Dancing Statistics

A still image from the project Communicating Psychology to the Public through Dance, produced by Lucy Irving, Elise Phillips, and Andy Field supported by the British Psychological Association and IdeasTap. Four videos: my favorite Frequency Distributions, Sampling and Standard Error, Variance, and Correlation.

In this image from the first of the videos the dancers start our in one large unorganized group, some dancing with very slow movements, some with very quick movements, and as one would then expect more with movements of a more intermediate speed. As they dance they sort themselves out, from the slower movement dancers on the left, to the more rapidly moving dancers on the right, building up a sample from a bell-shaped distribution. Very clever.

There is a video about correlation with dancers performing the same movements together or nearly opposite movements together. As they mention in the text of the video, these movements are just co-occurrences, one does not cause the other: correlation is not causation.

There is a video about variation with dancers performing variations on the same set of movements.

Another is about sampling and standard error. In this dance, a single blue-shirted dancer performs his movements to indicate the four corners of a rectangle. He and the rectangle defined by his  movements are termed the population. Then several red-shirted dancers mark four corners in their own styles producing various quadrilaterals that estimate the rectangular shape of the blue-shirted dancer.

I do think calling that initial, single blue-shirted dancer a population could be misleading, especially since at the beginning of this video the text mentions "a large group (a population)". This, of course, is the usual view: the large group is the population from which we observe samples to estimate it. But perhaps better, in this setting, would be to talk more generally about a statistical model. This is a model for a dancer's movements. These movements depend on the physical aspects of the dancer: height, limb length, reach, flexibility, etc. They also depend on artistic intent, style, technique, etc.

The blue-shirted dancer specifies the results of a certain collection of all of these aspects. This becomes a parameter, a target. The red-shirted dancers sample from the model of movements to estimate this parameter. The variety and range of their movements display the sampling variability as they attempt to match the governing shape (parameter) of the blue-shirted dancer. This view is more general than the viewing of sampling as from a fixed large group population.

Home Advice Lacks Skewness

I recently saw this TV commercial for Home Advisor a website that helps homeowners find home improvement professionals. The homeowners then report their costs for the repairs. The site displays a symmetric bell-shaped curve to show the distribution of these costs, irrespective of the shape of their actual distribution. The image above shows that the average cost for cleaning gutters is $180 and that "most homeowners" spent between $158 and $202. These values appear to mark the locations of  the inflection points for the curve. If we assume the curve describes a normal distribution of costs, these points lie at one standard deviation above and below the mean. Indicating that the standard deviation is $22. For a normal distribution, about 68% ( the website's "most homeowners") spent within $22 of the mean of $180. The minimum cost of $90 is about 4.1 standard deviations below the mean. Its placement on the graph seems appropriate. But the maximum cost of $300 is about 5.5 standard deviations above the mean. Maximum costs for other services sometimes exceed 5 and even 6 standard deviations above the mean, but are placed symmetrically with costs at about 4 standard deviations below the mean. The symmetric graphic hides the right skewness that we should expect in almost any monetary variable that is only bounded below by zero.

It would be better to show the actual  histogram of costs perhaps with a superimposed curve like we have seen previously with GetMarketPrice or TRUEcar.

Market Price

Market Price (getmarketprice.com) collects the current prices of items (over $100) selling on e-bay. It provides summary statistics (min, max, median, mean, and standard deviation) along with a histogram of prices and a scatterplot of  time left to bid against price. They also include a list of the raw data with links.