The Median Is Not the Message

intermediate3 min readmedicine · science · mathematics · statistics

A median does not describe an entire distribution or determine its mean. Taleb uses Gould’s account of survival statistics and a gamble with rare large losses to show why the spread and size of possible outcomes matter alongside a typical outcome.

THE IDEA

THE MEDIAN IS NOT THE MESSAGE

The essayist and scientist Steven Jay Gould (who, for a while, was my role model), was once diagnosed when he was in his forties with a deadly form of cancer of the lining of the stomach. The first piece of information he received about his odds of making it was that the median survival for the ailment is approximately eight months; information he felt akin to Isaiah’s injunction to King Hezekiah to put his house in order in preparation for death.

Now, a medical diagnosis, particularly one of such severity, can motivate people to do intensive research, particularly those prolific writers like Gould who needed more time with us to complete a few book projects. The further research by Gould uncovered a very different story from the information he had initially been given; mainly that the expected (i.e., average) survival was considerably higher than eight months. It came to his notice that expected and median do not mean the same thing at all. Median means roughly that 50% of the people die before eight months and 50% survive longer than eight months. Taleb emphasizes the long-lived survivors, comparing them with an average lifespan of 73.4 or so years from insurance mortality tables. A long right tail can raise mean survival above the median, but it does not guarantee a normal lifespan for any particular survivor.

There is asymmetry. Those who die do so very early in the game, while those who live go on living very long. With asymmetric outcomes, the median alone does not tell us the mean survival. This prompted Gould, who thus understood the hard way the concept of skewness, to write his heartfelt piece “The Median Is Not the Message.” His point is that the concept of median used in medical research does not characterize a probability distribution.

I will simplify Gould’s point by introducing the concept of mean (also called average or expectation) as follows, by using a less morbid example, that of gambling. I will give an example of both asymmetric odds and asymmetric outcomes to explain the point. Asymmetric odds means that probabilities are not 50% for each event, but that the probability on one side is higher than the probability on the other. Asymmetric outcomes mean that the payoffs are not equal.

Assume I engage in a gambling strategy that has 999 chances in 1,000 of making $1 (event A) and 1 chance in 1,000 of losing $10,000 (event B), as in Table 6.1, which is not included in this excerpt. My expectation is a loss of close to $9 (obtained by multiplying the probabilities by the corresponding outcomes). The frequency or probability of the loss, in and by itself, is totally irrelevant; it needs to be judged in connection with the magnitude of the outcome. Here A is far more likely than B. Odds are that we would make money by betting for event A, but it is not a good idea to do so.

When to use it

Interpreting medical prognosis: A patient or doctor misinterprets median survival as the expected outcome, leading to overly pessimistic decisions.
Evaluating investment strategies: A strategy with high probability of small gains but low probability of large losses might appear attractive based on median returns but actually have a negative expected value.
Assessing insurance risk: Insurers need to consider both probability and magnitude of rare but catastrophic events, not just median losses.
Reporting scientific findings: Journalists and researchers should clarify whether they are reporting the median or the mean, especially in skewed distributions.

FROM THEORY TO EVERYDAY LIFE

In practice

Gould's Cancer Diagnosis

Taleb recounts Gould learning that his cancer diagnosis carried a median survival of 8 months. That median is not an individual deadline and does not describe the long-surviving part of the distribution. Gould’s response illustrates why a prognosis needs more context than a single summary statistic.

Gambling with Asymmetric Outcomes

A gambling strategy has 999 chances in 1,000 of gaining $1 and 1 chance in 1,000 of losing $10,000. The expected loss is about $9, despite the high probability of gain. Relying on the median outcome (a gain of $1) would be dangerously misleading.

✳

A MOMENT TO REFLECT

Put it to work.

When you hear a statistic about survival rates, are you hearing the median or the mean? How might that affect your decision?
In what situations might the median be misleading as a measure of central tendency?
How can you check whether a distribution is skewed before relying on the median?