1. Meaning of Mean Deviation
Mean deviation measures how far the values in a data set spread away from a central value. It finds the average of the absolute deviations, which means it looks at the size of the differences without considering their sign.
Mean deviation can be calculated from the mean or the median. It gives a simple idea of how scattered the values are.
2. Mean Deviation About Mean
When deviations are taken from the mean, the formula is:
\( MD_{\bar{x}} = \dfrac{\sum |x - \bar{x}|}{n} \)
Here:
- \( x \) = each data value
- \( \bar{x} \) = mean of the data
- \( |x - \bar{x}| \) = absolute deviation
- \( n \) = number of values
2.1. Example
Consider the data: 4, 6, 8
Mean = \( \dfrac{4 + 6 + 8}{3} = 6 \)
Deviations:
- |4 − 6| = 2
- |6 − 6| = 0
- |8 − 6| = 2
Total deviation = 2 + 0 + 2 = 4
\( MD = \dfrac{4}{3} \approx 1.33 \)
3. Mean Deviation About Median
Mean deviation can also be found using the median instead of the mean. This is often more stable when the data has extreme values.
\( MD_{M} = \dfrac{\sum |x - M|}{n} \)
Here \( M \) is the median of the data.
3.1. Example
Consider the data: 3, 5, 9
Median = 5
Deviations:
- |3 − 5| = 2
- |5 − 5| = 0
- |9 − 5| = 4
Total deviation = 2 + 0 + 4 = 6
\( MD = \dfrac{6}{3} = 2 \)