
Example 2:
Consider the following set of data arranged in descending order: 8, 7, 5, 3, 3, 2. Since there are n=6 data values we have an even number of data points. Therefore we take the average of the two middlemost data values to be the median \text{Median }= (3^{rd}\text{ term } + 4^{th}\text{ term })/2 = (5+3)/2 = 4. We now calculate the differences of the data values from the median,
| x_i | x_i-\text{Median} = x_i - 4 | |x_i-\text{Median}| |
| 8 | 4 | 4 |
| 7 | 3 | 3 |
| 5 | 1 | 1 |
| 3 | -1 | 1 |
| 3 | -1 | 1 |
| 2 | -2 | 2 |
We now arrange the absolute differences in the last column in ascending order as follows, 1, 1, 1, 2, 3, 4. The median of the above values is the average of the two middlemost terms. Therefore, we have that Median Absolute Deviation = (1+2)/2 = 1.5 .
