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Much along the lines of the "real-life examples of common distributions" I'm interested if anyone has any pedagogical examples used to teach negative skewness?

There are plenty of canonical examples of symmetric or normal distributions used in teaching - even if ones like height and weight don't hold up under closer biological scrutiny! Maybe blood pressure is nearer normality. I like the historical example of astronomical measurement errors - it's intuitive they are no more likely to lie in one direction than another, and small errors are more likely than large.

There are also dozens of common pedagogical examples for positive skewness. People's incomes; mileage on used cars for sale; reaction times in a psychology experiment; costs of houses; number of accident claims made by an insurance customer; number of children in a family. The physical reasonableness of such examples often stems from distributions that are bounded below (usually by zero), with low values being plausible or even common, while very large (sometimes, orders of magnitude) higher values are also known to occur.

But for negative skew, clear and vivid examples do not abound, perhaps due to the lack of real-life distributions with a clear upper bound. A rather bad-taste example I was taught at school was "number of fingers". Most have ten, but it's not uncommon to lose one or more in an accident. (The upshot was that "99% of people have a higher-than-average number of fingers"!) Polydactyly rather complicates that issue, so ten is not a strict upper bound. And since both missing and extra fingers are relatively rare events, students may not have a clear intuition which of the effects predominates.

Instead, I usually use a binomial distribution with high p. But students often find "number of satisfactory components in a batch is negatively skewed" less intuitive than the complementary fact that "number of faulty components in a batch is positively skewed". (The textbook has an industrial theme; I actually use number of cracked and intact eggs in a packet of 12.) It seems students feel that "success" should be rare. Indeed some binomial cumulative probability tables require $p \leq 0.5$ so enforce this, and I've more often found students have a difficulty with questions that require them to take $p > 0.5$ than with ones which counter-intuitively require them to label contracting a terrible disease as a "success"!

Another option is to point out that if $X$ has a positively skewed distribution, then $-X$ is negatively skewed, but to attempt to place this in a practical context ("negative house prices are negatively skewed") would be doomed to pedagogical failure.

Perhaps the problem with fingers or intact eggs as examples is that they are (a) very artificial, (b) have a sharp cut-off at the mode. Good positive skew examples like incomes often have a smooth distribution, for which students have enough life-experience to be aware of the shape. Are there similar data sets, either well-known or easily grasped, which effectively illustrate negative skewness?

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It is not apparent that negating a variable will be a "pedagogical failure," because there is the option of adding a constant without changing the shape of the distribution. Many skewed distributions involve proportions $X$ for instance, and the complementary proportions $1-X$ are usually just as natural and easy to interpret as the original proportions. Even with house prices $X$ the values $C-X$ where $C$ is a maximum house price in the area could be of interest and is not difficult to understand. Also consider using logs and negative power transformations to create negative skew. –  whuber Mar 7 at 17:30
I agree that $C-X$ in the case of house prices would be a little contrived. But $1/X$ would not: it would be "amount of house you can buy per dollar." I suspect that in any reasonably homogeneous area this would have a strong negative skew. Such examples could teach the deeper lesson that skewness is a function of how we express the data. –  whuber Mar 7 at 20:13
@whuber It wouldn't be contrived at all. Maximum and minimum potential prices in a market arise naturally as those reflecting different evaluations by market participants. Among the buyers, there is conceivably one that would pay maximum price for a given house. And among the sellers there is one that would conceivably accept minimum price. But this information is not public and so actual observed transaction prices are affected by the existence of incomplete information. (CONT'D) –  Alecos Papadopoulos Mar 8 at 14:04
CONT'D ... The following paper by Kumbhakar and Parmeter (2010) models exactly that (permitting also the case of symmetry), and with an application on the house market:link.springer.com/article/10.1007/s00181-009-0292-8#page-1 –  Alecos Papadopoulos Mar 8 at 14:04
Age at death is negatively skewed in developed countries. –  Nick Cox Mar 12 at 0:59

5 Answers 5

up vote 4 down vote accepted

Scores on easy tests, or alternatively, scores on tests for which students are especially motivated, tend to be left skew.

As a result, the SAT/ACT scores of students entering sought after colleges (and even more so, their GPAs) tend to be left skew.

e.g. There's plenty of examples at collegeapps.about.com e.g. a plot of University of Chicago SAT/ACT and GPA is here.

Similarly GPAs of graduates are often left-skew.

e.g. see Fig 5 here

(It's not hard to find other, similar examples.)

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For an introductory stats class I think this example works well pedagogically - it is something students are likely to have real-life experience of, can reason about intuitively, and can confirm against widely available data sets. –  Silverfish Mar 11 at 0:17

In Stochastic Frontier Analysis, and specifically in its historically initial focus, production, the production function of a firm/production unit in general, is specified stochastically as

$$q = f(\mathbf x) + u-w$$

where $q$ is the actual output produced by the firm, and $f(\mathbf x)$ is its production function (which is understood more as an input-output relation rather than a mathematical expression reflecting "engineering" relations) with $\mathbf x$ being a vector of production inputs (capital, labor, energy, materials, etc). The production function in Economic Theory represents maximum output, given technology and inputs, i.e. it embodies full efficiency. Then $u$ is a zero-mean normal disturbance on the production process, and $w$ is a non-negative random variable representing deviation from full efficiency due to reasons that the econometrician may not know, but he can measure through this set up. This random variable is usually assume to follow a half-normal or exponential distribution. Assuming the half normal (for a reason), we have

$$u \sim N(0, \sigma_u^2),\;\; w\sim HN\left(\sqrt {\frac 2{\pi}}\sigma_2, \left(1- \frac 2{\pi}\right)\sigma_2^2\right)$$

where $\sigma_2$ is the standard deviation of the "underlying" normal random variable whose absolute value is the Half-normal.

The composite error-term $\varepsilon = u-w$ is characterized by the following density

$$f_{\varepsilon}(\varepsilon) = \frac 2{s_2}\phi\left(\varepsilon/s_2\right)\Phi\left((-\frac {\sigma_2}{\sigma_u})\cdot(\varepsilon/s_2)\right),\;\; s_2^2 = \sigma^2_u + \sigma^2_2$$

This is a skew-normal density, with location parameter $0$, scale parameter $s_2$ and skew parameter $(-\frac {\sigma_2}{\sigma_u})$, where $\phi$ and $\Phi$ are the standard normal pdf and cdf respectively. For $\sigma_u =1, \;\; \sigma_2 = 3$, the density looks like this: enter image description here

So negative skewness is, I'd say,the most natural modelling of the efforts of human race itself: always deviating from its imagined ideal -in most cases lagging behind it (the negative part of the density), while in relatively fewer cases, transcending its perceived limits (the positive part of the density) . Students themselves can be modeled as such a production function. It is straightforward to map the symmetric disturbance and the one-sided error to aspects of real life. I cannot imagine how more intuitive can one get about it.

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This answer seems to echo @Glen_b's suggestion of grad GPA. Highly motivated human behavior aimed at an elusive ideal certainly fits that scenario! Efficiency in general is a great example. –  Nick Stauner Mar 8 at 10:03
@Nick Stauner The important point here is that we consider "actual minus target" signed, not the "distance" in absolute values. We keep the sign in order to know whether we are above or below the target. The intuition here is, exactly as you write, that "highly motivated" behavior will push "actual" closer to "target", creating asymmetry. –  Alecos Papadopoulos Mar 8 at 11:12

Asset price changes (returns) typically have negative skew - many small price increases with a few large price drops. The skew seems to hold for almost all types of assets: stocks prices, commodity prices, etc. The negative skew can be observed in monthly price changes but is much more evident when you start looking at daily or hourly price changes. I think this would be a good example because you can show the effects of frequency on skew.

More details: http://www.fusioninvesting.com/2010/09/what-is-skew-and-why-is-it-important/

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I like this example a lot! Is there an intuitive way of explaining it - essentially, "downside shocks are more likely (or at least, likely to be more severe) than upside shocks"? –  Silverfish Mar 7 at 19:17
@Silverfish I would phrase it as extreme negative market outcomes are more likely than extreme positive market outcomes. Markets also have asymmetric volatility. Market volatility generally increases more following negative returns than positive returns. This is often modeled with Garch models, such as GJR-Garch (see Arch wikipedia entry). –  John Mar 7 at 19:21
I also saw an explanation that bad news is released in bunches. I have not used GJR-GARCH. I attempted to use multifractal Brownian motion (Mandelbrot) to model asymmetry, but was unable to make it work. –  wcampbell Mar 7 at 21:15
This is at best simplistic. For example, I just took a data set of daily returns on 31 equity indexes. More than half of them have positive skew (using Pearson's skewness) and over 70% are positive on the measure 3 * (mean - median) / stdev. For commodities you tend to see even more positive skew, as supply and demand shocks can both drive prices up rapidly (e.g. oil, gas and corn in recent years). –  Chris Taylor Mar 8 at 13:01

Negative skewness is common in flood hydrology. Below is an example of a flood frequency curve (South Creek at Mulgoa Rd, lat -33.8783, lon 150.7683) which I've taken from 'Australian Rainfall and Runoff' (ARR) the guide to flood estimation developed by Engineers, Australia.

There is a comment in ARR:

With negative skew, which is common with logarithmic values of floods in Australia, the log Pearson III distribution has an upper bound. This gives an upper limit to floods that can be drawn from the distribution. In some cases this can cause problems in estimating floods of low AEP, but often causes no problems in practice. [Extracted from Australian Rainfall and Runoff - Volume 1, Book IV Section 2.]

Often floods, at a particular location, are considered to have an upper bound called the 'Probable Maximum Flood' (PMF). There are standard ways of calculating a PMF.

enter image description here

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+1 This example nicely shows how arbitrary the question actually is: when you measure floods in terms of peak discharge, they will be positively skewed, but measured in log discharge, they (apparently) are negatively skewed. Similarly, any positive variable can be re-expressed in a simple way that skews its distribution negatively (merely by taking a suitably negative Box-Cox parameter). It all comes down to what is meant by "easily grasped," I suppose--but that's a question about the students, not about statistics. –  whuber Mar 11 at 23:08

Nick Cox accurately commented that "age at death is negatively skewed in developed countries" which I thought was a great example.

I found the most convenient figures I could lay my hands on came from the Australian Bureau of Statistics (in particular, I used this Excel sheet), since their age bins went up to 100 year olds and the oldest Australian male was 111 , so I felt comfortable cutting off the final bin at 110 years. Other national statistical agencies often seemed to stop at 95 which made the final bin uncomfortably wide. The resulting histogram shows a very clear negative skew, as well as some other interesting features such as a small peak in death rate among young children, which would be well suited to class discussion and interpretation.

Age at death of Australian males in 2012

R code with raw data follows, the HistogramTools package proved very useful for plotting based on aggregated data! Thanks to this StackOverflow question for flagging it up.


deathCounts <- c(565, 116, 69, 78, 319, 501, 633, 655, 848, 1226, 1633, 2459, 3375, 4669, 6152, 7436, 9526, 12619, 12455, 7113, 2104, 241)
ageBreaks <- c(0, 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 110)

myhist <- PreBinnedHistogram(
    breaks = ageBreaks,
    counts = deathCounts,
    xname = "Age at Death of Australian Males, 2012")
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