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Often introductory applied statistics texts distinguish the mean from the median (often in the the context of descriptive statistics and motivating the summarization of central tendency using the mean, median and mode) by explaining that the mean is sensitive to outliers in sample data and/or to skewed population distributions, and this is used as a justification for an assertion that the median is to be preferred when the data are not symmetrical.

For example:

The best measure of central tendency for a given set of data often depends on the way in which the values are distributed.... When data are not symmetric, the median is often the best measure of central tendency. Because the mean is sensitive to extreme observations, it is pulled in the direction of the outlying data values, and as a result might end up excessively inflated or excessively deflated."
—Pagano and Gauvreau, (2000) Principles of Biostatistics, 2nd ed. (P&G were at hand, BTW, not singling them out per se.)

The authors define "central tendency" thus: "The most commonly investigated characteristic of a set of data is its center, or the point about which observations tend to cluster."

This strikes me as a less-than forthright way of saying only use the median, period, because only using the mean when the data/distributions are symmetrical is the same thing as saying only use the mean when it equals the median. Edit: whuber rightly points out that I am conflating robust measures of central tendency with the median. So it is important to keep in mind that I am discussing the specific framing of the arithmetic mean versus the median in introductory applied statistics (where, mode aside, other measures of central tendency are not motivated).

Rather than judging the utility of the mean by how much it departs from the behavior of the median, ought we not simply understand these as two different measures of centrality. In other words being sensitive to skewness is a feature of the mean. One could just as validly argue "well the median is no good because it is largely insensitive to skewness, so only use it when it equals the mean."

(The mode is quite sensibly not getting involved with this question.)

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Personally, I like to include both measures, mean and median, which will give the reader not only some information about central tendency, but also an idea of how skewed the data are. –  Benjamin May 4 at 21:18
    
Surely, and adding in the numerical and visual elements of a box, dot and whisker plots moves even a little closer to presenting a distribution itself. –  Alexis May 4 at 21:44
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At one point you misinterpret: the median is not the only statistic that is robust to a few extreme observations. Thus the mean is indicted on the basis of an (often) undesirable characteristic and not by any comparison to the median. But I also get a glimmer of your concern, and perhaps it is related to the implicit conflation of asymmetry and existence of outliers that occurs in this quotation. That is regrettably ill-conceived, because although having outliers sometimes implies asymmetry, the converse is not often true. –  whuber May 5 at 19:15
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Readers here will find the following thread of interest: If the mean is so sensitive, why use it in the first place? –  gung May 5 at 19:56
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In light of the definition given for "central tendency", it seems clear why the mean would not be a useful measure in the presence of skew or outliers. Whether or not you really want to estimate this notion of central tendency seems to be another matter! –  jsk May 5 at 20:58

4 Answers 4

up vote 10 down vote accepted

I disagree with the advice as a flat out rule. (It's not common to all books.)

The issues are more subtle.

If you're actually interested in making inference about the population mean, the sample mean is at least an unbiased estimator of it, and has a number of other advantages. In fact, see the Gauss-Markov theorem - it's best linear unbiased.

If your variables are heavily skew, the problem comes with 'linear' - in some situations, all linear estimators may be bad, so the best of them may still be unattractive, so an estimator of the mean which is not-linear may be better, but it would require knowing something (or even quite a lot) about the distribution. We don't always have that luxury.

If you're not necessarily interested in inference relating to a population mean ("what's a typical age?", say or whether there's a more general location shift from one population to another, which might be phrased in terms of any location, or even of a test of stochastic dominance), then casting that in terms of the population mean is either not necessary or likely counterproductive (in the last case).

So I think it comes down to thinking about:

  • what are your actual questions? Is population mean even a good thing to be asking about in this situation?

  • what is the best way to answer the question given the situation (skewness in this case)? Is using sample means the best approach to answering our questions of interest?

It may be that you have questions not directly about population means, but nevertheless sample means are a good way to look at those questions ... or vice versa - the question might be about population means but sample means might not be the best way to answer that question.

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In real life, we should choose a measure of central tendency based on what we are trying to find out; and yes, sometimes the mode is the right thing to use. Sometimes it's the Winsorized or trimmed mean. Sometimes the geometric or harmonic mean. Sometimes there is no good measure of central tendency.

Intro books are written badly, they teach that there are cookbook rules to apply.

Take income. This is often very skewed and sometimes has outliers; sure enough, we usually see "median income" reported. But sometimes the outliers and skewness are important. It depends on context and requires thought.

I wrote more on this

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Peter, thank you so much for the link to your post. I wish that the intro texts took the 1 to 2 pages of space necessary to provide as thoughtful a consideration as you provided there. –  Alexis May 4 at 20:51
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I haven't written one but I want to insert a little defence of introductory texts. Any introductory text that tried to give a fully nuanced view that experienced professionals would recognise as such would be flamed by almost all intended recipients; indeed it would not even get published. –  Nick Cox May 5 at 17:23
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A substantive comment: when values are additive such that totals make (e.g.) physical sense, the mean is a a natural summary regardless of the distribution of the individual values. –  Nick Cox May 5 at 17:25
    
Nick, these are precisely what I was thinking. For example, if there are resources or risks attached to, say age, then policy makers might be interested in per-capital costs, which are one way of expressing total (additive) cost. Of course, I do not mean to malign the median and substantive interpretation of it either. –  Alexis May 5 at 19:16
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@NickCox I think that introductory texts can do a lot better than they do. For mean vs. median it's not even a mathematical argument - it's a substantive one. Introductory texts need to tell the person reading them that they are not really qualified to do data analysis. –  Peter Flom May 5 at 19:40

I think that what's missing from the question as well as both the answers so far is that the discussion of mean vs median in introductory statistics books generally occurs early on in a chapter about how to numerically summarize a distribution. As opposed to inferential statistics, this is generally about producing descriptive statistics that would be a useful way to convey information about the distribution of the data numerically as opposed to graphically. Contexts in which this arises is the descriptive statistics section of a report or journal article in which there generally is not room for graphical summaries of all the variables in your dataset. If the distribution is skewed, it seems sensible in this context to choose the median over the mean. If the distribution is symmetric without outliers, then the mean is generally preferred over the median as it will be a more efficient estimator.

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Your point about descriptive versus inferential statistics is worthwhile. But you are effectively saying (for descriptive statistics) "only use the mean when it is the same as the median." If the distribution is skewed, then the median does a poor job of representing the concept of per capita, right? So isn't it just as valid to take the position "only use the median when it equals the mean?" That's just as arbitrary, and seems to direct attention away from the substantive meaning of these measures (for folks learning them). –  Alexis May 5 at 5:23
    
@Alexis Though the sample mean and sample median are both estimating the same aspect of a symmetric population, they will rarely be equal for a dataset that is approximately symmetric. But yes, for all practical purposes they should be similar in which case it really doesn't matter which one to report. In choosing a numerical summary measure of the distribution to replace a graphical display, the goal is not represent the concept of per capita. In other contexts, I agree with you and the others that the appropriate measure of center should depend on the research question. –  jsk May 5 at 7:00
    
The goal is not to represent the concept of per capita? Says who? Why presuppose that's not the goal? –  Alexis May 5 at 12:48
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I don't see that it matters whether you are doing inferential or descriptive statistics in this instance. If the appropriate descriptive measure of central tendency is the median, then inferences should be drawn about the median; if the mean, then the mean. If no descriptive measure makes sense, then no inferential measure will make sense either. –  Peter Flom May 5 at 18:39
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@PeterFlom What about in cases where the end goal is not inference? I agree that the appropriateness of a descriptive statistic depends entirely on the reason for producing the statistic. The notion that it is possible that "no descriptive measure makes sense" seems to imply that a descriptive statistic cannot be inherently meaningful. I would argue that in almost all cases, the median makes sense as a measure of the center of the distribution by definition. Whether or not it makes sense for other purposes is another question. –  jsk May 5 at 20:14

Even when data are skewed (e.g., health care costs calculated alongside a clinical trial, where few patients totalled zero cost because they die just after the enrollment, and few patients accrued tons of cost due to side effects of a given health care programme under investigation), mean may be preferred to median for at least one pratical reason: multiplying the mean cost for the number of patients gives health care decision-makers the budget impact of the health care technology under study.

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