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 .

$$\mu = \frac{\sum_{i=1}^{N} X_i}{N}, \qquad \bar{X} = \frac{\sum_{i=1}^{n} X_i}{n}$$

The population mean μ and the sample mean X̄ are calculated the same way: both divide by the number of observations. Unlike the sample variance, the mean never divides by n − 1.

A statistic is robust when it stays reliable even if the data contain outliers or do not follow the distribution you assumed. The median is extremely robust: you could move almost half the observations as far as you like without shifting it. A single outlier can move the mean by any amount.

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 mode is unimodal ; one with several is multi-modal (modes can also be local peaks, such as a cluster of small gains and a separate cluster of losses); and some data sets have no mode at all. Unlike the mean and the median, the mode allows for more than one center.

Returns are rarely repeated to the last decimal, so for grouped data analysts use the modal interval : the bin with the highest frequency, which is always the tallest bar of the histogram.

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
  • Robust to outliers robust: best center for skewed data
• Mode Use it the most frequent value
  • Unimodal, multi-modal one mode, several, or none
  • Modal interval grouped data: the tallest histogram bar
• Outlier-resistant means Memorize still average, with the extremes handled
  • Trimmed mean drop the extremes, n falls
  • Winsorized mean replace extremes with the cutoff 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 that differs markedly from the rest. It can come from a measurement or data-entry error, a sampling error, or simply the natural variability of returns. Before choosing a measure, ask one question: is it real?

For a valid outlier there are three choices: leave it in, remove it (a trimmed mean) or replace it (a winsorized mean).

Extreme value in the datafirst ask whether it is valid

• Error fix or delete it
• Valid outlier Use it three choices
  • Leave it keep all data; report the median beside the mean
  • Remove it trimmed mean
  • Replace it 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.

Compute these statistics on returns, not on price levels: prices drift over time, so their mean and spread keep changing (lesson 7 explains why).

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 (in practice, with a chosen percentile value). n is unchanged.

In practice the cut is a percentage. An x% trimmed mean drops x/2% of the observations at each end: a 10% trimmed mean of 500 daily returns drops the 25 lowest and the 25 highest and averages the other 450. A 90% winsorized mean sets the lowest 5% equal to the value at the 5th percentile and the highest 5% equal to the value at the 95th percentile, then averages all 500.

Winsorizing keeps the information that extreme values occurred, which trimming throws away. It sits between the mean’s sensitivity and the median’s robustness. On a skewed series both pull the average toward the center: when the outliers are large losses, the trimmed and winsorized means come out above the raw mean.

Exam tip Trimmed: fewer numbers, smaller divisor. Winsorized: same count, clipped ends. A “5% trimmed mean” drops 2.5% from each end; a “95% winsorized mean” caps 2.5% at 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
Data that cluster in more than one place	The mode (or modes)

How do you get these in a spreadsheet?

Know the idea

Measure	Excel	Google Sheets
Mean	AVERAGE(range)	AVERAGE(range)
Median	MEDIAN(range)	MEDIAN(range)
Mode	MODE.SNGL(range); MODE.MULT for several modes	MODE(range)
Trimmed mean	TRIMMEAN(range, percent)	TRIMMEAN(range, percent)

TRIMMEAN’s percent is the total share cut from both ends together: TRIMMEAN(range, 0.1) drops 5% at each end. Neither program has a built-in winsorized mean; you cap the extremes yourself and then take the average.

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 the compound growth rate over several periods, the geometric mean is the right tool; it is covered in Learning Module 1, Returns of Financial Assets and Instruments. A portfolio return is also a mean, but a weighted one: each asset’s return times its weight, summed.