The trimmed mean is a family of measures of central tendency . The -trimmed mean of of values is computed by sorting all the values, discarding % of the smallest and % of the largest values, and computing the of the remaining values.
For example, to calculate the -trimmed mean for a set of values , steps are the following:
- Step 1. Sort the values: .
- Step 2. Discard 20% of the largest values - i.e. one (20% of 5) largest value (12); discard 20% of the smallest values - i.e. one smallest value (8). Now we have a set of 3 values:
- Step 3. Compute the mean of the 3 values: the mean value of is 10.
Thus the -trimmed mean of 5 values is 10.
In contrast to the arithmetic mean , the trimmed mean is a robust measure of central tendency. For example, a small fraction of anomalous measurements with abnormally large deviation from the center may change the mean value substantially. At the same time, the trimmed mean is stable in respect to presence of such abnormal extreme values, which get "trimmed" away.
For example, in the set of 5 values discussed above, replace one value by a large number, say, "12" by "1000". Then compute the mean of the 5 values, and the -trimmed mean. The replacement does not affect the trimmed mean (because the extreme value is discarded on step 2), but it changes the mean significantly - from 10 to 207.
The trimmed mean, as a family of measures, includes the arithmetic mean and the median as the most extreme cases. The trimmed mean with the minimal degree of trimming ( %) coincides with the mean; the trimmed mean with the maximal degree of trimming ( %) coincides with the median.
One popular example of a trimmed mean is judges´ scores in gymnastics, where the extreme scores are often discarded before computing the score for a particular performance.