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In my last post, I described three situations where the average of a sequence of numbers is not representative enough to be useful: in the presence of severe outliers, in the face of multimodal data distributions, and in the face of infinite-variance distributions.  The post generated three interesting comments that I want to respond to here.

First and foremost, I want to say thanks to all of you for giving me something to think about further, leading me in some interesting new directions.  First, chrisbeeleyimh had the following to say:

“I seem to have rather abandoned means and medians in favor of drawing the distribution all the time, which baffles my colleagues somewhat.”

Chris also maintains a collection of data examples where the mean is the same but the shape is very different.  In fact, one of the points I illustrate in Section 4.4.1 of Exploring Data in Engineering, the Sciences, and Medicine is that there are cases where not only the means but all of the moments (i.e., variance, skewness, kurtosis, etc.) are identical but the distributions are profoundly different.  A specific example is taken from the book Counterexamples in Probability, 2nd Edition by J.M. Stoyanov, who shows that if the lognormal density is multiplied by the following function:

f(x) = 1 + A sin(2 pi ln x),

for any constant A between -1 and +1, the moments are unchanged.  The character of the distribution is changed profoundly, however, as the following plot illustrates (this plot is similar to Fig. 4.8 in Exploring Data, which shows the same two distributions, but for A = 0.5 instead of A = 0.9, as shown here).  To be sure, this behavior is pathological – distributions that have finite support, for example, are defined uniquely by their complete set of moments – but it does make the point that moment characterizations are not always complete, even if an infinite number of them are available.  Within well-behaved families of distributions (such as the one proposed by Karl Pearson in 1895), a complete characterization is possible on the basis of the first few moments, which is one reason for the historical popularity of the method of moments for fitting data to distributions.  It is important to recognize, however, that moments do have their limitations and that the first moment alone – i.e., the mean by itself – is almost never a complete characterization.  (I am forced to say “almost” here because if we impose certain very strong distributional assumptions – e.g., the Poisson or binomial distributions – the specific distribution considered may be fully characterized by its mean.  This begs the question, however, of whether this distributional assumption is adequate.  My experience has been that, no matter how firmly held the belief in a particular distribution is, exceptions do arise in practice … overdispersion, anyone?)

The plot below provides a further illustration of the inadequacy of the mean as a sole data characterization, comparing four different members of the family of beta distributions.  These distributions – in the standard form assumed here – describe variables whose values range from 0 to 1, and they are defined by two parameters, p and q, that determine the shape of the density function and all moments of the distribution.  The mean of the beta distribution is equal to p/(p+q), so if p = q – corresponding to the class of symmetric beta distributions – the mean is ½, regardless of the common value of these parameters.  The four plots below show the corresponding distributions when both parameters are equal to 0.5 (upper left, the arcsin distribution I discussed last time), 1.0 (upper right, the uniform distribution), 1.5 (lower left), and 8.0 (lower right).

The second comment on my last post was from Efrique, who suggested the Student’s t-distribution with 2 degrees of freedom as a better infinite-variance example than the Cauchy example I used (corresponding to Student’s t-distribution with one degree of freedom), because the first moment doesn’t even exist for the Cauchy distribution (“there’s nothing to converge to”).  The figure below expands the boxplot comparison I presented last time, comparing the means, medians, and modes (from the modeest package), for both of these infinite-variance examples: the Cauchy distribution I discussed last time and the Student’s t-distribution with two degrees of freedom that Efrique suggested.  Here, the same characterization (mean, median, or mode) is summarized for both distributions in side-by-side boxplots to facilitate comparisons.  It is clear from these boxplots that the results for the median and the mode are essentially identical for these distributions, but the results for the mean differ dramatically (recall that these results are truncated for the Cauchy distribution: 13.6% of the 1000 computed means fell outside the +/- 5 range shown here, exhibiting values approaching +/- 1000).  This difference illustrates Efrique’s further point that the mean of the data values is a consistent estimator of the (well-defined) population mean of the Student’s t-distribution with 2 degrees of freedom, while it is not a consistent estimator for the Cauchy distribution.  Still, it also clear from this plot that the mean is substantially more variable for the Student’s t-distribution with 2 degrees of freedom than either the median or the modeest mode estimate.

Another example of an infinite-variance distribution where the mean is well-defined but highly variable is the Pareto type I distribution, discussed in Section 4.5.8 of Exploring Data.  My favorite reference on distributions is the two volume set by Johnson, Kotz, and Balakrishnan (Continuous Univariate Distributions, Vol. 1 (Wiley Series in Probability and Statistics) and Continuous Univariate Distributions, Vol. 2 (Wiley Series in Probability and Statistics)) , who devote an entire 55 page chapter (Chapter 20 in Volume 1) to the Pareto distribution, noting that it is named after Vilafredo Pareto, a mid nineteenth- to early twentieth-century Swiss professor of economics, who proposed it as a description of the distribution of income over a population.  In fact, there are several different distributions named after Pareto, but the type I distribution considered here exhibits a power-law decay like the Student’s t-distributions, but it is a J-shaped distribution whose mode is equal to its minimum value.  More specifically, this distribution is defined by a location parameter that determines this minimum value and a shape parameter that determines how rapidly the tail decays for values larger than this minimum.  The example considered here takes this minimum value as 1 and the shape parameter as 1.5, giving a distribution with a finite mean but an infinite variance.  As in the above example, the boxplot summary shown below characterizes the mean, median, and mode for 1000 statistically independent random samples drawn from this distribution, each of size N = 100.  As before, it is clear from this plot that the mean is much more highly variable than either the median or the mode.

In this case, however, we have the added complication that since this distribution is not symmetric, its mean, median and mode do not coincide.  In fact, the population mode is the minimum value (which is 1 here), corresponding to the solid line at the bottom of the plot.  The narrow range of the boxplot values around this correct value suggest that the modeest package is reliably estimating this mode value, but as I noted in my last post, this characterization is not useful here because it tells us nothing about the rate at which the density decays.  The theoretical median value can also be calculated easily for this distribution, and here it is approximately equal to 1.587, corresponding to the dashed horizontal line in the plot.  As with the mode, it is clear from the boxplot that the median estimated from the data is in generally excellent agreement with this value.  Finally, the mean value for this particular distribution is 3, corresponding to the dotted horizontal line in the plot.  Since this line lies fairly close to the upper quartile of the computed means (i.e., the top of the “box” in the boxplot), it follows that the estimated mean falls below the correct value almost 75% of the time, but it is also clear that when the mean is overestimated, the extent of this overestimation can be very large.  Motivated in part by the fact that the mean doesn’t always exist for the Pareto distribution, Johnson, Kotz and Balakrishnan note in their chapter on these distributions that alternative location measures have been considered, including both the geometric and harmonic means.  I will examine these ideas further in a future post.

Finally, klr mentioned my post on useless averages in his blog TimelyPortfolio, where he discusses alternatives to the moving average in characterizing financial time-series.  For the case he considers, klr compares a 10-month moving average, the corresponding moving median, and a number of the corresponding mode estimators from the modeest package.  This is a very interesting avenue of exploration for me since it is closely related to the median filter and other nonlinear digital filters that can be very useful in cleaning noisy time-series data.  I discuss a number of these ideas – including moving-window extensions of other data characterizations like skewness and kurtosis – in my book Mining Imperfect Data: Dealing with Contamination and Incomplete Records Again, thanks to all of you for your comments.  You have given me much to think about and investigate further, which is one of the joys of doing this blog.