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Lesson 1 of 8 · 14 min

Mean, median, mode and outliers

A measure of central tendency condenses a return series into one number for where it is centred; which measure you trust depends on outliers and on the type of data.

In short

The arithmetic mean weights every observation equally. It is the standard summary, but one extreme value can pull it far off.
The median is the middle of the sorted data. It ignores how extreme the extremes are, so outliers do not move it.
The mode is the most frequent value. It is the only centre measure for categorical data, and for continuous returns you report the modal interval.
Outliers can be kept, deleted (trimmed mean) or replaced by boundary values (winsorized mean).
If the trimmed or winsorized mean is above the plain mean, the extreme values were mostly large losses.

From raw returns to a summary

A list of a few hundred daily returns is impossible to read. The first step is a frequency distribution: choose return intervals (bins) and count how many observations fall into each. A histogram draws those counts as bars, so you can see where returns cluster and how far they stretch.

A frequency table can carry four columns: the absolute frequency (count per bin), the relative frequency (count ÷ total), and the running totals of each, the cumulative absolute and cumulative relative frequency. The cumulative relative frequency tells you what share of observations lies at or below the top of a bin.

A measure of central tendency then condenses the sample into one number that says where the data are centred. For returns, it is the sample's estimate of a typical or expected return. Measures of location are the wider family: they add quantiles, which pin down other points of the distribution (next lesson).

We nearly always work with a sample (the returns we observed), not the whole population. The statistics in this module are therefore sample statistics, written $\bar X$ for the sample mean and $s$ for the sample standard deviation.

The 48 monthly returns as a histogram (bins in %). The tallest bar, 0% to 2%, is the modal interval and also holds the median; the three bars left of zero add up to 18 months, or 37.5%.

The arithmetic mean

The arithmetic mean adds all observations and divides by their number. Every value gets the same weight and all the information in the sample is used, which is why it is the default summary of returns.

A useful property: the deviations from the mean always sum to zero. Values above the mean exactly offset values below it. That is why you cannot measure risk by averaging plain deviations (lesson 3).

The weakness is the flip side of using every value. A single extreme observation drags the mean toward it: one crash day in an otherwise quiet year can turn a positive average daily return negative.

Picture the mean as the balance point of the data on a see-saw. Push one observation far out and the balance point must move toward it, even though every other observation stayed where it was.

The median and the mode

The median is the middle value once the data are sorted. With an odd number of observations it sits at position $(n+1)/2$ . With an even number there is no single middle, so you average the values at positions $n/2$ and $(n+2)/2$ .

Half the observations lie below the median and half above, whatever their size. That makes it robust to outliers: change the best year from 12% to 120% and the median does not move. The price is that it ignores the size of the values, and it is less convenient mathematically than the mean. It is the better centre measure for skewed data.

The mode is the most frequently occurring value. One most frequent value makes the distribution unimodal, two make it bimodal, three trimodal; if all values differ there is no mode. It is the only centre measure that works for nominal or categorical data, such as the most common fund style in a database.

Returns are continuous, so exact repeats are rare. Instead, group the data and report the modal interval: the bin with the highest frequency, which is always the tallest bar of the histogram. If two or three bins tie for the highest count, the distribution is bimodal or trimodal.

Which centre measure, when?
Measure	Uses	Strength	Weakness
Arithmetic mean	All values, equally weighted	Uses all the information; easy to work with	Pulled toward outliers
Median	Rank order only	Unaffected by outliers; best for skewed data	Ignores the size of values
Mode	Frequencies	Works for categorical data	May not exist or may not be unique

Dealing with outliers

An outlier is an observation far from the rest. It may be a genuine rare event, a recording error, or a value from a different population. Check for and fix errors first; sometimes transforming the variable (for example taking logs) or choosing another variable removes the problem.

If the extremes are legitimate, there are three options. Keep them: right when you want the whole distribution reflected, and it avoids a subjective call on what counts as extreme. Delete them: the trimmed mean drops a stated percentage of the lowest and highest values and averages the rest. Replace them: the winsorized mean resets the extreme values to boundary values and then averages all observations.

The naming convention matters. A 5% trimmed mean removes the lowest 2.5% and the highest 2.5% and averages the middle 95%. A 95% winsorized mean sets the bottom 2.5% of values equal to the 2.5th percentile value and the top 2.5% equal to the 97.5th percentile value. Trimming lowers the number of observations; winsorizing keeps it.

Comparing the versions is informative. If the trimmed or winsorized mean is above the plain mean, the extremes that were cut or capped were mostly large negative returns. If it is below, large positive returns were inflating the plain mean.

Trimmed vs winsorized mean
	Trimmed mean	Winsorized mean
Extreme values are	Removed	Replaced by the boundary value
Number of observations	Falls	Unchanged
5% / 95% version	Drop the bottom 2.5% and top 2.5%	Cap the bottom and top 2.5% at the 2.5th and 97.5th percentiles

The ten NOK index days: one −6.2% day drags the plain mean below zero. Trimming or capping that day lifts the centre to about 0.3%, close to the median, which tells you the distorting extreme was a loss.

Formulas

Sample mean: $\bar X = \dfrac{\sum_{i=1}^{n} X_i}{n}$
Add all n observations and divide by n. Every observation has equal weight.
Median position: $n \text{ odd: } \dfrac{n+1}{2} \qquad n \text{ even: average of positions } \dfrac{n}{2} \text{ and } \dfrac{n+2}{2}$
Positions refer to the data sorted from lowest to highest.

Worked examples

Example 1 Mean, median and mode from raw returns

A EUR equity fund earned these annual returns (%): 6, −4, 9, 3, 6, 12, 1, 7. What are the mean, median and mode, and what does their relationship suggest?

Example 2 Median bin and modal interval from a frequency table

48 monthly returns are grouped into bins (%): −6 to −4: 2; −4 to −2: 5; −2 to 0: 11; 0 to 2: 16; 2 to 4: 10; 4 to 6: 4. Which bin holds the median, what is the modal interval, and what share of months were negative?

Example 3 Trimmed vs winsorized mean (exam level)

Ten daily returns of a NOK index (%), sorted: −6.2, −0.8, −0.3, 0.1, 0.4, 0.5, 0.7, 0.9, 1.2, 2.5. What are the arithmetic mean, the 20% trimmed mean and the 80% winsorized mean, and what do they reveal?

On the exam

Short calculations (median with an even n, a trimmed or winsorized mean) and reading a histogram or frequency table for the median bin or the number of modes. Concept items ask which measure suits skewed data or what a gap between the plain and trimmed mean implies.

Taking the median without sorting first, or using only one middle value when n is even.
Mixing up the outlier methods: trimming deletes extremes (n falls), winsorizing replaces them (n stays).
Calling a histogram unimodal when two or three bins tie for the highest frequency.
Reading a 5% trimmed mean as cutting 5% from each end; it cuts 2.5% from each end.

Check yourself

Q1A fund's annual returns (%) are 7, −3, 4, 10, 2 and 5. The median return is closest to:
Q2Which measure of central tendency is the only one that can be used with categorical data, such as fund style labels?
Q3Eight monthly returns (%) are −9, −1, 0, 1, 2, 2, 3 and 6. A winsorized mean that resets the lowest and highest values to their nearest neighbours is closest to:
Q4An index's 5% trimmed mean daily return is higher than its arithmetic mean daily return. This most likely indicates that the full sample contains:
Q5A histogram of 60 monthly returns has six bins holding 4, 15, 9, 15, 11 and 6 observations. The distribution is best described as:

0/5Check yourself

Answer the 5 questions: +10 XP for each one you get right first time.

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