The Wear of Love

As part of the Virginia Tourism Corporation's promotion campaign, this LOVE artwork has been placed in the Robert Reed waterfront park in Chincoteague, Virginia. It displays four, 10 foot tall Adirondack chairs. The wooden chairs spell out the LOVE with the symbols 
L O ♥ E.

Of course, when posing for pictures, tourists prefer to sit in the   ♥   chair over the others as evidenced by the greater frequency of wear on the chairs as people climb up and rub off the paint.

This view looks at the letters in reverse. The nearest chair is E, the next is  ♥   with the most wear, then O and L. Note that the pattern of wear is in a bell-shape, with more wear near the middle of the seat and less towards the edges.

Feel the Curve

Here is a relief model of a normal curve that was developed to aid teaching statistics to the visually impaired. Students would trace their fingers along the raised impression of the curve and its divisions into standard deviation intervals to gain experience and understanding of the normal curve and how it describes the normal distribution of measurements along its horizontal axis, distinctions that we have seen repeatedly on this blog. In this figure, the lines representing one standard deviation above and below the mean seem to fall a bit short of the curve's two points of inflection, where they should naturally fall. But this is an excellent concept for aiding those with visual impairments.

YADDA BBQ 2

Here's another picture of the entry door to Sonny's BBQ in Sanford, Florida. You can see the vertical wear pattern of the many hands that have held open the door. The most frequent use is on the brown wood of the door. Those hands have left grease and grime that has mixed with the finish on the door producing a mottled white central portion of a bell-shaped curve. The lower tail of the bell-curve comes from the polishing that hands have done on the door's copper entry plate. The break between the mottled white pattern and the polished copper tail seems to fall at the inflection point of the bell-curve. Properties of the normal curve tell us that the wear beyond the inflection point in the copper tail accounts for about 16% of the total wear. The corresponding upper inflection point seems to fall just were the mottled white pattern ends. This also indicates about 16% of wear in the upper tail. These divisions likely correspond to the normal distribution of the height of hands as they open the door, perhaps just below shoulder level.

Top of the Heap? Mediocre Still

Here is a cartoon from Rhymes with Orange. A student looking at a bell-curve of SAT scores says, "My strategy? Shoot for the top of the bell curve. Then I can look down on everybody." The student clearly has the wrong idea. He seems to think that the peak of the bell curve puts him on the top of the heap. For him, higher on the curve is better than everyone. This mistake we've seen before in this blog here and here. But just perhaps the cartoonist, Hilary Price, has the idea of the bell curve correct. She shades in the letter C on the side. Rather than seeing this as an illustration of the student's multiple choice answer to an SAT question we could imagine that she has assigned to the student a score of C, a traditional average grade, that would be the most common or the most likely grade. This is exactly what the height of the bell-curve represents for such a mid-range grade. The bell curve is tallest for the most commonly occurring grades, not for the highest grade one might strive for. That grade is at the extreme right, where the curve is low. As we've seen before the "Top of the heap is mediocre."

Wear Pattern in "Bedrock"

This is a symmetric and bell-shaped pattern of wear on the entry door to the restrooms at Rocky Gap Casino and Resort in Flintstone, MD (yes, Flintstone). We've seen this type of wear often, for example, here and here. People use the handle to open and pass through door, but many, likely on exit, place their hands around the edge of the door to pass through or hold it open for others. They can't reach too high and when they do it seems only fingers are used, leaving little wear.  It's also  uncomfortable to hold it open too low, again likely only with fingers. So most of the wear and likely many whole hands are used between these extremes causing much more wear. Top to bottom, little use, greater use, then little use generates the bell-shaped pattern wear.

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.

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.

Top of the Curve = Middle of the Pack

Here's a normal distribution design from a t-shirt (thanks Jun). Unlike the possible confusion we have seen from bell-curve t-shirts, this one, from shirtwoot! shows that the designer got the bell-curve's relation to the normal distribution correct. The bell-curve is not the normal distribution. The bell-curve only describes the normal distribution. The normal distribution is a specific random arrangement of measurements on a number line. This arrangement ranges from small measurements on the left of the number line, up to large measurements on the right. The curve represents the frequency or density of observations along the line. Where the curve is low, the observations have a very low density. We would expect few measurements there, occurring sparsely in those regions. Where the bell-curve is higher, we expect a dense arrangement, with observations piling up to a peak in the middle. But the location of this peak on the bell-curve lies in the middle of our number line range. The peak corresponds to the measurement that occurs most often, where its measurement is most common. It is literally the peak of mediocrity.

This shirt design gets it right. At this peak, you have not surmounted all around you to become the best. On the contrary, you've truly "reached mediocrity".

Naughty or Nice

Christmas is over and according to cap-news Santa is updating his "Naughty List Algorithm". Exactly how is this Naughty/Nice scale quantified? What continuous random variable could be measured here?