1. Mean, median and mode: why the mean can mislead you with financial data

Level 3: Descriptive and inferential statistics applied to markets

A strategy with 24 months of track record. The median says it earns 1.5% a month. The mean says it earns 0.19%.

Both figures are correct. They are computed on the same data. And the distance between them is, in itself, the most informative of the three.

Three ways to summarize with a single number

When you summarize a series of returns with a single number, that number hides decisions.

• The arithmetic mean adds up all the values and divides by how many there are. It is sensitive to every data point, extremes included.
• The median sorts the data from smallest to largest and takes the middle one, or the average of the two middle ones if there is an even number of observations. Only position matters to it, not magnitude.
• The mode is the value that repeats most often.

The mode deserves a caveat. In continuous series of daily returns there are almost never two identical values, so there it loses its usefulness.

It reappears, however, when you work with data grouped into intervals —for example, how many days the return fell between 0% and 0.5%— or with discrete variables, such as the number of winning trades per week.

When mean and median disagree

The difference between mean and median is what really matters in markets, because returns are rarely symmetric.

Let us go back to the example we opened with. A strategy with 24 months of track record: in 23 of them it earns 1.5%, and in one it loses 30%, a month of crisis.

• The median of those 24 data points is 1.5%. The bad month is just one observation among many and does not shift the central position.
• The mean, by contrast, collapses. Total sum = 23 × 1.5 − 30 = 4.5, divided by 24 = 0.19% per month.

An investor looking only at the median would think the strategy is consistently good. One looking only at the mean would think it barely generates anything.

Neither of the two numbers, on its own, tells the whole story. You need to see both and understand why they differ so much, because that difference is itself information about the shape of the distribution: negative skew, a long left tail.

The opposite case

The opposite case exists too.

Think of a strategy of buying options far out of the money, that is, betting on extreme moves. It loses a small amount almost every month and, very occasionally, wins an enormous amount.

There the median is negative, because most months lose. But the mean can be positive if the winning month more than makes up for all the losing ones put together.

Reporting only the mean would give the impression that you normally win, when the normal thing, month by month, is to lose a little.

Arithmetic and geometric are not the same thing

There is a second trap, more technical and specific to returns: the difference between the arithmetic mean and the geometric mean.

The arithmetic mean of a series of periodic returns is not the same thing as the real compounded return that an investor who reinvests would obtain.

Look at it with numbers. If an asset rises 50% one year and falls 50% the next, the arithmetic mean of those two years is 0%, that is ((50 − 50) / 2).

But the real capital ended up worse than it started: 100 goes to 150 and then to 75, a cumulative loss of 25%. The geometric mean —the one that does reflect compound growth— would be negative, consistent with that real loss.

The greater the volatility of a series, the wider the gap between the two. This phenomenon is sometimes called volatility drag.

The consequence is concrete: two strategies with the same arithmetic mean return but different volatility do not end up generating the same compounded wealth. For an equal arithmetic mean, the more volatile one compounds worse.

This is key when assessing prospectuses or backtests that present a mean annual return without making clear which of the two they are talking about. The arithmetic one almost always looks better on paper than the geometric one when there is volatility involved, and it is easy, deliberately or not, to pick the one that favors the sales pitch.

The median takes the punches

The median has another practical advantage worth exploiting in financial analysis: it is resistant to extreme values. Statisticians say it has a high breakdown point.

A single monstrous data point can move the mean disproportionately, but it barely moves the median. It makes no difference whether it is a data-capture error or a real event such as a crash.

That is why, when you suspect outliers in a series —a badly adjusted dividend, an uncorrected split, a failure at the data provider— comparing mean and median is a quick first diagnosis.

If they differ a great deal, something asymmetric or extreme is going on, and it is worth investigating before you trust either of the two numbers.

The mode, recycled

One last nuance about the mode. Although in continuous returns it is of no use as it stands, the idea it represents —which particular value repeats most frequently— does have real applications in markets.

Closing prices, for example, cluster in a non-random way at round numbers or at particular ticks. It is a well documented phenomenon that reflects the behavior of limit orders and the psychology of whoever sets target prices.

Detecting that kind of clustering, by looking at which return or price bucket concentrates the most observations, is in spirit applying the logic of the mode to financial data. Even if technically you are working with data grouped into intervals rather than exact repeated values.

None of them replaces the distribution

In quantitative analysis practice, none of the three measures of central tendency replaces looking at the whole distribution. The module on outliers and the earlier levels on fat tails are the natural continuation of this idea.

But knowing how to read the disagreement between mean, median and mode is the first warning sign that a single-figure summary may be hiding the true shape of the risk.