Sample Moments: formula sheet

Key formulas

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.

Sample mean

X̄ = (1 ÷ n) × Σ Xᵢ, for i = 1 to n

Add all observations and divide by n.

Unbiasedness

E(X̄) = μ

Holds for i.i.d. data with finite mean, for any sample size.

Variance of the sample mean

Var(X̄) = σ² ÷ n

Requires independent observations with common variance σ².

Standard error (σ known)

SE = σ ÷ √n

Standard deviation of X̄.

Estimated standard error

SE = s ÷ √n

Use when σ is unknown; s uses n − 1 in the denominator.

Standardised sample mean

Z = (X̄ − μ) ÷ (σ ÷ √n)

Approximately standard normal by the CLT for large n.

Confidence interval for the mean

X̄ ± critical value × SE

Use z for large samples or known σ; t with n − 1 degrees of freedom otherwise.

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.

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.

Sample covariance

s_xy = Σ(xᵢ − x̄)(yᵢ − ȳ) ÷ (n − 1)

Uses n − 1 for the sample. Population covariance divides by n. Equivalent form: [Σxᵢyᵢ − n·x̄·ȳ] ÷ (n − 1).

Sample correlation

r = s_xy ÷ (s_x × s_y)

Unit-free, between −1 and +1. The n − 1 factors cancel, so you can use sums of squares and cross-products directly.

Correlation from sums

r = Σ(xᵢ − x̄)(yᵢ − ȳ) ÷ √[Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)²]

Fastest when you have deviations. No need to divide by n − 1 at all.

Covariance from correlation

s_xy = r × s_x × s_y

Use when a question gives r and the standard deviations.

Variance as self-covariance

Cov(x, x) = Var(x)

Covariance of a variable with itself is its variance.

Scaling property

Cov(aX + b, cY + d) = ac × Cov(X, Y); Corr(aX + b, cY + d) = Corr(X, Y) for ac > 0

Correlation is unchanged by positive linear rescaling. It flips sign if ac < 0.

Coskewness S(X,X,Y)

S(X,X,Y) = E[(X − μX)²(Y − μY)] ÷ (σX² σY)

Standardized by σX² and σY. Sample version averages the products over observations.

Coskewness S(X,Y,Y)

S(X,Y,Y) = E[(X − μX)(Y − μY)²] ÷ (σX σY²)

The mirror version. Two coskewness measures exist for a pair.

Cokurtosis K(X,X,Y,Y)

K(X,X,Y,Y) = E[(X − μX)²(Y − μY)²] ÷ (σX² σY²)

Equals 1 if X and Y are independent. Other versions are K(X,X,X,Y) and K(X,Y,Y,Y).

Cokurtosis K(X,X,X,Y)

K(X,X,X,Y) = E[(X − μX)³(Y − μY)] ÷ (σX³ σY)

Reduces to the kurtosis of X when Y = X.

Special case X = Y

S(X,X,X) = skewness; K(X,X,X,X) = kurtosis

Normal distribution: skewness 0, kurtosis 3.

Counts of distinct comoments (two variables)

Coskewness: 2; Cokurtosis: 3

Excluding the pure skewness and kurtosis of each variable.

Bias

Bias(θ̂) = E(θ̂) − θ

Unbiased when bias equals zero.

Sample mean

X̄ = (1/n) Σ Xᵢ

A linear estimator with every weight equal to 1/n.

Expected value of sample mean

E(X̄) = μ

Holds for iid draws (really only needs each Xᵢ to have mean μ).

Variance and standard error of sample mean

Var(X̄) = σ² ÷ n; SE = σ ÷ √n

Needs iid (or uncorrelated) observations. Falls as n rises, which gives consistency.

Linear estimator conditions

θ̂ = Σ wᵢXᵢ; unbiased if Σ wᵢ = 1; Var = σ² Σ wᵢ²

Minimised at wᵢ = 1/n, so the sample mean is BLUE.

Mean squared error

MSE = Var(θ̂) + Bias²

A biased estimator can have lower MSE than an unbiased one.

Consistency

θ̂ → θ in probability as n → ∞

Sufficient: bias → 0 and variance → 0.

Quick revision

• 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.

Common mistakes

• Dividing the sample variance by n 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 Fix: The first raw moment is the mean. The first central moment is always zero.
• Using the data standard deviation as the standard error. Fix: Always divide by √n when the question is about the mean.
• Dividing by n instead of √n. Fix: Variance of X̄ uses n; standard error is its square root, so it uses √n.
• Dividing by n when the data is a sample 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 Fix: Check a tiny example first, such as 1 and 3: s = 1.4142 while the population value is 1.
• Reporting kurtosis when the question asks for excess kurtosis 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 Fix: Skewness and kurtosis are always ratios. Compute σ̂ first and divide by its cube or fourth power.
• Dividing by n instead of n − 1 for sample covariance. Fix: The word "sample" means n − 1. If the question gives only a population, divide by n.
• Mixing n and n − 1 when computing correlation from covariance and standard deviations. Fix: Use the same divisor for all three. Or use the sums-of-squares form where the divisor cancels.

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).
• Questions often test the distinction between standard deviation and standard error. Check which one the question asks for.
• Expect conceptual items on unbiasedness: it holds for any n and does not need normality.
• Watch for 'how does SE change' questions; use the √n scaling instead of recomputing.