Skewed Fast Food

This is a view of the side of a small counter at a fast food restaurant in Snow Hill, Maryland. Patrons have slid this chair back and forth to sit at or leave the adjacent table. This chair movement has marred the paneling of the counter into a pattern that is skewed to the right: much more wear on the left with decreasing use and wear as the chair is set closer to the table. Of course, on the right, the chair's wear pattern is truncated since it must stop short of the table. On the left, we've got no wall to reveal the chair's position. The chair's wear is censored. What remains is a right skewed pattern of the frequency of use and wear. The pattern somewhat resembles the pattern of a sample from a right skewed exponential distribution.

YADDA - Donut Bakery

Yet Another Door Distribution Again. This time on a bakery door just east of Cambridge, Maryland. Here we see a skewed frequency distribution of scratches, perhaps from keys, with greatest concentration around the door handle with progressively fewer scratches extending higher up the door. Many fewer scratches are below the door. Perhaps the handle is too low as customers handle their keys and hold or open the door, while also eating the delicious donuts the bakery sells.

Scattered Dimples

Here is a bivariate, skewed distribution of finger wear and motion in taking a receipt from a gas pump in Greenfield, Indiana. Most wear is from fingers pushing down on the gas receipt as it exits the dispenser. There is some left-to-right variability in this placement, and many customers have dragged their fingers further downward to capture the receipt. This leaves an elongated, skewed pattern of wear from top-to-bottom. The dimpled surface of this pump also leaves the impression of a skewed scatterplot of individual points. We have seen wear patterns on gasoline pumps before.

YADDA Garage Gate

Yet Another Door Distribution Again (YADDA), this one on a gate in a garage stairwell showing a skewed distribution of hand placement opening the door. We've seen and described such patterns often,  (YADDA in our list of labels). Since opening a hinged door is easiest with a long lever arm, more wear is shown near this gate's right-hand edge with less towards the door's pivot, resulting in this skewed pattern of wear.

Kitchen Distribution

Here is a well-used cutting board. Every morning it protects the counter top from an errant knife cutting slices of bread for breakfast. The loaf is most often placed so that the cuts fall near the middle of the board, with the knife's blade repeatedly marring the front edge of the cutting board. Less often the cuts continue and extend off to the right or left of the middle. In this contest right seems to win out. This leaves us with a frequency distribution of the knife's marks skewed a bit to the left: the fewest marks along the left of the front edge, the most along the middle, and then a bit fewer marks on the right of the edge. This is a bell-shaped, although somewhat skewed to the left, pattern that we've seen often.

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.

Salty Residuals

Washington, this Winter, has not been wearing "on his smiling face a dream of Spring". On the contrary, Spring has been continually cold and damp much like the movie Groundhog Day. Just last week we had an unusual mid-March snowstorm. Groundskeepers spread salt along campus walkways to speed the snow melt. Their spreader sprayed the salt left and right as they drove down the path. After melt, the salt remained attracting moisture, absorbing - not reflecting - the light that fell on it, leaving a dark, wet residual on the asphalt.

At every point down the walk we can see the horizontal spread of the dark residuals, much concentrated near the center of the walk with lesser concentrations to the right and left. Horizontally, what remains is a bell-shaped distribution of salt deposition. The distribution has this same, consistent shape at each place down the walk. In this angled, perspective view the distributions look more skewed to the right. But the symmetry can be seen more accurately in the rotated image below.

This rotated image now shows the distributions in vertical slices as we move horizontally along the walk. The distributions are centered along the same horizontal line with the same shape and degree of vertical spread.

This is exactly the image of ideal residuals from a simple linear regression fit to data plotted against an explanatory variable from a uniform design. Of course, a different design placement of the explanatory variable would vary the pattern horizontally, but not so vertically. Other design patterns could arise from the spreader moving faster or slower down path leaving a more uneven, erratic deposition of salt - more at some steps along the walk than at others. But the assumptions for such a regression model still require identical, vertical normal distributions of scatter around a straight line of means irrespective of the horizontal position down the path. For an even, uniform walk down the path, our salty residuals model and reflect the ideal behavior of regression residuals.

YADDA Sears - Yet Another Door Distribution Again

Bell-shaped wear pattern on the exit door at a Sears store. Lots of wear in the middle, less on the right-hand edge. The distribution appears skewed to the right. Would we see more of a left-hand tail if the handle was bigger or does the door open better by pushing on it more to the left than to the right?

See another door distribution in our video here.

Law of Large Crowds (Monty Pythony)

A NOVA video about the Wisdom of Crowds. In his paper "Vox Populi" published in Nature, March 7, 1907, Sir Francis Galton investigated this wisdom at a Fat Stock and Poultry Exhibition writing:

In these democratic days, any investigation into the trustworthiness and peculiarities of popular judgments is of interest. The material about to be discussed refers to a small matter, but is much to the point. A weight-judging competition was carried on at the annual show of the West of England Fat Stock and Poultry Exhibition recently held at Plymouth. A fat ox having been selected, competitors bought stamped and numbered cards, for 6d. each, on which to inscribe their respective names, addresses, and estimates of what the ox would weigh after it had been slaughtered and "dressed." Those who guessed most successfully received prizes.

Galton received the 787 cards with the estimates and, then he tabulated results. The actual "dressed" weight of the ox was 1198 lbs. The median of the crowd's guesses was 1207 lbs., so the crowd was off by only 9 pounds out of 1198 or less than 0.8%. This was better than any individual guess, even from the experts. Galton also found the upper quartile (q3) of the guesses to be 1236 and the lower quartile (q1) to be 1162. Thus the inter-quartile range is 74. He modeled the data with a normal distribution and estimated its standard deviation as 1/2 of this inter-quartile range or 37, (he calls this the probable error, p.e.). We could be a bit more accurate by using 3/4 of the inter-quartile range. He notes the skewness of the distribution of guesses, with the lower portion best modeled with a normal distribution with a standard deviation of 45 and the upper portion best modeled with a standard deviation of 29. He graphs horizontally the cumulative distribution function of the fitted normal (solid line) along with the cumulative relative frequencies of the data (dotted line).

His overall finding:

This result is, I think, more creditable to the trust-worthiness of a democratic judgment than might have been expected.

Climate Dice Back in the Day

There's been much on blogs about "Climate Dice" and how humans are perhaps loading them. See the article by Andrew Revkin (New York Times) and more recently by Paul Krugman (New York Times) discussing the findings by James Hansen et al. on public perceptions of climate change, these links via Andrew Gelman. Basically, the data show that hotter extremes are becoming more likely.

Investigating the variability in climate, with analogies to random tosses of dice, has a long history, and not always considered in the correct way. C. F. Marvin, former chief of the US Weather Bureau, can be seen using such random methods in this article from Popular Science 1932 page 46.  Describing this methods a bit more, this newspaper article from 1931 begins with an especially lyrical view of his work :

Common dice, inventions of the ancients and purveyors of financial distress to unlucky moderns, have risen to a new dignity. After having been rolled in many places, ranging from the cobblestones of side streets to the green covered tables in palaces of chance, they are now being tossed with analytical earnestness by the hands of science.

In this new field the "galloping dominoes" are being used as a means of increasing man's working knowledge of the weather and its pranks. The greenback and silver involved when glassy eyed gamblers seek to get something for nothing are supplanted by graphs and slide rules in this new environment, where scientists seek the answer to the high sounding question, "Are meteorological sequences fortuitous?"

I think here one should read "random" for "fortuitous". The article's final question is the title of a paper by Marvin the appeared in December 1930 Monthly Weather Review (pdf). In that paper he uses such random methods to simulate graphs of precipitation and compares the results to actual records questioning "who could pick out the natural from the chance order"?

He does complicate things by confusing a sample and a population. He marvels at the fact that the products of the results on four tossed dice produces a histogram of measurements that is  sparse and skewed to the right, resembling a sample record of actual precipitation. Of course, this is a population of all the possible products and its sparseness will always be present. As he notes, there is no way to fill in between the products that can only be produced with four dice. This has no relation to the sampling variability and sparse histograms, of many shapes, that could result come from a skewed but continuous population of measurements. Of course, it was 1931!

Bell-Shaped Animals

Adapted from a graphic in Life magazine October 10, 1955 showing speeds of various animals. The artist has displayed the animals sampled on a horizontal scale of speed. There are few slow animals on the left, many more medium speed animals in the middle, and few fast animals on the right. The outline of the cloud of animals shows the essence of a roughly bell-shaped frequency distribution with a little skewness to the right. But take out the elephant and much of the skewness goes away also.

A Skewed Runway

This is a composite picture obtained from the United States Geological Survey. The top picture is a view of the runway called 1L at Washington-Dulles International Airport. Look at the tire skid pattern from the landing airplanes.

The real runway is many times longer than it is wide. The images shown here have been stretched across the width of the runway and shrunk along the length to better see the pattern of use. Aside from this mild distortion, the tire skid pattern has not been altered.

The aiming point for landing pilots is indicated by two broad white rectangular marking stripes about 1,000 feet from the end of the runway. Touchdown zone markers are groups of one, two, or three rectangular bars every 500 feet arranged on either side of the runway center line.

This runway at Washington-Dulles airport is so long (11,500 feet) that pilots need not land exactly on the aiming point for safe operation. This along with the fact that they definitely don’t want to land short of the runway accounts for the skewed tire skid pattern. Note that although the skid pattern is skewed along the length of the runway it is symmetric across the width of the runway.

The skewness along the length of the runway shows that most of the airplanes land within 1,000 feet of the aiming point, but some land, as indicated by the skid pattern, much further down the runway. The symmetry across the width of the runway is, of course, due to the two sets of wheels of the landing gear and how accurately the pilots hit the centerline of the runway.

The second runway picture shows the entire 1L/19R runway (albeit distorted to fit the page). Notice now the U-shaped distribution of tire skids, as we see the accumulated skid marks from airplanes landing from both directions.