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FRM Exam Part I · Sample Moments

Population vs Sample Moments for FRM Part I

Updated 11 October 2026 · Fact-checked

Moments are summary measures of a distribution: mean, variance, skewness and kurtosis. Population moments are fixed parameters of the true distribution. Sample moments are estimates computed from data. Raw moments use powers of X; central moments use powers of deviations from the mean. To solve questions, identify which type is asked, then apply the right formula and divisor.

Understand Population vs Sample Moments

A moment is an expected value of a power of a random variable. Moments describe the shape of a distribution in a few numbers. The first moment gives location, the second gives spread, the third gives asymmetry and the fourth gives tail weight.

The population moment belongs to the true distribution. It is a fixed number, called a parameter, and you almost never observe it. The sample moment is a number computed from the data you have. It is a statistic, so it changes from sample to sample. You use it to estimate the population moment.

There are two families. A raw moment is about zero: the k-th raw moment is E[Xᵏ]. The first raw moment is the mean. A central moment is about the mean: the k-th central moment is E[(X − μ)ᵏ]. The second central moment is the variance. The first central moment is always zero.

Skewness and kurtosis are standardized central moments. You divide the third and fourth central moments by σ³ and σ⁴. This makes them unit-free. A normal distribution has skewness 0 and kurtosis 3.

Sample versions replace the expectation with an average over the data. The sample mean is the simple average. The sample variance divides by n − 1, not n, so that it is unbiased. Higher sample moments have their own estimators, which you will meet in later topics. Here the key skill is telling parameters from estimates and raw from central.

Key formulas to remember

k-th population raw moment
μ′ₖ = E[Xᵏ]
k = 1 gives the mean μ.
k-th population central moment
μₖ = E[(X − μ)ᵏ]
μ₁ = 0 always. μ₂ = σ², the variance.
Variance from raw moments
σ² = E[X²] − (E[X])²
Second central moment = second raw moment minus the squared mean.
Population skewness
Skew = E[(X − μ)³] ÷ σ³
Third central moment standardized. Zero for symmetric distributions with a finite third moment.
Population kurtosis
Kurt = E[(X − μ)⁴] ÷ σ⁴
Normal = 3. Excess kurtosis = Kurt − 3.
Sample mean
X̄ = (1 ÷ n) Σ Xᵢ
Unbiased estimator of μ.
Sample variance
s² = Σ (Xᵢ − X̄)² ÷ (n − 1)
Divisor n − 1 makes it unbiased for σ². Dividing by n gives a biased estimate.
Sample raw moment
m′ₖ = (1 ÷ n) Σ Xᵢᵏ
Average of the k-th powers of the observations.

How to solve Population vs Sample Moments questions

Use this routine for any question on population versus sample moments.

  1. 1Decide whether the data describe the whole distribution (population) or a sample from it.
  2. 2Identify the moment asked: mean, variance, skewness or kurtosis, and whether it is raw or central.
  3. 3Write the matching formula. Use E[ ] for a population and averages for a sample.
  4. 4For central moments, compute the mean first, then the deviations from it.
  5. 5For a sample variance, divide the sum of squared deviations by n − 1. For a population variance, divide by N.
  6. 6Standardize if skewness or kurtosis is asked: divide by σ³ or σ⁴.
  7. 7Check reasonableness: variance is not negative, the first central moment is zero, normal kurtosis is 3.

Quickest way: Raw-to-central shortcut

When to use it: When the question gives E[X] and E[X²] and asks for variance, or gives a small data set and asks for the variance.

  1. Variance = E[X²] − (E[X])². No deviations needed.
  2. For a small data set, compute the sum of squares Σ X² and use Σ (Xᵢ − X̄)² = Σ Xᵢ² − n X̄².
  3. Divide by n − 1 for a sample, or by n if the data are the full population.
  4. Scan the options: eliminate negative variances and any answer that divides by the wrong n.

Common mistakes in Population vs Sample Moments

  • Dividing the sample variance by n

    The population formula is more familiar and the wording is missed.

    Fix: If the data are a sample and the question wants an estimate of σ², divide by n − 1.

  • Calling the first central moment the mean

    The first raw moment is the mean, so the names get mixed up.

    Fix: The first raw moment is the mean. The first central moment is always zero.

  • Confusing raw second moment with variance

    Both involve squares.

    Fix: Variance = E[X²] − μ². Subtract the squared mean from the raw second moment.

  • Treating a sample statistic as a fixed parameter

    The sample number looks exact.

    Fix: A sample moment is an estimate and varies between samples. The population moment is fixed.

  • Forgetting to standardize skewness and kurtosis

    The third and fourth central moments are computed and then reported directly.

    Fix: Divide by σ³ or σ⁴. Remember normal kurtosis is 3, not 0.

Worked examples

Example 1

A random variable X has E[X] = 4 and E[X²] = 25. What is the population variance and what is the second central moment?

Show the solution
  1. The second central moment is E[(X − μ)²], which is the variance.
  2. Variance = E[X²] − (E[X])² = 25 − 4² = 25 − 16 = 9.
  3. So the second central moment is also 9.

Answer: Variance = 9, and the second central moment = 9 (standard deviation 3).

Example 2

A sample of five daily returns (in %) is 2, 4, 6, 8, 10 (in order, no ties). Compute the sample mean and the unbiased sample variance.

Show the solution
  1. Sample mean = (2 + 4 + 6 + 8 + 10) ÷ 5 = 30 ÷ 5 = 6.
  2. Deviations: −4, −2, 0, 2, 4.
  3. Squared deviations: 16, 4, 0, 4, 16. Sum = 40.
  4. Unbiased sample variance = 40 ÷ (5 − 1) = 40 ÷ 4 = 10.

Answer: Sample mean = 6%, sample variance = 10 (%²). Dividing by n would give 8, which is the wrong estimator here.

Exam tips

  • Read whether the question says population or sample before choosing n or n − 1.
  • Know the identity variance = E[X²] − μ². It saves time on moment-based questions.
  • Remember the first central moment is zero. It is a common trap option.
  • For skewness and kurtosis, check that the answer is standardized. Compare against 0 and 3 for the normal benchmark.
  • Use the calculator's statistics mode to get X̄ and sample standard deviation quickly, but check which standard deviation key it uses (sample or population).

Practice questions from Sample Moments

Population vs Sample Moments: frequently asked questions

What is the difference between population and sample moments?

Population moments are fixed parameters of the true distribution. Sample moments are statistics calculated from observed data to estimate them. Sample moments vary from one sample to another.

What is the difference between raw and central moments?

Raw moments use powers of X itself, E[Xᵏ]. Central moments use powers of deviations from the mean, E[(X − μ)ᵏ]. The first raw moment is the mean, and the second central moment is the variance.

Why does the sample variance divide by n − 1?

The sample mean is estimated from the same data, which makes deviations slightly too small. Dividing by n − 1 corrects this so that the estimator is unbiased for the population variance.

Are skewness and kurtosis moments?

They are standardized central moments. Skewness is the third central moment divided by σ³. Kurtosis is the fourth central moment divided by σ⁴, and the normal distribution has kurtosis of 3.