Mean, Median and Mode

An average is supposed to describe a typical value, but there are several kinds, and they can tell very different stories. The mean adds everything up and divides; the median picks the middle value; the mode is the most common value. In trading, where results are often skewed by a few big wins or losses, choosing the wrong average can make a strategy look much better or worse than it really is. Returns over time also need a special average, the geometric mean.

The three main averages#

Worked example#

Mean vs median in skewed data#

Trend following and option buying often produce right skewed results: many small losses and a few large wins. Option selling and many mean reversion strategies often produce left skewed results: many small wins and occasional large losses. See Skewness and Kurtosis.

Arithmetic vs geometric mean for returns#

For returns that compound over time, the arithmetic mean overstates what you actually earn. The geometric mean gives the true average growth rate.

arithmetic mean = (r1 + r2 + ... + rn) / n
geometric mean = [(1 + r1) × (1 + r2) × ... × (1 + rn)]^(1/n) - 1

A useful approximation: geometric mean ≈ arithmetic mean minus half the variance. Higher volatility lowers compound returns, a drag sometimes called volatility drag.

Weighted averages#

Sometimes values deserve different weights, such as portfolio returns weighted by position size, or VWAP, which weights prices by volume. See VWAP.

weighted mean = Σ (weight × value) / Σ weights

Which average to use#

Averages in your journal#

When reviewing trades, report the mean, the median and the share of total profit coming from your best few trades. If removing the top three trades turns a profitable year into a losing one, your results depend on rare outliers, which is fine for some strategies but should be expected and planned for. See Trading Journal.

Frequently asked questions#

What is the difference between mean and median?#

The mean is the sum divided by the count and is affected by outliers; the median is the middle value and is resistant to outliers.

Why use the geometric mean for returns?#

Because returns compound, and the geometric mean shows the actual average growth rate, while the arithmetic mean overstates it when returns vary.

Which average should I use for trading results?#

Use the mean for expected value, the median to see a typical trade, and the geometric mean to measure compound growth.

Next, learn how spread is measured in Variance and Standard Deviation.