Plastic Feet Peaks

Here is an example of a remarkably symmetric pattern of wear and use, but most assuredly not bell-shaped. The pattern on the top of this trash bin shows two prominent areas of wear at the left and right side of the opening. These two areas show greater wear than a large fairly uniform area of use in the center between the peaks. The two extreme areas of use tell us something about the modes of customers’ and restaurant workers’ actions.

Fast food is often delivered on plastic serving trays. As diners leave, they collect the assorted packaging and wrappings from their meals and deposit them in the trash bin near the exit. The diners then return their serving trays to the top of the trash bin. The plastic trays have small raised ridges on bottom of each corner. These small ridged “feet” act to provide a tiny gap between stacked trays to make them easier to separate.

When the top of the trash bin is empty, trays are returned by sliding them back along the front edge of the bin. The plastic feet on the bottom of the trays scrape along the top of the bin. This leaves prominent peaks in the wear pattern on the bin. As the trays are slid further the central portion of the trays sag and also scrape the bin to produce the pattern of use showing almost uniform wear between these two peaks.

Of course we would expect to produce this type of wear mainly when the top of the bin is empty, allowing the sliding tray to wear down the top. Later trays may not produce any wear along these edges if they are just placed on top of trays already in position. But here is where the restaurant’s workers contribute to the pattern.

After awhile, the trays stack up and must be returned for, what is hoped, a good washing. As they are retrieved, the pile of trays is slid forward to be picked up. This produces the uniform center wear and the peaks along the right-hand and left-hand edges as the trays and their feet again scrape the top of the bin. These actions produce the pattern of nearly equal left and right peaks of use with more uniform wear in between, resulting in a symmetric, but bi-modal frequency distribution.

Earliest Living Histogram Revisited (and Reversed)

While preparing for my talk at the Joint Statistical Meetings in Montreal this week, I had the occasion to consider again the Earliest Living Histogram that I posted in 2008. This image appeared on page 450 of Popular Science Monthly, September 1901 in a paper "The Statistical Study of Evolution," by C.B. Davenport. Forty University of Chicago students are arranged by height in bins of two inch width. When viewed from above we see what was much later called a "living histogram" of the heights of this sample of men.  It is described in more detail in Graphical Methods for Presenting Facts (1914) (Figure 141) by W.C. Brinton where, on page 165, he writes:

In Fig. 141 a group of men have been arranged in different rows. There is only one man in the shortest class at the left, and only one man in each of the tallest two classes at the right. Most of the men are of that height shown by the row to the right of the center of the diagram. A glance at the photograph taken looking down on this group of men shows that there are more men shorter than the most frequent height than there are men taller.

Davenport's original publication of this photograph also contains another image of the forty students:

Davenport says they are "arranged (approximately) in order of height." Examine the tallest few students shown here:

Note the shading of the hats that the tallest five men are holding: Gray, White, Black, White, and Black, reading from tallest downward (right to left). Now consider their arrangement in classes by height in the image above. Note that the five men on the far left are now wearing hats with shading Gray, White, Black, White, and Black.

The histogram image is reversed!

The taller men are shown on the left and the shorter on the right. This is exactly the reverse of the description given by Brinton and it reverses his reasoning and conclusions about the frequency of tall and short men in this sample. But there is more evidence. Reversing this image along a properly oriented and indicated number line we get the image below along with counts of the men standing in two of the histogram classes.

The five numbered men in dark hats, on the right, stand in a row of about the same extent as the seven numbered, smaller men in the row on the left. The men on the right are not just taller but also broader, and the smaller men on the left take up much less space in this regard. Another indication that this is the proper view of this histogram. The original published histogram should be reversed.
The correctly oriented Earliest Living Histogram is shown below:

Shop Worn

This is the checkout counter of the gift shop at the Mansion at Strathmore part of the Strathmore Arts Center in Bethesda, Maryland. The gift shop has seen many patrons. Standing to pay for purchases they rub against the edge of the counter. This results in a region of more centrally located wear with less wear extending to the right and left. Yet another, common, bell-shaped distribution pattern of wear. Here's another view.

Things Shouldn't Be So Hard

Here is "a worn-out place" where restaurant servers stand to collect plates and silverware to set up the tables for new patrons. Many feet have stepped, dirtied, or worn away these kitchen floor tiles. More wear at a central target and less wear towards the edges. This is a frequency patten that we have seen often.

It brings to mind the poem "Things Shouldn't Be So Hard," by former US Poet Laureate and MacArthur Fellow, Kay Ryan (via the NYTimes). Considering all the worn and marked things we have illustrated in these postings, Ryan's poem could be the defining wish for this blog.

THINGS SHOULDN'T BE SO HARD

A life should leave
deep tracks:
ruts where she
went out and back
to get the mail
or move the hose
around the yard;
where she used to
stand before the sink,
a worn-out place;
beneath her hand
the china knobs
rubbed down to
white pastilles;
the switch she
used to feel for
in the dark
almost erased.
Her things should
keep her marks.
The passage
of a life should show;
it should abrade.
And when life stops,
a certain space—
however small —
should be left scarred
by the grand and
damaging parade.
Things shouldn't
be so hard.

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.

Happiness in Negative Space

This image is from an Institute of Contemporary Art, Philadelphia exhibit called The Happy Show by Stefan Sagmeister that just ended its tour at the Institute of Museum of Contemporary Art, Los Angeles Pacific Design Center. In this display, included in the exhibit, people were instructed to select one gumball from column that indicated their level of happiness (1 to 10). The bar chart of the happiness level of those that participated is shown in the negative space above each column, showing how many gumballs were removed. Median happiness here looks to be about a 7. We've seen a similar exhibit before.

Beall Shaped

The Beall-Dawson House in Rockville, Maryland is an 1815 Federal-style home built for Upton Beall, then Clerk of the Circuit Court. It has been owned since 1960 by the Montgomery County Historical Society. Almost 200 years of use has left its mark. Below is the wooden threshold leading into the home's main parlor. The threshold shows the pattern we have seen often, the most wear in the middle where the frequency of use is greatest, trailing off to lesser usage and wear at the edges. A bell- (Beall?)-shaped distribution of wear.