FRM Exam Part I · Sample Moments

Sample Variance and Standard Deviation Explained

Updated 11 October 2026 · Fact-checked

Sample variance measures how spread out your observations are around the sample mean. Compute s² = Σ(xᵢ − x̄)² ÷ (n − 1). Dividing by n − 1 makes s² an unbiased estimator of population variance. Sample standard deviation is s = √s², in the same units as the data.

Understand Sample Variance and Standard Deviation

Variance measures dispersion. For each observation, take its distance from the mean, square it, and average the results. Squaring stops positive and negative deviations from cancelling and gives larger deviations more weight.

If you know the whole population, you use the population variance: σ² = Σ(xᵢ − μ)² ÷ N. In risk work you rarely know μ. You have a sample of returns and must estimate both the mean and the variance from the same data.

This causes a problem. The sample mean x̄ is the point that minimises the sum of squared deviations for your sample. So deviations from x̄ are, on average, smaller than deviations from the true mean μ. If you divide by n, you underestimate the true variance. That is bias.

The fix is to divide by n − 1. Once x̄ is calculated, only n − 1 deviations are free to vary, because all n deviations must sum to zero. This is the degrees of freedom idea: you used up one degree of freedom estimating the mean. With the n − 1 divisor, E[s²] = σ², so s² is unbiased (this holds for independent, identically distributed observations).

The sample standard deviation s = √s² is the usual measure of volatility. Note that s is slightly biased downward as an estimator of σ, even though s² is unbiased, because the square root is a concave function. For large n the difference is small, and the n versus n − 1 gap shrinks too.

Key formulas to remember

Sample mean: x̄ = (1 ÷ n) × Σxᵢ; Computed first. Needed for every deviation.
Sample variance (unbiased): s² = Σ(xᵢ − x̄)² ÷ (n − 1); Use this by default when the data is a sample and the mean is estimated.
Sample standard deviation: s = √s²; Same units as the data. Take the root last.
Population variance: σ² = Σ(xᵢ − μ)² ÷ N; Use only when you have every member of the population, or when μ is known.
Shortcut for the sum of squares: Σ(xᵢ − x̄)² = Σxᵢ² − n × x̄²; Handy with a calculator. Rounding errors can grow if the numbers are large.
Bias of the divide-by-n estimator: E[Σ(xᵢ − x̄)² ÷ n] = σ² × (n − 1) ÷ n; The n divisor understates variance by the factor (n − 1) ÷ n.
Variance of the sample mean: Var(x̄) = σ² ÷ n, so standard error = s ÷ √n; For independent observations. Links to confidence intervals.

How to solve Sample Variance and Standard Deviation questions

Use this routine for any question on sample variance or standard deviation.

1Decide whether the data is a sample or the full population. Words like 'estimate', 'sample' or 'historical observations' point to a sample.
2Calculate the sample mean x̄ = Σxᵢ ÷ n.
3Find each deviation (xᵢ − x̄) and square it. Check that the raw deviations sum to zero.
4Add the squared deviations to get the sum of squares.
5Divide by n − 1 for sample variance, or by N for population variance.
6Take the square root if the question asks for standard deviation.
7Check units and scale. If returns are in percent, variance is in percent squared.
8If asked about bias, remember: dividing by n underestimates σ²; dividing by n − 1 gives an unbiased s².

Quickest way: Calculator statistics mode shortcut

When to use it: Use when you have a short list of data points (about 10 or fewer) and need s fast.

Enter the data in the statistics mode of your approved calculator.
Read the sample standard deviation key (often labelled s or Sx), not the population one (σ or σx).
Square it if the question wants variance.
If you have no calculator mode, use Σxᵢ² − n × x̄² then divide by n − 1.
To convert between the two divisors: s² = population variance × n ÷ (n − 1).

Common mistakes in Sample Variance and Standard Deviation

Dividing by n when the data is a sample
The population formula looks simpler and is the one learned first.
Fix: If the mean is estimated from the same data, divide by n − 1. Use n only if the question says population or gives the known mean.
Reading the wrong calculator key
Calculators show both σ (divide by n) and s (divide by n − 1) and they look alike.
Fix: Check a tiny example first, such as 1 and 3: s = 1.4142 while the population value is 1.
Saying the sample standard deviation is unbiased
Students know s² is unbiased and assume the root keeps that property.
Fix: s² is unbiased for σ², but s slightly underestimates σ because the square root is concave.
Forgetting to take the square root
Attention is spent on the sum of squares and the question asks for volatility.
Fix: Re-read the question for the word standard deviation or volatility. Variance is in squared units.
Mixing up the variance of the data with the variance of the mean
Both use σ² and n, so the formulas blur together.
Fix: Variance of the data is s². Variance of the sample mean is s² ÷ n. Standard error is s ÷ √n.
Calling n − 1 a rule that fixes all bias
It is taught as a universal correction.
Fix: It corrects bias from estimating the mean for independent observations. It does not fix problems like autocorrelation or outliers.

Worked examples

Example 1

A risk analyst records five daily returns (in %): 2, 4, 4, 6, 9. Compute the sample variance and sample standard deviation.

Show the solution

Sample mean: x̄ = (2 + 4 + 4 + 6 + 9) ÷ 5 = 25 ÷ 5 = 5.
Deviations: −3, −1, −1, 1, 4. They sum to 0, which is a good check.
Squared deviations: 9, 1, 1, 1, 16. Sum = 28.
Sample variance: s² = 28 ÷ (5 − 1) = 28 ÷ 4 = 7.
Sample standard deviation: s = √7 = 2.6458.

Answer: s² = 7 (%²) and s ≈ 2.65%.

Example 2

A sample of 10 observations has a sample variance of 18 computed with divisor n − 1. What is the variance if the divisor n is used instead? Which estimate is unbiased?

Show the solution

Recover the sum of squares: Σ(xᵢ − x̄)² = s² × (n − 1) = 18 × 9 = 162.
Divide by n: 162 ÷ 10 = 16.2.
Check with the factor: 18 × 9 ÷ 10 = 16.2.
The n − 1 estimate (18) is unbiased for σ². The divide-by-n estimate (16.2) understates it on average.

Answer: The divisor-n variance is 16.2. The estimate of 18 using n − 1 is the unbiased one.

Exam tips

Look for the words sample, estimate or historical data. They signal n − 1.
Questions often give a variance computed one way and ask you to convert it. Use the sum of squares as the bridge.
Remember that s² is unbiased but s is not. This is a favourite conceptual question.
Check the deviations sum to zero before squaring. It catches arithmetic slips cheaply.
If the true mean is given, the divisor is n, because no degree of freedom was used on the mean.

Practice questions from Sample Moments

Sample Variance and Standard Deviation in other exams

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

CA FoundationStandard Deviation and Variance

Sample Variance and Standard Deviation: frequently asked questions

Why do we divide by n − 1 in sample variance?

The sample mean is fitted to the same data, so deviations from it are smaller than deviations from the true mean. Dividing by n would understate variance on average. Using n − 1 removes this bias, since only n − 1 deviations are free once the mean is known.

What is the difference between population variance and sample variance?

Population variance uses every member of the population and the true mean μ, with divisor N. Sample variance uses a subset and the estimated mean x̄, with divisor n − 1. The sample version is an estimator of the population value.

How do I calculate sample standard deviation?

Find the mean, square each deviation from it, sum them and divide by n − 1. Then take the square root. On a calculator, use the s or Sx key, not σ.

Is the sample standard deviation an unbiased estimator?

No. The sample variance with n − 1 is unbiased, but its square root slightly underestimates the true standard deviation. The bias is small for large samples.