By ordering the original observations, and taking away the first smallest observations and the first largest obserations, the trimmed mean takes the arithmetic average of the resulting data. The idea of a trimmed mean is to eliminate outliers, or extreme observations that do not seem to have any logical explanations in calculating the overall mean of a population.
For example, suppose 10 new lightbulbs are drawn from a population of 100 to find the average lifetime of a typical lightbulb, measured in number of hours. The measurements are 802, 854, 823, 428, 815, 840, 833, 809, 843, 821. The (arithmetic) mean of the measurement is
with sample standard deviation = 127.1, whereas the 1st trimmed mean gives:
with sample standard deviation = 14.6, greatly reducing the sample deviation.
Another robust estimtor of a mean is the winsorized mean. Like the trimmed mean, the winsorized mean eliminates the outliers at both ends of an ordered set of observations. Unlike the trimmed mean, the winsorized mean replaces the outliers with observed values, rather than discarding them. The formal definition of the th winsorized mean is:
From the definition, we see that the winsorized mean is the average of the observations where the first smallest values are replaced by the th smallest value, , and the first largest values are replaced by the th largest value, .
From the above example, the 1st winsorized mean is
with sample standard deviation = 16.1, fairly close to the answer given by the trimmed mean.
|Date of creation||2013-03-22 14:42:02|
|Last modified on||2013-03-22 14:42:02|
|Last modified by||CWoo (3771)|