FRM Exam Part I · Sample Moments

Skewness and Kurtosis: Formulas, Meaning and Jarque-Bera Test

Updated 11 October 2026 · Fact-checked

Skewness is the third standardized moment and measures asymmetry. Kurtosis is the fourth standardized moment and measures tail heaviness. To solve a question, find the mean and standard deviation, average the cubed or fourth-power deviations, and divide by σ³ or σ⁴. A normal distribution has skewness 0 and kurtosis 3, so excess kurtosis is kurtosis minus 3.

Understand Skewness and Kurtosis

Moments describe the shape of a distribution. The mean is the first moment and the variance is the second central moment. Skewness and kurtosis are the third and fourth moments, rescaled by the standard deviation so they have no units. That lets you compare a USD return series with a EUR one.

Skewness tells you whether the distribution leans to one side. Cubing a deviation keeps its sign, so large positive deviations push skewness up and large negative ones push it down. Positive skew means a longer right tail, with occasional large gains. Negative skew means a longer left tail, with occasional large losses. Equity returns often show negative skew, which matters for risk managers.

Kurtosis tells you how heavy the tails are. Raising deviations to the fourth power makes extreme observations dominate the sum. A normal distribution has kurtosis of exactly 3. Excess kurtosis is kurtosis minus 3, so the normal has excess kurtosis of 0. A leptokurtic distribution has kurtosis above 3: fatter tails and, for a given variance, a more peaked centre. A platykurtic distribution has kurtosis below 3: thinner tails. A mesokurtic distribution has kurtosis near 3.

Financial returns are usually leptokurtic. Extreme losses happen more often than a normal model predicts, so a VaR based on normality can understate tail risk.

To test whether a sample looks normal, use the Jarque-Bera (JB) test. It combines sample skewness and excess kurtosis into one statistic. Under the null hypothesis of normality, JB follows a chi-squared distribution with 2 degrees of freedom. A large JB means you reject normality.

Key formulas to remember

Skewness (population): S = E[(X − μ)³] ÷ σ³; Third central moment divided by the cube of the standard deviation. Zero for any symmetric distribution whose third moment exists.
Kurtosis (population): K = E[(X − μ)⁴] ÷ σ⁴; Fourth central moment divided by σ⁴. The normal distribution has K = 3.
Sample skewness: Ŝ = [(1/n) Σ (xᵢ − x̄)³] ÷ σ̂³; Use the same σ̂ the question gives. Some texts use an n divisor for σ̂² and others use n − 1. Read the question's definition.
Sample kurtosis: K̂ = [(1/n) Σ (xᵢ − x̄)⁴] ÷ σ̂⁴; Same divisor convention for σ̂ as in sample skewness.
Excess kurtosis: Excess kurtosis = K − 3; Positive means leptokurtic (fat tails). Negative means platykurtic (thin tails). Zero matches the normal.
Jarque-Bera statistic: JB = n × [Ŝ² ÷ 6 + (K̂ − 3)² ÷ 24]; Some GARP texts use (n − 1) in place of n. Use the form given. Compare with a chi-squared distribution with 2 degrees of freedom; the 5% critical value is 5.991.

How to solve Skewness and Kurtosis questions

Use this sequence for any question on sample skewness, kurtosis or normality.

1Identify what is asked: skewness, kurtosis, excess kurtosis, an interpretation, or a Jarque-Bera test.
2Compute the sample mean x̄ = Σx ÷ n.
3Compute each deviation (x − x̄). Then compute σ̂² using the divisor the question specifies, and take the square root to get σ̂.
4For skewness, cube each deviation, average them, and divide by σ̂³. For kurtosis, raise each deviation to the fourth power, average, and divide by σ̂⁴.
5Subtract 3 from kurtosis if the question asks for excess kurtosis or uses JB.
6For JB, substitute Ŝ and K̂ − 3 into the formula and compare with the chi-squared critical value (2 degrees of freedom). Reject normality if JB exceeds it.
7State the interpretation in words: direction of skew, fat or thin tails, and the risk implication such as VaR understating tail losses.

Quickest way: Sign-and-benchmark shortcut

When to use it: Use it when an option set can be narrowed by logic, or when the question gives moments and asks for an interpretation or JB.

Check signs first. A long left tail means negative skew. Kurtosis above 3 means fat tails.
Compare to benchmarks: skewness 0, kurtosis 3, excess kurtosis 0. This answers most interpretation questions without calculation.
If the question gives Ŝ and K̂, go straight to JB. Compute Ŝ² ÷ 6 and (K̂ − 3)² ÷ 24 separately, add them, then multiply by n.
If you must compute from data, work in deviations from the mean. Choose small datasets' cubes and fourth powers by hand, and use the calculator's memory to store σ̂.
Compare JB with 5.991 and decide. Do not look up the p-value.

Common mistakes in Skewness and Kurtosis

Reporting kurtosis when the question asks for excess kurtosis
The two numbers differ only by 3, so it is easy to stop one step early.
Fix: Read the last line of the question. If it says excess, subtract 3. Write 'K − 3' on your scratch paper as a reminder.
Using unstandardized moments
Students average (x − x̄)³ or (x − x̄)⁴ and forget to divide by σ̂³ or σ̂⁴.
Fix: Skewness and kurtosis are always ratios. Compute σ̂ first and divide by its cube or fourth power.
Reading the sign of skewness backwards
People look at where the bulk of the data sits rather than the tail.
Fix: Skew is named for the longer tail. Negative skew means the longer tail is on the left, so large losses are more likely than large gains of the same size.
Treating kurtosis as peakedness only, or thinking high kurtosis means high variance
Textbook pictures stress the tall peak.
Fix: For FRM, think tails. Kurtosis is scaled by variance, so it measures tail weight relative to the spread, not the spread itself.
Using the wrong degrees of freedom or tail in the Jarque-Bera test
Students confuse JB with a t-test or use n − 1 degrees of freedom.
Fix: JB uses a chi-squared distribution with 2 degrees of freedom (one for skewness, one for excess kurtosis). Reject normality when JB is above the critical value.
Assuming zero skewness and kurtosis of 3 prove normality
Those are the normal's values, so students treat them as proof.
Fix: They are necessary signs, not sufficient. Other distributions can match both. Use the language 'consistent with normality' unless the test question says otherwise.

Worked examples

Example 1

A sample of five monthly returns (in %) is 1, 2, 3, 4, 10. Using an n divisor for the variance, compute the sample skewness, kurtosis and excess kurtosis.

Show the solution

Mean: (1 + 2 + 3 + 4 + 10) ÷ 5 = 20 ÷ 5 = 4.
Deviations: −3, −2, −1, 0, 6.
Squares: 9, 4, 1, 0, 36. Sum = 50. Variance = 50 ÷ 5 = 10. σ̂ = √10 = 3.1623.
Cubes: −27, −8, −1, 0, 216. Sum = 180. Average = 180 ÷ 5 = 36. σ̂³ = 10 × 3.1623 = 31.623. Skewness = 36 ÷ 31.623 = 1.138.
Fourth powers: 81, 16, 1, 0, 1,296. Sum = 1,394. Average = 278.8. σ̂⁴ = 10² = 100. Kurtosis = 278.8 ÷ 100 = 2.788.
Excess kurtosis = 2.788 − 3 = −0.212.

Answer: Skewness ≈ 1.14 (right-skewed because of the 10), kurtosis ≈ 2.79 and excess kurtosis ≈ −0.21 (slightly platykurtic). With only five points, treat these as illustration, not strong evidence.

Example 2

Daily returns on a EUR equity index over n = 250 days have sample skewness −0.40 and sample kurtosis 4.5. Using JB = n × [Ŝ² ÷ 6 + (K̂ − 3)² ÷ 24], what is the JB statistic, and is normality rejected at 5% (chi-squared, 2 df, critical value 5.991)? Options for the statistic: A) 3.01 B) 15.05 C) 30.10 D) 60.21

Show the solution

Skewness term: Ŝ² ÷ 6 = 0.16 ÷ 6 = 0.026667.
Excess kurtosis = 4.5 − 3 = 1.5. Kurtosis term: 1.5² ÷ 24 = 2.25 ÷ 24 = 0.09375.
Sum = 0.026667 + 0.09375 = 0.120417.
JB = 250 × 0.120417 = 30.10.
Compare: 30.10 is greater than 5.991, so reject the null of normality.

Answer: C) JB ≈ 30.10, which exceeds 5.991. Reject normality: the returns show negative skew and fat tails, so a normal-based VaR would likely understate tail risk.

Exam tips

Know the benchmarks cold: normal skewness 0, kurtosis 3, excess kurtosis 0. Many questions are answered by comparing with them.
Check whether the question gives kurtosis or excess kurtosis before using any formula. Options are often built around the 3 gap.
For JB, remember the 2 degrees of freedom and the 5.991 critical value at 5%. Check whether the question's formula uses n or n − 1.
Link interpretations to risk. Negative skew and high kurtosis mean large losses are more likely than a normal model says, and VaR or ES from a normal assumption may be too low.
Do the arithmetic in two parts, skewness term then kurtosis term, and keep four decimals until the last step to avoid rounding errors.

Practice questions from Sample Moments

Skewness and Kurtosis in other exams

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

Skewness and Kurtosis: frequently asked questions

What does excess kurtosis mean in finance?

Excess kurtosis is kurtosis minus 3, so it compares a distribution's tails with the normal's. A positive value means fatter tails than the normal, so extreme returns are more likely. Most financial return series have positive excess kurtosis.

How do I calculate sample skewness?

Find the mean and standard deviation, cube each deviation from the mean and average the cubes, then divide by the standard deviation cubed. Use the divisor convention the question specifies. A positive result means a longer right tail and a negative result a longer left tail.

What is the difference between leptokurtic and platykurtic?

A leptokurtic distribution has kurtosis above 3, with fatter tails than the normal. A platykurtic distribution has kurtosis below 3, with thinner tails. A mesokurtic distribution, like the normal, has kurtosis of about 3.

How does the Jarque-Bera test use skewness and kurtosis?

It combines squared sample skewness and squared excess kurtosis into one statistic, scaled by the sample size. Under normality it follows a chi-squared distribution with 2 degrees of freedom. If JB exceeds the critical value, such as 5.991 at 5%, you reject normality.