Question

Consider the discussion in our Devore reading in this unit involving an important distinction between mean and median that uses the concept of a trimmed mean to highlight an important continuum between the two. Presuming that the mean and median are different values for a distribution, the mean can be taken to indicate a 0% trim, and the median can be taken to approach a 50% trim (with effectively 100% of the values removed). These two values define a continuum of trimmed mean values that would fall between the two. Discuss why the mean and median of the distribution always approach each other as we take trimmed means at higher and higher percentages (e.g., 10%, 20%, 30% ...). In particular, describe what is happening to the kurtosis and skewness of the distribution as we trim off more and more data. Speculate on whether or not you might expect to see an optimum point in that process at some value between the mean and median. (Hint: You should!) Why might this matter?

Answer 1

Explanation:

A trimmed mean is a method of averaging that removes a small designated percentage of the largest and smallest values before calculating the mean. After removing the specified observations, the trimmed mean is found using a standard arithmetic averaging formula. The use of a trimmed mean helps eliminate the influence of data points on the tails that may unfairly affect the traditional mean.

trimmed means provide a better estimation of the location of the bulk of the observations than the mean when sampling from asymmetric distributions;

the standard error of the trimmed mean is less affected by outliers and asymmetry than the mean, so that tests using trimmed means can have more power than tests using the mean.

if we use a trimmed mean in an inferential test , we make inferences about the population trimmed mean, not the population mean. The same is true for the median or any other measure of central tendency.

I can imagine saying the skewness is such-and-such, but that's mostly a side-effect of a few outliers, the fact that the 5% trimmed skewness is such-and-such.

I don't think that trimmed skewness or kurtosis is very much used in practice, partly because

If the skewness and kurtosis are highly dependent on outliers, they are not necessarily useful measures, and trimming arbitrarily solves that problem by ignoring it.

Problems with inconvenient distribution shapes are often best solved by working on a transformed scale.

There can be better ways of measuring or more generally assessing skewness and kurtosis, such as the method above or L-moments. As a skewness measure (mean ? median) / SD is easy to think about yet often neglected; it can be very useful, not least because it is bounded within [?1,1][?1,1].

i expect to see the optimum point in that process at some value between the mean and median.