Post Modern Bell Wear

Watch a fun video The Evolution of Tap Dance by Post Modern Jukebox with tap dancer Sarah Reich. Note the bell-shaped distribution pattern of wear on her tap platform: most wear in the middle and less towards the edges.

Stepping Up

This is a ladder leading to an elevated playhouse for kids visiting Brookside Gardens in Maryland. The ladder reveals a frequency distribution of foot placement wear. On most steps we see more wear in the middle of the ladder rung and less wear towards the left and right edges. This leaves a bell-shaped pattern of use.

But a rung near the bottom has a bi-modal pattern, showing the wear resulting from both left and right feet. This doesn't persist on higher rungs. There the wear seems more central. So why not on the lower rung as well? Perhaps central steps on higher rungs feel safer. A care that is not that needed closer to the ground.

Guitar Fret Wear?

Here is an image from imgur of the fret board on a 1956 Fender Stratocaster guitar showing the frequency distribution of playing wear (thanks Scott). But many on Reddit disagree, calling it faked. Not being a guitar player, I can't judge. What do you think?

Menu Basket Queue

This is a view of a basket of menus at a restaurant in Chincoteague, Virginia. Notice the pattern of marks left as groups of menus scratch the wall when they are returned to the basket. As customers are seated at the restaurant, they are given menus that are removed from the right hand side of the basket. After ordering, the menus are returned to the basket and placed to the right of the remaining menus. When a single is menu returned, it nicks the wall at a location that depends on how many menus are currently in the hands of customers. If few menus are out with the customers, more remain in the basket and the wall marks of this returning menu will be further to the right. If many menus are out with customers, say just before the lunch rush, this returning menu will make a mark on the wall further to the left.

But it is not often that a single menu is returned alone. It is much more likely that a group of menus will be returned to the basket in a bunch. The size of the bunch that is returned is random depending on the size of the party seated. Each of the menus in the bunch makes a mark on the wall as they are returned to the basket. What we see is a steady state distribution of the number of seated customers, with menus in hand, waiting for their order to be taken.

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.

A Distribution on a Cylinder

Here's a utility pole at a traffic intersection in Aspen Hill, Maryland. The pole has served as display for the many yard sales, community meetings, and businesses that have had their advertising flyers posted on the pole. The flyers have long since been removed. Only their staples and nails remain. These accumulated staples show a distribution of the heights of flyer postings. 

The close-up view below shows the distribution of individual staples and nails on the cylinder of the pole. The staples are distributed both around the pole and vertically up and down the pole. Vertically, it's too difficult to put flyers high on the pole and few staples can be found there. Flyers very low on the pole wouldn't be easily seen by those passing by, so few staples are also found there. Most staples and nails are at a comfortable shoulder and viewing height. If we imagine the height of a staple above the ground is our random variable, we find few staples with small height, few with large height, and many more with a medium height. This is a bell-shaped pattern up and down the pole that we have seen often.

Horizontally, the staples are distributed circularly around the pole. They would also have greatest frequency towards the traffic and lesser frequency on the backside of the pole. This is likely also a bell-shaped distribution, wrapped around a circle. We have seen such a distribution before connected with the characteristic function of a random variable. We've also seen a distribution on a pole at the Rodin museum in Paris.

Salary Dots

From Flowing Data, an interactive dotplot showing the distribution of annual salaries in various fields. Selections can be made for the 1960s (above), the 1980s, 2000s, and 2014. As a time range is selected, the dots representing the annual salaries of 50 randomly selected people, dynamically redistributed themselves to reflect the times salary frequency distribution. Compare the dramatic change in spread from 1960s above to the 2000s below.

YADDA Boston

Yet Another Door Distribution Again, this time at the Summer Shack Restaurant in Boston. Not many patrons grab the door near the handle, not many reach it much higher. Most grab the door, and wear away its paint, at a comfortable, likely shoulder height. A bell-shaped frequency distribution results. Thanks Laura.

The Fourth of July

Oliver Stone's 1989 film Born on the Fourth of July tells the story of Ron Kovic, US Marine and anti-war activist. Kovic was portrayed by Tom Cruise, as advertised in the film's poster above. The poster's dominate color is black with, as expected, red, white, and blue, but also oranges and yellows in the face tones. Photoshop reveals the poster's colors in the swatches below.

Vijay Pandurangan has considered these colors along with those of many other films. Below are his findings for the year 1989:

But he has done more. He has looked at the colors in movie posters from 1914 to 2012 and produced an interactive image where you can select any year within this range and see the pie chart of movie posters from that year. Here is a still image of his interactive one.

He also produces an interactive image (still image below) with lightness and saturation ignored.

Movie posters seem to have gotten bluer over time. We've seen movie poster colors here before. Thanks for the link Nick.

On The Road ... Map

On a trip to visit family, we stopped at a gas station in Hammondville, Alabama. On the wall was a map of the US with this wear pattern of customers touching where they were on the map. The many touchers have worn though the paper map down to the underlying supporting board. It seems that many have traced their path of travel extending southwest to Birmingham, AL and northwest to Chattanooga, TN likely along the connecting route US 11, passing through Hammondville, or along the parallel interstate 59 a bit further west. What remains is a roughly ellipsoidal bivariate frequency distribution of wear with a greater frequency of wear centered on Hammondville and lesser frequencies of wear in ellipsoidal contours around it.

So Young, So Old

Nathan Yau at Flowing Data has produced an interactive graphic to compare your age with others. What percentage of the US population is younger than you? What percentage is older? In the static image above the US median age appears to be about 37 years old. Based on a 5-year American Community Survey from 2014, his interactive graphs lets you slide the line to match any age and see the percentage of Americans older or younger.

March Madness: It Didn't Happen Yet Again

It didn't happen yet again. In last month's NCAA "March Madness" men's college basketball championship, a number of 16 seeds have again failed to best the number 1 seeds. The histogram above shows the score differences in such matchings since 1985. It closely matches a normal distribution, allowing for us to estimate the probability that such an upset could happen as the area under the approximating bell-curve that falls beyond zero. Our estimated probability that a number 16 seed would beat a number 1 seed is 0.0208. It has risen a bit since our last view.

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.

Nuptial Age

Flowing Data has produced an interactive graphic showing the distribution of the age of marriage of Americans. Smoothed relative frequency distributions are shown for women (in green) and men (in orange) with selections possible by employment, education, race, or whether or not it is a first marriage. The data are from the American Community Survey marriages from 2009 to 2014. My guess is that similar frequency distributions from earlier decades would have modes that move more to the left towards younger ages for both women and me. What is the overall youngest median age of marriage for Americans? More data needed.

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.

Scuff and Wear

This is an doorway threshold between two rooms of a thrift store in Easton, Maryland. People step on or over scraping their feet on both sides of the edges of the raised threshold. More frequent steps and wear in the middle and fewer steps and less wear on the left and right edges all scuff away the black painted wood revealing a bell-shaped frequency distribution of wear.

YADDA on TV

Seen on an episode of Major Crimes on TNT, a wear pattern of hand placement in holding a door open. Not many hands are placed low, many more higher, and not many are placed still higher. The frequency of hand wear shows us, yet again, the bell-shaped pattern of wear. Yet Another Door Distribution Again.

Glove-Up

Before assisting patients, hospital professionals glove-up from this supply rack in the patient's room. From the gloves that are missing, it appears that Medium and Small gloves are used more frequently than Large or XLarge. Medium and Small workers outnumber others at this hospital.

Independence in the Trumans' Wallpaper

  This is a view inside the kitchen of President Harry S Truman’s home in Independence, Missouri, courtesy of the National Park Service. Notice the pattern of wear in the wallpaper from pulling the chain to turn on the wall lamp. Here's a close up:

When Harry Truman left the Presidency he retired to a very modest and quiet life. He lived with his wife Bess in her family’s home in Independence, Missouri. He read five newspapers a day, no doubt many at the table shown in this photograph. One can imagine their morning routine of taking a seat at the kitchen table with a cup of coffee leaning against the kitchen wall and reaching up to pull the chain to turn on the lamp above the table. In so doing perhaps knuckles hit the wall or the chain rubbed the wallpaper and wore it through. This resulted in a clustered pattern of wear from the pulling the lamp’s chain and releasing it. But the wear on the wallpaper suggests that this targeting was not always exact and the release, in hitting the wall was not always consistent. These small, accumulated errors left a record over their many morning reads.

Such a targeting routine gives rise to the normal, bell-shaped pattern of wear that we have seen before. Only this time it develops in two dimensions. With a few assumptions we can derive the normal probability distribution that models and describes these actions.

In reaching for the lamp’s chain, perhaps the Mr. Truman’s morning grogginess or intense attention to the news of the day, caused him to miss the target, reaching just a little too far to the left or a little too far to the right to grab the chain. Likewise, his marks indicate that his reach was sometimes a little too high or a little too low.

Let us first assume that these small errors, right and left or high or low are independent of one another. This independence means that if his reach was too high one morning, this had no affect on how the reach left its mark when the light was turned off later in the day or on the next morning. He didn’t repeat the same too-high reach the next morning nor did he overly compensate and leave a mark too low the next.

But this Independence goes beyond Missouri, it goes further than day to day variations. It applies more importantly to each individual action of turning on the lamp. We assume that at each targeting of the lamp’s chain their right to left targeting is independent of the up and down targeting. They are not consistently grabbing slightly up and simultaneously to the right nor slightly down and at the same time to the left. On the contrary, independence would dictate that the other two possibilities of slightly up and to the left or slightly down and to the right are equally represented motions. These independent actions leave their marks on the wallpaper in the clustered pattern of roughly circular shape. The pattern has no tendency to tilt up or down to the right or the left.

Next, since turning on a lamp is such a routine and repeated task, it is much more likely that their targeting error was small rather than large. It would be very unlikely that a reach would leave a mark far from the target chain. It is much more likely that they left a mark resulting from a small error in targeting. So our second assumption is that, the bigger the error, the less its chance of occurrence. Small errors are much more likely.

Now, imagine that a high reach is just as likely as a low reach. Likewise, a reach to the right of the target is just as likely as a reach to the left. This would say that the marks fall symmetrically around the target and that the probability of a particular size error to the right is the same as the probability of the same size error to the left.

But here we make an even stronger assumption. Let us assume that errors at any given distance from the target have the same chance of occurrence in whatever direction they may land. This would mean that not just horizontal, right and left errors or vertical, up and down errors are considered. Targeting errors along any tilted diagonal are also possible. After all, Mr. Truman could have occasionally reached a little to the right and a little too high falling northeast of the target. As we’ve said it does not appear that they did this consistently, but however far in this tilted direction a mark was eventually made, we assume that the chance of such an occurrence is the same as the chance of an equally distance mark in a purely horizontal or vertical direction. What matters is not the direction of the error whether up, down, right, left, or diagonally. What matters is only how far the error is from the target. The chance of occurrence of any error depends only on how far it is from the target.

We have these assumptions: 1) independence of horizontal and vertical targeting, 2) smaller errors are more likely than larger ones, 3) errors the same distance from the target have the same chance of occurrence and finally 4) the resulting probability density function that describes the results of these targeted actions is always positive, that is, no targeted misses are excluded from possibility. With these assumptions the bivariate normal probability density function can be derived.

In fact, this result has a long history. It has been derived many times and used in many contexts. It is most notably attributed to Herschel (1850), but was developed much earlier by Adrain(1805).

Let x represent the horizontal position and y represent the vertical position of a targeting mark.  Let f(x) (or f(y)) denote the probability density of the horizontal (or vertical) position. Independence tell us that the probability of the joint positions of x and y, denoted by their joint probability distribution, g(x,y), can be represented as the product of probability distributions for x and y individually. That is, g(x,y) = f(x) f(y). But we also have the assumption that the probability distribution of the joint position of x and y depends only on the distance from the target origin. So that,

for some function h. If we let y = 0 then we see that h(x) = f(x) f(0).

Now define

Then

 

            But the well known solution to such a functional equation is given by the linear function 

k(x) = cx, for some constant c. Then

or 

We have a probability density (i.e. one that integrates to 1) only if we use our second assumption that larger errors are less likely to occur than smaller ones. This says that we must have c to be negative. We can write such a negative constant as 

for some standard deviation   . Then the function f(x) takes the form 

 

This is exactly the probability density function of a normal random variable. To find f(0) we note that the area under a probability density must be one. This results in a properly scaled probability density for our horizontal (or vertical) position:

 

            This is the normal, bell-shaped probability distribution, centered at the origin and having a standard deviation of . The two-dimensional wallpaper wear pattern can then be considered a sample from the bivariate normal probability distribution          

 

This same quantitative argument was also used by James Clerk Maxwell in 1860 in his study of the kinetic theory of gasses. 

There is more to see here. Notice the stains on the wall above each chair, about at head level. Was this the result of resting groggy, early morning heads? Next, close inspection might discern greater variability and therefore a wider spread of marks on the wallpaper in the left to right direction compared to the up and down direction. This concerns the behavior of those individual directions separately. One may be spread out a bit more than the other, that is, the horizontal direction might have a larger standard deviation than the vertical direction. Although this changes the expression of the bivariate normal density, with our assumptions, it is still normally distributed. The key requirement is the independence of the directions. Independence is concerned with how the up and down or left and right directions of action behave together and leave their marks. We would doubt independence of these individual motions only if they consistently left marks in a tilted directional pattern, rather than just the possible stretched direction seen here.

Finally, the National Park Service Rangers tell me that President Truman and his wife picked out this wallpaper in 1971. Mr. Truman died in 1972. His wife Bess Truman likely sat at the same table, until her death in 1982. A large portion of the wallpaper wear shown here is most likely due to her turning the lamp on and off.