Michigan’s Graduate Student Instructors (GSIs) receive student evaluations for every class they teach. The undergrads are invited to rate their GSIs using Likert Scales (aside: Likert had a degree in economics from the University of Michigan). Students see something like this:

The Registrar compiles this data, encoding “Strongly Disagree” as 1, “Strongly Agree” as 5, etc. They then provide GSIs with what is called their “Median score”, which is actually a slightly complicated interpolated form of the median.

The application of medians to discrete values was perplexing, so I asked the Office of the Registrar to provide me with the exact formula they use. They did so, and I reproduce it here in case any GSI in the future ever wonders as I once did. (I checked: the Registrar has no objection to this being made public knowledge.)

Their formula, which requires some lingo that is explained below, is:

Calculate the range the median must lie in, e.g. somewhere between 2 and 3. Let’s call this range “the interval”. The Lower Real Limit is the midway point of the interval, i.e. 2.5.

The total number of observations (in the whole sample) is denoted \(N\). Denote as \(a\) the number of responses below the Lower Real Limit. In this example, \(a\) would be equal to the numbers of 1s and 2s.

Recall that our interval is the range between 2 and 3. The upper-limit of this is 3. Finally, denote as \(b\) the number of observations that choose the upper-limit of this interval, i.e. the number of people that chose 3.

Now for an example, suppose you have the following scores:

By inspection, the median cannot be greater than 4 and cannot be lower than 3. Therefore our interval is 3-4, with a Lower Real Limit of 3.5. There are 32 responses, so \(N=32\). Eleven people chose a score below the Lower Real Limit, so \(a=11\). Fourteen people chose the upper-limit of the interval, so \(b=14\). Plugging this into our equation above:

\(\text{Median } = 3.5 + \frac{0.5(32) – 11}{14} = 3.86\).

While watching football this morning, I also kept an eye on the betting odds. As soon one team scored, the odds changed.

Instead of introducing the efficient market hypothesis to students as a theory about stocks and bonds, we should perhaps start by drawing parallels to sports-betting. It is hard to consistently make money betting on Premier League results. It’s not impossible, but it takes a lot of skill and/or luck to pull it off.

The same logic applies to beating the general rate of return in a market. Financial markets tend to adjust prices so that there are no obvious bargains. This is not a claim that sporting events never throw up shock victories, nor does it imply that markets never crash. If anything, it suggests that predicting a stock market crash is about as difficult as predicting a shock sporting victory.

Like comparative advantage, it’s hard for undergrads to grasp this idea. I have found that the sporting analogy works well as a stepping stone.

(While on the topic of the EMH: there’s a fun contrast between the predicament of Cassandra, who could perfectly predict the future but nobody would believe her, and the EMH. The more people believe in EMH, the less likely they are to actively search for arbitrage opportunities. This will tend to leave market-beating returns uncollected, nullifying the EMH. The converse also holds. In this respect, EMH is a self-defeating prophecy.)