CFA Level I Exam · Statistical Characteristics of Asset Returns

Measures of Dispersion and Risk: Variance, Standard Deviation and CV

Updated 7 October 2026 · Fact-checked

Measures of dispersion show how widely returns spread around their mean, and finance treats that spread as risk. The main ones are range, mean absolute deviation, variance, standard deviation, target downside deviation and coefficient of variation. To solve questions, find the mean, compute squared deviations, divide by n or n − 1, then take the square root.

Understand Measures of Dispersion and Risk

Average return tells you where returns are centred. It does not tell you how far a single year can miss that centre. Dispersion measures that miss. Two funds can both average 8% a year, but one may swing between 7% and 9% while the other swings between −20% and +36%. Investors call the second fund riskier.

The simplest measure is the range: maximum minus minimum. It uses only two values, so one extreme point distorts it. The mean absolute deviation (MAD) averages the absolute distance of each value from the mean. Absolute values stop positive and negative deviations from cancelling out.

Variance squares each deviation instead, then averages. Squaring removes signs and punishes large misses more. The catch is that variance is in squared units (% squared), which is hard to interpret. Standard deviation is the square root of variance and returns to the original units. It is the most common risk measure in the curriculum.

Average-based measures treat upside and downside the same, but investors mostly dislike losses. Target downside deviation uses only observations at or below a chosen target. When the target is the mean, it is called semi-deviation, which is simply a downside deviation measured around the mean. The coefficient of variation (CV) divides standard deviation by the mean, giving risk per unit of return. It lets you compare assets with different average returns.

A key distinction is population versus sample. If your data is the entire population, divide by N. If it is a sample used to estimate the population, divide by n − 1. Dividing by n − 1 corrects for the fact that deviations around the sample mean are slightly too small.

Key formulas to remember

Range: Range = Maximum value − Minimum value; Uses only two observations, so it is sensitive to outliers.
Mean absolute deviation (sample): MAD = Σ|Xᵢ − X̄| ÷ n; Take absolute values of deviations from the sample mean. Divide by n.
Population variance: σ² = Σ(Xᵢ − μ)² ÷ N; Use only when you have every member of the population.
Sample variance: s² = Σ(Xᵢ − X̄)² ÷ (n − 1); Use for samples. Dividing by n − 1 makes it an unbiased estimator of population variance.
Standard deviation: σ = √σ² and s = √s²; Same units as the data, so easy to compare with the mean.
Target downside deviation: √[ Σ over Xᵢ ≤ B of (Xᵢ − B)² ÷ (n − 1) ]; B is the target. Only observations at or below B contribute to the sum, but the divisor is still n − 1, where n is the total number of observations. If B equals the mean, the result is the semi-deviation, a downside deviation measured around the mean.
Coefficient of variation: CV = s ÷ X̄; Risk per unit of mean return. A lower CV means less risk per unit of return. Interpret with care when the mean is near zero or negative.

How to solve Measures of Dispersion and Risk questions

Use this order for any dispersion question. It works for variance, standard deviation, MAD, downside deviation and CV.

1Read the question for the keyword: population or sample, absolute or squared, target or mean, and which statistic is asked.
2Compute the mean of the data (or note the target if given).
3Write each deviation from the mean (or target). For downside deviation, keep only values at or below the target.
4Absolute-value the deviations for MAD. Square them for variance, standard deviation or downside deviation.
5Sum them and divide: by n for MAD, by N for population variance, by n − 1 for sample variance and for target downside deviation.
6Take the square root if the question asks for standard deviation or downside deviation. For CV, divide standard deviation by the mean.
7Check units and sense: the standard deviation should be in the same units as the data and typically smaller than the range.
8 Pick the option that matches, remembering that options run smallest to largest.

Quickest way: Calculator shortcut and elimination

When to use it: Use when the data has 5 to 10 observations and you need sample standard deviation or CV fast.

On the TI BA II Plus press 2nd, DATA, then enter each value with ENTER and the down arrow. Clear old data first with 2nd, CLR WORK.
Press 2nd, STAT and cycle with the down arrow until you see Sx. Sx is the sample standard deviation (n − 1). σx is the population standard deviation (n).
Square Sx to get sample variance. Divide Sx by X̄ (shown in the same list) to get CV.
Estimate first: the standard deviation must be smaller than the range, so any option larger than the range can be eliminated. Use the range only as a rough sense check.
Use the sample versus population cue in the stem. For the same data, sample variance is larger than population variance (unless all values are identical), which removes one option.

Common mistakes in Measures of Dispersion and Risk

Dividing by n when the data is a sample.
Students remember the plain 'average' idea and forget the n − 1 correction.
Fix: Treat data as a sample unless the question says it is the entire population. Use Sx, not σx, on the calculator.
Reporting variance when the question asks for standard deviation, or vice versa.
The last step (square root) gets skipped under time pressure.
Fix: Underline the requested statistic in the stem. Take the square root as a deliberate last step.
Forgetting absolute values in MAD.
Deviations from the mean sum to zero, so skipping the absolute value gives zero.
Fix: Write |Xᵢ − X̄| in every row before adding.
Dividing downside deviation by the count of below-target observations.
It seems logical to average only the observations used.
Fix: Divide by n − 1 where n is the total number of observations. Values above the target add zero to the sum but stay in the count.
Comparing standard deviations of assets with very different means.
A higher standard deviation looks automatically riskier.
Fix: Compute CV for each asset. Lower CV means less risk per unit of return.
Mixing units, such as 4% and 0.04.
Percentages and decimals get switched mid-calculation.
Fix: Pick one form at the start. Keep variance in the matching squared units.

Worked examples

Example 1

A fund had annual returns of 4%, 8%, 12% and 16% over four years. Treat these as a sample. What is the sample standard deviation? (A) 4.47% (B) 5.16% (C) 6.67%

Show the solution

Mean = (4 + 8 + 12 + 16) ÷ 4 = 10%.
Deviations: −6, −2, +2, +6.
Squared deviations: 36, 4, 4, 36. Sum = 80.
Sample variance = 80 ÷ (4 − 1) = 26.67 (%²).
Sample standard deviation = √26.67 = 5.16%.
Option A is the population figure (80 ÷ 4 = 20, √20 = 4.47). Option C (6.67%) is not a valid standard deviation for this data. It is only a distractor loosely derived from the variance (26.67 ÷ 4 = 6.67).

Answer: (B) 5.16%

Example 2

Asset X has a mean return of 6% and a standard deviation of 9%. Asset Y has a mean return of 10% and a standard deviation of 12%. Which statement is correct? (A) X has the lower CV, so less risk per unit of return (B) Y has the lower CV, so less risk per unit of return (C) Both have the same CV

Show the solution

CV of X = 9 ÷ 6 = 1.50.
CV of Y = 12 ÷ 10 = 1.20.
Y has the lower CV, so it carries less risk per unit of mean return.
X has the lower standard deviation, but that alone does not decide the comparison.

Answer: (B) Y has the lower CV, so less risk per unit of return

Exam tips

Look for the words 'sample' and 'population' first. They decide the divisor and often decide which distractor is wrong.
When a question asks which asset is riskier per unit of return, think CV, not standard deviation.
For target downside deviation, count all observations in the divisor and use n − 1 as in the curriculum formula.
Variance has squared units. If an option looks too large to be a standard deviation, it may be the variance.
With no penalty for guessing, always answer. Eliminate options using the sample versus population relationship, then pick.

Practice questions from Statistical Characteristics of Asset Returns

Measures of Dispersion and Risk in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

ACCA Applied KnowledgeMeasures of Dispersion

Measures of Dispersion and Risk: frequently asked questions

What is the difference between population and sample variance?

Population variance divides the sum of squared deviations by N and uses the true mean μ. Sample variance divides by n − 1 and uses the sample mean. The n − 1 divisor corrects the downward bias that comes from measuring deviations around the sample mean.

How do I calculate mean absolute deviation for the CFA exam?

Find the mean, take the absolute value of each observation's distance from the mean, add them up and divide by n. MAD is usually smaller than the standard deviation for the same data because squaring weights large deviations more.

What does the coefficient of variation tell you?

CV is standard deviation divided by the mean. It shows how much risk you take per unit of return, so you can compare investments with different average returns. A lower CV is better for the same return objective.

What is target downside deviation?

It measures only the dispersion of returns at or below a chosen target. You square each shortfall from the target, sum them, divide by n − 1 and take the square root. It fits investors who care about losses rather than all variation.