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I have read a book saying that statistical properties of the mode and the median are difficult to determine compared to the mean. I am not entirely sure why is so.

Is it correct that because the mean can be put into mathematical functions more easily compared to the mode and median? What does it mean by statistical properties? What does it mean by determining statistical properties?

If anyone could provide more explanations or examples of some sort.

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    $\begingroup$ It would be much easier to interpret the book's meaning if you provided a larger quote with some of the surrounding context. $\endgroup$
    – Nick Stauner
    Commented Jan 21, 2014 at 7:32
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    $\begingroup$ also a complete reference can help $\endgroup$
    – Maarten Buis
    Commented Jan 21, 2014 at 8:41

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For 'statistical properties' read 'computing the distribution of', or 'computing some aspect of the distribution of' (such as variance, say).

In particular, they're probably referring to the sampling distribution of a sample statistic.

Sample means have some quite nice properties*, so that in many cases, for example, it's relatively easy to compute the mean, variance (and covariances) of the distribution of sample means, and asymptotically, we have the central limit theorem which (speaking very loosely) tells us about distributions of means in large samples.

* means of sums "add", and variances of sums of independent variables also "add" (i.e. the mean of a sum is the sum of the means, the variance of a sum of independent variables is a sum of the variances, which makes the mean and variance of the distributions of sample means - usually - quite easy to find.)

By contrast, sample medians (and other quantiles) are often more difficult to work with, and don't have nice linear properties like that. Nevertheless, sometimes we can make progress in finite samples, and asymptotically (i.e. in very large samples) they tend to behave relatively more 'nicely' - there's a nice result about non-extreme quantiles (in effect, a kind of central limit theorem) and a different kind of theorem relating to extremes.

Modes are much worse. Generally speaking, they really don't have very 'nice' properties; for example, it's relatively easy for modes to 'jump about' in somewhat surprising ways when you take larger samples, and even asymptotically the variance of a mode doesn't decrease as $1/n$, so they may tend to be pretty 'noisy' things to deal with. Even the basic step of defining a sample mode for continuous data can be a tricky business.

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  • $\begingroup$ I was just wondering, would presence of outliers make some difference? $\endgroup$
    – Abhimanyu Arora
    Commented Jan 21, 2014 at 9:02
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    $\begingroup$ @Abhimanyu - when computing such properties as I am describing, one starts with a specified distribution; obviously such a distribution doesn't have outliers (whatever there is is from the specified distribution). Clearly, however, a distribution having skewness or heavy tails will have a different distribution for sample means, medians and modes than one not having skewness or heavy tails. Of course, a sample may have outliers relative to that specified distribution, so the data is actually coming from a different distribution than the one assumed. In that sense, yes. ... (ctd) $\endgroup$
    – Glen_b
    Commented Jan 21, 2014 at 9:32
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    $\begingroup$ (ctd) ... it does make a difference, and it doesn't affect means, medians and modes equally. We're now talking more about robustness to some contamination/mixture but where we still want to have our inference relate to the uncontaminated part (the major component of the mixture); means are quite sensitive to extreme outliers, while medians (and modes, in many cases) are typically relatively insensitive to them, at least when they don't occur in large proportions. $\endgroup$
    – Glen_b
    Commented Jan 21, 2014 at 9:40
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    $\begingroup$ @AbhimanyuArora No, not on a sample, on population distribution $\endgroup$
    – Glen_b
    Commented Jan 21, 2014 at 14:16
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    $\begingroup$ It refers to the population distribution of the sample statistic. Sample statistics have their own distributions (this fact is crucial to understanding a lot of statistics). $\endgroup$
    – Glen_b
    Commented Jan 23, 2014 at 0:22

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