Module 2 · Descriptive statistics Core

Summarize a return series by its centre, spread and shape.

Centre

The mean is the arithmetic average. The median is the middle value when sorted; it is not distorted by extreme values. The mode is the most frequent value. When the mean is well above the median, a few large values are pulling it up.

Spread

• Range: maximum minus minimum.
• Variance: the average squared deviation from the mean. For a sample, divide by n minus 1.
• Standard deviation: the square root of variance, in the same units as the data. In finance this is volatility.
• Percentiles: the value below which a given share of observations fall. The 5th percentile of returns is a simple measure of a bad outcome.

Coefficient of variation

Standard deviation / mean, a measure of risk per unit of return. A fund with a mean of 8% and a standard deviation of 12% has a coefficient of 1.5.

Shape

• Skewness measures asymmetry. Negative skew means a long left tail: many small gains and occasional large losses, typical of equity indexes and of strategies that sell insurance or options. Positive skew means many small losses and occasional large gains, typical of lottery-like bets and venture investing.
• Kurtosis measures the weight of the tails. A normal distribution has kurtosis of 3. Excess kurtosis (kurtosis minus 3) above zero means fat tails: extreme outcomes happen more often than the normal curve predicts. Daily stock returns have strongly positive excess kurtosis. The spreadsheet function KURT reports excess kurtosis.

Why shape matters

Two investments with the same mean and standard deviation can have very different risks. One with negative skew and fat tails hides its danger in rare events that a short sample may not contain.

Looking at the data

Always plot a histogram and a time-series chart before trusting summary numbers. One data error or one extraordinary month can dominate a mean and a standard deviation.

More worked examples

Quartiles and the interquartile range

The 25th and 75th percentiles bracket the middle half of the data. If they are 1% and 9%, the interquartile range is 8 percentage points. Unlike the standard deviation, it ignores the tails entirely, which makes it robust to outliers and blind to crashes.

Downside deviation

Investors dislike losses more than gains, so some measures count only returns below a target. With returns of -4%, 2%, 6%, -2% and 3% and a target of 0, the shortfalls are 4 and 2. Squared and averaged over all five observations: (16 + 4) / 5 = 4, so the downside deviation is 2%. The ordinary sample standard deviation of the same returns is 4.0%, inflated by the 6% gain that nobody minds. The Sortino ratio (RS 302, Module 8) uses downside deviation.

Comparing risk per unit of return

Fund A has a mean return of 6% and a standard deviation of 9%; fund B, 10% and 20%. A's coefficient of variation is 1.5 and B's is 2.0, so A delivers more return per unit of volatility, although B delivers more return in total. The Sharpe ratio (RS 302) does the same comparison after subtracting the risk-free rate.

Excess kurtosis in practice

Spreadsheet KURT returns excess kurtosis. A reading of 4 means kurtosis of 7, far fatter tails than a normal distribution's 3. Daily stock index returns often show excess kurtosis between about 5 and 20 over long samples, driven by a handful of crash days. Monthly returns have much less, because extremes are diluted over a month.

Summary statistics can hide the story

Two series with identical means and standard deviations can look completely different on a chart: one steady with a single crash, the other noisy throughout. Report the maximum drawdown and the worst month alongside the standard deviation.

Put it into practice