Work out measures of dispersion and explain what they show

CFA® Level I · Quantitative Methods · Statistical Characteristics of Asset Returns · QM.2.b

The idea

Dispersion says how far observations sit from their centre, which is what turns an average into something useful. Variance is the mean squared deviation; standard deviation is its square root and shares the units of the data, which is why it is what gets reported. A sample divides by n - 1 rather than n, because using the sample mean to measure deviations from that same mean understates spread. Two portfolios with identical mean returns and different standard deviations are not the same investment, and in finance that difference is the definition of risk.

Formulas

Sample variance

s² = Σ(xᵢ − x̄)² ÷ (n − 1)

xᵢ: each observation
x̄: the sample mean
n: the number of observations

Dividing by n − 1 rather than n is what makes it a SAMPLE variance. A population variance divides by n.

Coefficient of variation

CV = s ÷ x̄

s: the standard deviation
x̄: the mean

Comparing dispersion between series of different size or unit. Higher CV means more risk per unit of return.

A practice question

From the question bank, with the reason each wrong option is wrong.

Why is standard deviation usually reported rather than variance?

It is always smallerWhy not: Not always — a variance below 1 has a larger square root. And size is not the reason anyone prefers it.
It is expressed in the same units as the data — correct
It cannot be negativeWhy not: True, but so is variance: both are built from squared deviations. That is not what distinguishes them.

Squaring deviations to compute variance also squares the units, so a variance of returns is in percent-squared. Taking the root returns it to percent, where it can be compared with the mean.

A harder one

Annual returns are 4%, 8%, 8%, 10% and 15%. What is the sample variance?

16.00 — correct
12.80Why not: 64 ÷ 5 — the squared deviations divided by n. A sample variance divides by n − 1, because the mean was itself estimated from these same five numbers.
9.00Why not: That is the mean return, not a measure of spread around it.

The mean is 9.0%, and the squared deviations sum to 64. Dividing by n − 1 = 4 gives 16.00; dividing by n would understate the spread.

The hardest tier

Why does a sample variance divide by n − 1 rather than n?

To make the figure larger and more conservativeWhy not: It does make it larger, but that is the consequence, not the reason. Dividing by n − 1 makes the estimator unbiased; a correction chosen for conservatism would have no particular size.
Because one observation is always discarded as an outlierWhy not: Every observation is used. The n − 1 is a degrees-of-freedom adjustment, not a dropped data point.
Because the sample mean is itself estimated from the data, which understates deviations — correct

Deviations are measured from the sample mean, which sits closer to the data than the true mean does. Dividing by n − 1 corrects that bias, so the estimator is unbiased rather than merely larger.

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