Skip to content

FRM Part I · FRM Exam Part I

Sample Moments for FRM Part I: Complete Chapter Guide

Sample moments are statistics computed from data that estimate features of a distribution: the mean (centre), variance (spread), skewness (asymmetry) and kurtosis (tail weight). Cross moments such as covariance and correlation describe how two variables move together. To solve questions, write the formula, plug in the data, and watch the n versus n − 1 divisor.

What this chapter covers

This chapter, part of Quantitative Analysis, shows how you turn raw observations into numbers that describe a distribution. You start with the difference between population moments (true, usually unknown values) and sample moments (estimates from data). Then you work through the mean, variance, standard deviation, skewness and kurtosis. These are the first four moments.

Next the chapter moves to two variables. Covariance and correlation measure co-movement. Coskewness and cokurtosis extend the idea to asymmetry and tail behaviour shared between variables. The chapter closes with the Best Linear Unbiased Estimator (BLUE), which explains why the sample mean is a good estimator and what 'good' means.

The ideas here feed the rest of the paper. Hypothesis testing and confidence intervals rely on the sample mean and its standard error. Regression uses covariance, variance and the BLUE idea (OLS). Value-at-Risk and portfolio risk use volatility and correlation. Skewness and kurtosis help you judge when a normal assumption fails for returns.

Sample moments are the vocabulary of every later quantitative topic, so mistakes here carry into hypothesis testing, regression, time series and market risk questions. The questions are usually short calculations (variance from a small data set, correlation from covariance and standard deviations) or concept checks (what does excess kurtosis above zero imply, what makes an estimator BLUE). These are quick marks if your formulas are automatic, and with 100 questions in 4 hours, speed on this material frees time for harder questions elsewhere.

Sample Moments: topics in the order to study them

  1. 1Population vs Sample MomentsStart here: it defines the notation and the idea of estimating unknown population values from data.
  2. 2Sample Mean and Its PropertiesThe simplest estimator; its unbiasedness and standard error (σ ÷ √n) are used throughout the later chapters.
  3. 3Sample Variance and Standard DeviationBuilds on the mean and introduces the n − 1 divisor, the most tested detail in the chapter.
  4. 4Skewness and KurtosisThird and fourth standardised moments, which use the mean and standard deviation you have just learned.
  5. 5Covariance and CorrelationMoves from one variable to two, reusing the variance formula; the link to regression and portfolio risk is direct.
  6. 6Coskewness and CokurtosisExtends the cross-moment idea to higher orders, so it only makes sense after covariance and skewness.
  7. 7Best Linear Unbiased Estimator (BLUE)Last, because it uses earlier ideas (unbiased, variance of an estimator) to judge estimators and prepares you for regression.

How to prepare Sample Moments

Treat this chapter as a formula-and-interpretation drill. Aim to compute each statistic by hand on a small data set and then explain in one sentence what the result says.

  1. Write a one-page sheet with each formula: mean = Σx ÷ n; sample variance = Σ(x − mean)² ÷ (n − 1); skewness = E[(X − μ)³] ÷ σ³; kurtosis = E[(X − μ)⁴] ÷ σ⁴; covariance; correlation = Cov(X, Y) ÷ (σX × σY).
  2. Practise with 4 to 6 data points. Compute the mean, deviations, squared deviations and variance in a table, then take the square root for the standard deviation. Use the statistics mode on your calculator to check your answer.
  3. Learn the interpretation rules: normal kurtosis is 3 (excess kurtosis 0), positive skew means a longer right tail, correlation lies between −1 and 1, and correlation has no units while covariance does.
  4. Do the covariance and correlation questions both ways: find correlation from covariance and standard deviations, and recover covariance from correlation.
  5. Learn the BLUE conditions in words: best means lowest variance among linear unbiased estimators; know that OLS is BLUE under the Gauss-Markov assumptions you meet in regression.
  6. Finish with timed mixed questions, about one minute each, and log every error by type: formula, divisor, or interpretation.

Common mistakes in Sample Moments

  • Dividing by n instead of n − 1 when asked for sample variance

    Fix: Check whether the data is a sample or the full population before you calculate. If a sample is stated or implied, use n − 1.

  • Confusing kurtosis with excess kurtosis

    Fix: Read the question for the term used. Kurtosis of normal = 3; excess = kurtosis − 3 = 0.

  • Forgetting to take the square root, or reporting variance as volatility

    Fix: Underline what the question asks for (variance or standard deviation) before you start and check units at the end.

  • Treating zero correlation as independence

    Fix: Remember that correlation measures only linear dependence. Independence implies zero correlation, but not the reverse.

  • Mixing up covariance and correlation scales

    Fix: If a value is outside −1 to 1 it cannot be a correlation. Convert covariance by dividing by both standard deviations.

  • Treating BLUE as 'the best estimator of all kinds'

    Fix: Say it fully: best means minimum variance within the class of linear unbiased estimators, under stated assumptions.

Last-day revision: Sample Moments

  • Population moments are fixed parameters; sample moments are estimates from data.
  • Sample mean = Σx ÷ n, and it is an unbiased estimator of the population mean.
  • Standard error of the sample mean = σ ÷ √n for independent observations.
  • Sample variance divides by n − 1 so it is an unbiased estimator of population variance.
  • Standard deviation is the square root of variance and is in the same units as the data.
  • Skewness is zero for a symmetric distribution; negative skew means a longer left tail.
  • Kurtosis of a normal distribution is 3; excess kurtosis = kurtosis − 3.
  • Higher kurtosis (leptokurtic) means fatter tails and more extreme outcomes than the normal.
  • Covariance can be any value; its sign shows direction but its size depends on units.
  • Correlation = Cov(X, Y) ÷ (σX × σY) and is always between −1 and +1.
  • Zero correlation does not imply independence, because nonlinear dependence can remain.
  • An estimator is BLUE if it is linear, unbiased and has the smallest variance among such estimators.

Sample Moments practice questions

Sample Moments in other exams

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

Sample Moments: frequently asked questions

How many formulas do I need to memorise for sample moments?

Only a handful: mean, sample variance, standard deviation, skewness, kurtosis, covariance and correlation. Most are variations of averaging powers of deviations from the mean. Learn the pattern and the interpretation, not just the symbols.

Why does sample variance use n − 1?

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

Can I use a financial calculator for this chapter?

Yes. The statistics mode on an approved calculator gives the mean and standard deviations from entered data. Check whether it shows the sample (n − 1) or population (n) version, and pick the one the question needs.

What should I know about coskewness and cokurtosis?

Expect concept questions rather than heavy calculation. Know that they extend skewness and kurtosis to two variables and capture shared asymmetry and tail behaviour that covariance misses.