Stats in the Wild

Happy New Year! Here is an article from the Freakonomics blog tracking American’s (un)happiness.

Cheers

Finals are over and I hope to post more regularly again. Here is a quick picture.

This picture, by French engineer Charles Joseph Menard, graphically depicts Napolean’s fateful march to Russia. The width of the line represents how many troops Napolean had at each point on his way to Russia and what makes this graphic so great is just how many different variables are all displayed at once.

Edward Tufte says in his book, The Visual Display of Quantitative Information, “Minard’s graphic tells a rich, coherent storhy with its multivariate data, far more enlightening than just a single number bouncing along over time. Six variables are plotted: the size of the army, its location on a two-dimensional surface, direction of the army’s movement, and temperature on various dates during the retreat from Moscow”.

In the last line of the description below the graph, Edward Tufte says, “It may well be the best statistical graphic ever drawn,” which, in my opinion, may be the best claim ever made about a statistical graphic ever.

Cheers.

For Stanley:
Complete caption of the graphic:
“This classic of Charles Joseph Minard (1781-1870), the French engineer, shows the terrible fate of Napolean’s army in Russia. Described by E. J. Marey as seeming to defy the pen of the historian by its brutal eloquence, this combination of data map and time-series, drawn in 1861, portrays the devastating losses suffered in Napolean’s Russian campaign of 1812. Beginning at the left on the Polish-Russian border near the Niemen River, the thick band shows the size of the army (422,000 men) as it invaded Russia in June 1812. The width of the band indicates the size of the army at each place on the map. In September, the army reached Moscow, which was by then sacked and deserted, with 100,000 men. The path of Napolean’s retreat from Moscow is depicted by the darker, lower band, which is linked to a temperature scale and dates at the bottom of the chart. It was a bitterly cold winter, and many froze on the march out of Russia. As the graphic shows, the crossing of the Berezina River was a disaster, and the army finally struggled back into Poland with only 10,000 men remaining. Also shown are the movements of auxiliary troops, as they sought to protect the right flank of the advancing army. Minard’s graphic tells a rich, coherent story with its multivariate data, far more enlightening than just a single number bouncing along over time. Six variables are plotted: the size of the army, its location on a two-dimensional surface, direction of the army’s movement, and temperature on various dates during the retreat from Moscow. It may well be the best statistical graphic ever drawn.”

Voter turnout for this last election was as high as it has been in the last 60 years. Click here for a good graphical display of voter turnout since 1948 on Andrew Gelman’s blog.

Also check out the United States Election Project where they have data about past election going back about 50 years.

A friend of mine told me about this problem, so I went and looked it up. This is stats in the wildest of the wild.

So, during World War II, the Allies were trying to estimate the number of a certain kind of German tank. They needed this information to better plan their attacks and invasions. There were two sets of estimates made, one by intelligence and another by a group using statistical methods.

Estimates made using statistical methods in June 1940, June 1941 and August 1942 of the number of a certain type German tank were, respectively, 169, 244, and 327. The intelligence estimates for each of those same three periods of time were, respectively, 1000, 1550, and 1550. (from :Number of German tanks)

These estimates are drastically different, and depending on which estimate was believed, it is possible that battle plans may have been significantly affected. So who made the better estimates?

In most situations when we estimate something, we can never actual know what the true value is. However, as it turns out, after the war was over, German records became available and the actual number of tanks that they had at each of those three points in time became available. The actual number of tanks that the Germans had at the three points in time (June 1940, June 1941 and August 1942) when the estimates were made were, respectively, 122, 271, and 342. (Recall that the statistical estimates were 169, 244, and 327 for those three time periods.) The statistical estimates are astonishingly close. (As well as the intelligence estimates being alarmingly inaccurate.) So how did they do it?

The statistical group looked at the serial numbers of tanks that had been captured or destroyed by Allied troops, and they assumed that the serial numbers of the tanks were ordered from 1 to T where T is the number of tanks that the Germans had. So they assumed that if the Allies found a tank with serial number 200, that the Germans had at least (and almost surely more than) 200 tanks.

So if we assume that each serial number has equal probability of being observed our maximum likelihood estimate (our best guess) of T is simply the maximum serial number that we encounter on a destroyed tank. However, using the maximum encountered serial number to estimate T turns out to be an unbiased estimator. (If we always used the largest serial number as our estimate of T, we would be systematically underestimating T, because our largest observation is usually not the actual largest value.) So what we need is an unbiased estimator for T.

As it turns out the expected value of our estimator of T (the maximum observed serial number) is n/(n+1)*T (hence biased). So on the average the largest observed value will be smaller than actual T. To correct for this we simply multiply the largest observed value by (n+1)/n. This will give us an unbiased estimate for the number of tanks the Germans had, and this is how they reached their statistical estimates.

Example:
Say we observe 50 tank serial numbers and the largest observed serial number is 245. With all of the above assumptions, our unbiased estimate as to the number of tanks is 51/50*245=249.9.

If we observe 25 serial number and the largest is 110, our best guess is 26/25*110=114.4.

Here is a link to another blog post about the German tank problem.

Modern note: I saw online that someone was using this approach to try to estimate the number of servers that Google has. (More to come on that)

References:
Ruggles, R., and Brodie H. (1947) An empirical intelligence in World War 2. Journal of the American Statistical Association, 42:72-91.

Goodman, L. A. (1954), “Some Practical Techniques in Serial Number Analysis,”
Journal of the American Statistical Association, 49, 97–112.

Accoriding to this article from www.readwriteweb.com, they claim that “Errors By Bloggers Kill Credibility & Traffic, Study Finds”. Interesting. So how did they reach this conclusion?

From the article:
“The company [goosegrade.com] asked a demographically diverse group of respondents on Amazon’s Mechanical Turk website to fill out the survey and published the results today on the goosegrade.com company blog. The bulk of respondents spent some time reading blogs but were people who remained dependent on ‘mainstream sources’ for most of their news.”
(For an explanation of Mechanical Turk, the Wikipedia article is here.

Comment: How does goosegrade know these people were demographically diverse? The only people they asked were Mechanical Turk workers. That seems like a very specific group of people. So you should only be able to make inference about that group of people. They hardly speak for internet users in general, but goosegrade.com uses them to make inference about “internet users” when they should just be making inference about “mechanical turk workers who are being paid by gooseGrade.com”. Those two groups are drastically different.

gooseGrade.com says on their site (http://www.goosegrade.com/reader-perception-survey-results)

“Readers want gooseGrade. Here’s proof.
175 People polled.
ABSTRACT: It appears that grammar, spelling, factual, and other errors do affect reader opinion as well as how likely they are to share or link to an article. These errors also seem to dictate the readers opinion of the author’s skills as a writer. 65.86% of internet users say that a tool like gooseGrade would increase their confidence in the content they are reading. Filtering further shows that 9 out of 10 newspaper readers say that a tool like gooseGrade would increase their confidence in author’s content. This merrits further investigation of newspaper readers and could show a path for new media to take more market share.”

As I said before, I’m not sure the opinions of 175 (more on this below) mechanical turk workers are sufficient to make inference on all internet users. Furthermore, remember that all of these respondents were paid by goosegrade.com (although it was probably only a few cents.)

A note on their sample size: They claim a sample size of 175 internet users, but an examination of the raw data shows that there are only 161 unique IP address. 9 IP addresses are repeated twice and 1 IP address is repeated 5 times. These should be thrown out of the sample because it is likely that they are the same person.

The readwriteweb.com article concludes with:
“Below are a few of the charts, you can see the rest on the GooseGrade blog. The lesson here? It seems pretty clear. We bloggers are harming our own credibility and traffic with our inattention to details, not just in the facts, but in the basics of our writing. Let’s do better!”

Here is a promise I am willing to make. I’ll write better and make less grammatical errors if you apply statistics more fairer. (LOL)

Cheers.