Real life examples of distributions with negative skewness Inspired by "real-life examples of common distributions", I wonder what pedagogical examples people use to demonstrate negative skewness? There are many "canonical" examples of symmetric or normal distributions used in teaching - even if ones like height and weight don't survive closer biological scrutiny! Blood pressure might be nearer normality. I like astronomical measurement errors - of historic interest, they are intuitively no more likely to lie in one direction than another, with small errors more likely than large.
Common pedagogical examples for positive skewness include people's incomes; mileage on used cars for sale; reaction times in a psychology experiment; house prices; number of accident claims by an insurance customer; number of children in a family. Their physical reasonableness often stems from being bounded below (usually by zero), with low values being plausible, even common, yet very large (sometimes orders of magnitude higher) values are well-known to occur.
For negative skew, I find it harder to give unambiguous and vivid examples that a younger audience (high schoolers) can intuitively grasp, perhaps because fewer real-life distributions have a clear upper bound. A bad-taste example I was taught at school was "number of fingers". Most folk have ten, but some lose one or more in accidents. The upshot was "99% of people have a higher-than-average number of fingers"! Polydactyly complicates the issue, as ten is not a strict upper bound; since both missing and extra fingers are rare events, it may be unclear to students which effect predominates.
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 is industrially themed; I prefer cracked and intact eggs in a box of twelve.) Maybe students feel that "success" should be rare.
Another option is to point out that if $X$ is positively skewed then $-X$ is negatively skewed, but to place this in a practical context ("negative house prices are negatively skewed") seems doomed to pedagogical failure. While there are benefits to teaching the effects of data transformations, it seems wise to give a concrete example first. I would prefer one that does not seem artificial, where the negative skew is quite unambiguous, and for which students' life-experience should give them an awareness of the shape of the distribution.
 A: 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:

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.
A: 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/
A: 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.

A: Gestational age at delivery (especially for live births) is left skewed. Infants can be born alive very early (although chances of continued survival are small when too early), peak between 36-41 weeks, and drop fast. It is typical for women in the US to be induced if 41/42 weeks, so we don't usually see many deliveries after that point.
A: In fisheries there are often examples of negative skew because of regulatory requirements. For instance the length distribution of fish released in recreational fishery; because there is sometimes a minimum length that a fish must be in order for it to be retained all fish under the limit are discarded. But because people fish where there tends to be legal length fish there tends to be negative skew and mode towards the upper legal limit. The legal length does not represent a hard cut off though. Because of bag limits (or limits on the number of fish that can be brought back to the dock), people will still discard legal size fish when they have caught larger ones. 
e.g., Sauls, B. 2012. A Summary of Data on the Size Distribution and Release Condition of Red Snapper Discards from Recreational Fishery Surveys in the Gulf of Mexico. SEDAR31-DW11. SEDAR, North Charleston, SC. 29 pp.  
A: In the UK, price of a book. There is a "Recommended retail price" which will generally be the modal price, and virtually nowhere would you have to pay more. But some shops will discount, and a few will discount heavily.
Also, age at retirement. Most people retire at 65-68 which is when the state pension kicks in, very few people work longer, but some people retire in their 50s and quite a lot in their early 60s.
Then too, the number of GCSEs people get. Most kids are entered for 8-10 and so get 8-10. A small number do more. Some of the kids don't pass all their exams though, so there is a steady increase from 0 to 7. 
A: Some great suggestions have been made on this thread. On the theme of age-related mortality, machine failure rates are frequently a function of machine age and would fall into this class of distributions. In addition to the financial factors already noted, financial loss functions and distributions typically resemble these shapes, particularly in the case of extreme-valued losses, e.g., as found in BIS III (Bank of International Settlement) estimates of expected shortfall (ES), or in BIS II the value at risk (VAR) as inputs to regulatory requirements for capital reserve allocations.
A: 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.

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.
library(HistogramTools)

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")
plot(myhist)

A: Age of retirement in the U.S. is negatively skewed. The majority of retirees are older with a few retiring relatively young.
A: In random matrix theory, the Tracy Widom distribution is right-skewed. This is the distribution of the largest eigenvalue of a random matrix. By symmetry, the smallest eigenvalue has negative Tracy Widom distribution, and is therefore left-skewed. 
This is roughly due to the fact that random eigenvalues are akin to charged particles that repel each-other, and hence the largest eigenvalue tends to be pushed away from the rest. Here's an exaggerated picture (taken from here) : 

A: 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. 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. the histograms below of GPAs of white and black graduates at a for-profit university taken from Fig 5 of Gramling, Tim. "How five student characteristics accurately predict for-profit university graduation odds." SAGE Open 3.3 (2013): 2158244013497026.

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