Number 2 Pencil

Statistics Term of the Day - Measures of Central Tendency

I'm already getting good feedback from Devoted Readers about this new feature, so I guess I'll continue. I'd also like to put my $.02 in here and say that, hands down, the best textbook I've found for teaching undergraduate statistics in the social sciences is Graveter & Wallnau's Statistics for the Behavioral Sciences. I have the third edition; two more editions are currently out.

Also, I'll try to add in statistical notation images later when I get home and have the software for it.

Let's step back from percentiles and talk about measures of central tendency. Central tendency of what, you might ask? Of a distribution - a word many non-statisticians consider intimidating, but which just means a collection of observations; in testing, distributions are often sets of scores.

Measures of central tendency are one kind of descriptive statistics (measures that describe or summarize a distribution of scores.) Any measure of central tendency identifies a single score as something that is representative of the entire distribution. Needless to say, the correct measure of central tendency - the one that is most likely to give you a representative score - will depend on the type of data and the shape of the distribution.

The three most-often used measures of central tendency are the mode, median, and mean. The mode couldn't be simpler - it's the most commonly-occuring score in the distribution, and it's suitable for use on discrete (not "discreet"!) and continuous data, and any scale of measurement (ratio, interval, ordinal, and nominal). If more students score a 1060 on the SAT than any other score, that score is the mode; if more students respond "Yes" to a survey than "No" or "No opinion," "Yes" is the mode.

The median is, as we discussed yesterday, the 50th percentile of the distribution - the point at which half the observations are above and half are below. Medians are good for skewed or open-ended observations. If you're asking local families how many children live in each house, a median will give you a better representative number than the mean, which could be skewed too high by that little old lady who lives in the shoe. Medians are also good for ordinal data; i.e., data that are ranked in order but that do not have equal intervals between scores. Ranking of standings in a race are on an ordinal scale - you know you got 2nd place, but you might have been only a second behind the person in first, whereas you might have been an hour faster than the person in 3rd place. Ordinal scales preserve only the order of observations, and not the distances between them.

Finally, there's the mean, aka the average. Add up all the scores in the distribution and divide by the number of scores. You can thus also think of the mean as the score each individual would get if the total of the scores were divided equally among the population (this is what anti-testing types would like to see happen, to combat the "unfairness" of unequal scores). The mean is also a balance point of the distribution, but not with 50% of scores on each side. Instead, it's more like a seesaw, where a score way on one end is balanced out by a lot of lower scores on the other end.

With standardized tests, means, medians, and modes are all useful to know. If the distribution of the test scores is normal (e.g. the "bell curve"), the mean, median, and mode should be roughly equal; this equality is in fact part of the definition of a normal distribution.

If you add/subtract a constant value to each score, the mean will shift by that constant.

If you multiply/divide each score by a constant value, you can multiply/divide the old mean by that same constant, and you'll get the new mean.

Last but not least - be suspicious when the wrong central tendency measure is used (mean income of a group and median income of a group could be very different), and be very wary of measures of central tendency that are provided in the absence of measures of spread (that's tomorrow's term).

Update Devoted Reader Doug S. notes the following:

You might wish to discuss situations when none of these measures is appropriate. Specifically, datasets with multiple strong local maxima make any measure of the central tendency of the whole data set much less useful. Lack of utility doesn't seem to correlate with a reduced frequency of use though. After all, if I have a dataset, the mean (or median, or mode) must mean something (so to speak).

Good point, and something I should have said first thing when discussing descriptive statistics. Important Rule # 1: Always look at your data. Graphing data is ideal; non-normality, skewness, kurtosis - all those will show up with graphing. Descriptives statistics provide one kind of look, but there's nothing like nifty graphical representation of quantitative data.

Here's one example of what Doug is talking about (graph cheerfully stolen from a discussion of what "normality" means for the atmosphere, by Chuck Doswell:

This is a bimodal distribution - something like what you might see if you combined men's and women's heights into one distribution of measures. Often, bimodal distributions are an indication that you have more than one distribution of scores going on in your sample, and might need to separate them. Sometimes, though, this is what one population actually looks like. There can be any number of modes, or any number of bumps in the distribution's curve (those would be the local maxima).

You can calculate a mean and median for this distribution, sure - but they won't be very descriptive as a representative of your data. Important Rule # 2 - Just because you can calculate a statistic doesn't mean it's correct for your data.

Posted by kswygert at March 8, 2005 04:04 PM