Mean, median, mode and what outliers do to them

What are the three ways to find the middle?

Each answers “what is a typical return?” in a different way: the mean uses every value, the median uses position, and the mode uses frequency.

A measure of central tendency tells you where a data set is centered. For a return series it is, in effect, a summary of the return you might “expect” based on history. The arithmetic mean is the sum of the observations divided by the number of observations. When the data are a sample, the same calculation gives the sample mean .

$$\bar{X} = \frac{\sum_{i=1}^{n} X_i}{n}$$

The median is the middle value once the data are sorted. With an odd count it is the single middle observation. With an even count there is no single middle, so you average the two that straddle it.

$$n \text{ odd: position } \frac{n+1}{2}; \quad n \text{ even: average positions } \frac{n}{2} \text{ and } \frac{n}{2}+1$$

The mode is the value that appears most often. A data set with one peak is unimodal . With two equally frequent values it is bimodal , and it can have more modes or none at all. The mode is the only one of the three that works with categories (such as “sector” or “rating”), because it needs no arithmetic.

Measure of central tendencywhere a data set is centered

• Mean Memorize sum ÷ n, uses every value
  • Pulled by outliers drifts toward the tail
• Median Memorize the middle value after sorting
  • Resists outliers best center for skewed data
• Mode Use it the most frequent value
  • Unimodal, bimodal one, two (or more) peaks; may have none
  • Only measure for categories works with nominal data
• Outlier-resistant means Memorize still average, with the extremes handled
  • Trimmed mean drop the extremes, n falls
  • Winsorized mean replace extremes with nearest kept values, n stays

Drag the 14.0 and watch which marker moves. The mean follows the outlier. The median does not notice it.

What do outliers do?

An outlier is an observation far from the rest. Before choosing a measure, ask one question: is it real?

Extreme value in the datafirst ask whether it is real

• Recording error fix or delete it
• Real and important Use it keep it, report the mean and the median
• Real but mostly noise Use it use a trimmed or winsorized mean

If the 14.0 is a typing error (say the true figure was 1.4), fix it. If it is a real month, deleting it would hide something true about the asset, so keep it and say what it does. Replace 14.0 with 3.0 and the mean drops from 2.34% to 1.12%, while the median stays at 1.4%. That gap is the whole argument for reporting the median of a skewed series.

How do the trimmed and winsorized means handle extremes?

Both keep the averaging idea but blunt the outliers. They differ in what happens to the extreme values.

• The trimmed mean drops a set number (or percentage) of the lowest and highest observations, then averages what remains. n falls.
• The winsorized mean keeps every observation, but replaces the extremes with the nearest values that stay. n is unchanged.

Exam tip Trimmed: fewer numbers, smaller divisor. Winsorized: same count, clipped ends. A “5% trimmed mean” drops 2.5% from each end.

Which measure should you report?

Match the choice to the data and to what the outlier means.

Situation	Report
One real, important outlier you want to keep	The mean and the median together
Data errors	Fix them, then use the mean
Heavy tails where the extremes are mostly noise	A trimmed or winsorized mean
Skewed data (house prices, private-company returns)	The median
Categories, not numbers	The mode

Go deeper Why the median is the natural center for skewed data

In a distribution with a long right tail, a few very large values lift the mean above where most observations sit. The median depends only on ranks, so it is not lifted. That is why house prices and venture returns are quoted as medians. The same logic explains the order rule you will meet in the skewness lesson: with positive skew, the mean is above the median because the mean is dragged toward the tail.

Go deeper Other ways to average returns

The arithmetic mean is not the only mean. When you want compound growth over several periods, the geometric mean is the right tool, and when a fixed amount buys varying units, the harmonic mean is. Those are covered in the rates and returns module. A portfolio return is also a mean, but a weighted one: each asset’s return times its weight, summed.