Measures of Dispersion

Updated for the 2026-2027 CFA® Level I curriculum.

Two investments can earn the same average return and still behave very differently from one period to the next. Measures of dispersion help you see that difference by showing how tightly or widely the observations are spread.

For CFA Level I, you should know how to calculate the main measures of dispersion, interpret what each result means, and choose the measure that fits the investment problem.

Key Takeaways About Measures of Dispersion

• The range measures the distance between the highest and lowest observations.

• Mean absolute deviation gives the average absolute distance from the mean.

• Variance averages squared deviations from the mean.

• Standard deviation is the square root of variance and uses the same units as the original data.

• Downside deviation focuses on returns below a chosen target.

• The coefficient of variation compares dispersion relative to the mean.

• Population and sample calculations use different denominators.

• Measures of dispersion are most useful when interpreted alongside a measure of central tendency.

What You Need to Know for CFA Level I

For CFA Level I, focus on both calculation and interpretation.

You should be able to:

• Calculate the range and mean absolute deviation.

• Calculate population and sample variance.

• Convert variance into standard deviation.

• Explain why variance uses squared units.

• Interpret standard deviation as total variability around the mean.

• Calculate or interpret downside deviation relative to a target return.

• Use the coefficient of variation to compare relative dispersion.

• Choose between population and sample formulas based on the wording of the question.

• Compare investments with similar average returns but different levels of dispersion.

What Is Dispersion in Statistics and Investing?

Dispersion describes how closely observations cluster around the center of a dataset.

A dataset with low dispersion contains observations that stay relatively close to the mean. A dataset with high dispersion contains observations that are spread farther apart.

For an investor, that spread can reveal information that an average return leaves out. Consider two funds that both earn an average annual return of 8%. One fund may remain between 6% and 10% each year, while the other moves between large gains and substantial losses. Their average returns are identical, but their return patterns create different risk profiles.

Measures of central tendency describe the typical result. Measures of dispersion show how much individual results vary around it. Reading the two together gives you a more complete view of the data.

Main Methods of Measuring Dispersion

Range

The range is the difference between the highest and lowest observations.

It provides a quick view of the full observed spread. For example, if annual returns range from to , the range is 19 percentage points.

Because the calculation uses only two observations, the range provides limited information about the values between them. One extreme observation can also change the result substantially.

Mean Absolute Deviation

Mean absolute deviation, or MAD, measures the average absolute distance between each observation and the arithmetic mean.

Using absolute values prevents positive and negative deviations from cancelling each other out. The result remains in the same units as the original observations, which makes it fairly intuitive to interpret.

MAD is less common in advanced statistical calculations because absolute values are more difficult to manipulate algebraically than squared deviations.

Variance

Variance measures the average squared distance between each observation and the mean.

Squaring the deviations serves two purposes. It removes negative signs, and it gives greater weight to observations that sit farther from the mean. A return that is 10 percentage points from the mean contributes more to variance than a return that is only 2 percentage points away.

The resulting measure is expressed in squared units. If the observations are percentages, the variance is expressed in percentage points squared. This makes variance useful for statistical calculations but less intuitive for direct interpretation.

Standard Deviation

Standard deviation is the square root of variance.

Taking the square root returns the measure to the same units as the original data. If a fund has a mean return of 8% and a standard deviation of 6%, you can compare both figures directly.

Standard deviation includes observations above and below the mean. Large positive returns and large negative returns both increase the result. It therefore measures total variability rather than downside risk alone.

Downside Deviation

Downside deviation measures the variation of returns that fall below a chosen target, sometimes called the minimum acceptable return.

Suppose an investor requires a return of at least 5%. A return of 9% contributes nothing to downside deviation, while a return of produces an 8-percentage-point shortfall.

Observations above the target contribute zero to the numerator. They usually remain part of the denominator used by the applicable population or sample formula.

Downside deviation is useful when the investor is mainly concerned with returns that fail to reach a required level.

Coefficient of Variation

The coefficient of variation compares standard deviation with the mean return.

It expresses variability per unit of expected return, allowing you to compare investments that have different average returns.

For example, a standard deviation of 5% may appear low in isolation. It becomes less attractive when paired with an expected return of only 2% than when paired with an expected return of 10%.

The coefficient of variation is most meaningful when the mean is positive and comfortably above zero. A mean near zero can produce an unusually large and unstable ratio.

Sample vs Population Dispersion

The formula you use depends on whether the observations represent an entire population or a sample drawn from a larger population.

Population Measures

Use a population formula when the dataset includes every observation in the group being studied.

Population variance divides the sum of squared deviations by , the total number of observations in the population.

Sample Measures

Use a sample formula when the observations represent only part of a larger population.

Sample variance divides by . This adjustment accounts for the degree of freedom used when the sample mean is estimated from the same data.

Question wording usually indicates the correct formula:

• “The complete set of possible returns” suggests a population.

• “A sample of historical returns” suggests a sample.

• “Returns recorded over the last five years” may require context, so read the full question carefully.

Using instead of produces a lower variance and standard deviation. On a short exam calculation, that denominator can determine which answer choice is correct.

How to Choose a Measure of Dispersion

The appropriate measure depends on the information the analyst needs.

A larger number does not automatically make one investment worse. The interpretation depends on the expected return, the investor’s objectives, the shape of the distribution, and the source of the variability.

For example, a growth-oriented investor may accept greater standard deviation in exchange for higher expected return. An investor with a strict minimum-return objective may focus more closely on downside deviation.

Measures of Dispersion Formulas

Each block below can be pasted into a Notion block equation.

Range Formula

Where:

• = highest observation

• = lowest observation

Mean Absolute Deviation Formula

Where:

• = mean absolute deviation

• = observation

• = arithmetic mean

• = number of observations

Population Variance Formula

Where:

• = population variance

• = observation

• = population mean

• = number of observations in the population

Sample Variance Formula

Where:

• = sample variance

• = observation

• = sample mean

• = number of observations in the sample

Population Standard Deviation Formula

Sample Standard Deviation Formula

Sample Target Downside Deviation Formula

Where:

• = sample target downside deviation

• = observation

• = target or minimum acceptable return

• = number of sample observations

When is above , the expression equals zero. Only observations below the target contribute a squared shortfall.

Coefficient of Variation Formula

Where:

• = coefficient of variation

• = sample standard deviation

• = sample mean

Worked Example: Comparing Investments With the Same Mean

Consider the following annual returns:

• Asset A:

• Asset B:

Both assets have an arithmetic mean return of 8%.

Asset A Sample Standard Deviation

First, calculate the mean:

The deviations from the mean are:

Square each deviation:

The squared deviations sum to .

Because these returns are treated as a sample, divide by , which equals 4:

Take the square root:

Asset B Sample Standard Deviation

Asset B also has a mean of 8%.

Its deviations from the mean are:

The squared deviations are:

Their sum is .

Interpretation

Both assets earned an average return of 8%, but Asset A had a standard deviation of approximately 6.67%, compared with 1.58% for Asset B.

Asset A’s returns moved much farther from the mean. Asset B produced more consistent annual results during the period.

The standard deviation does not tell you which investment an investor should choose on its own. It shows that Asset A carried substantially more return variability, which the investor can evaluate alongside expected return and investment objectives.

Asset A Downside Deviation

Suppose the investor sets a minimum acceptable return of 8%.

Asset A’s returns of 5% and fall below the target. Their shortfalls are:

The remaining returns contribute zero.

Using the sample target downside deviation formula:

This result focuses on Asset A’s shortfalls below the investor’s 8% target. Its total standard deviation of 6.67% also includes variability from returns above the target.

Common Exam Traps

• Using when the question describes a sample and requires .

• Using when the observations represent the full population.

• Forgetting to square deviations when calculating variance.

• Reporting variance as though it were expressed in the original return units.

• Forgetting to take the square root when the question asks for standard deviation.

• Treating standard deviation as a downside-only measure.

• Dividing downside deviation by only the number of observations below the target.

• Assuming the range describes how all observations are distributed.

• Using the coefficient of variation when the mean is close to zero.

• Comparing dispersion without also considering the expected or average return.

Practice Question

An analyst records the following sample of annual fund returns:

What is the sample standard deviation?

1. 3.63%

2. 4.06%

3. 16.50%

FAQs About Measures of Dispersion