Describe how a sample moment differs from a population moment.
The idea
A population moment is a fixed constant defined by the distribution. The mean is E(X), the second raw moment is E(X²), and central moments subtract the true mean first. A sample moment is the same idea computed on data, an average of x, x² or deviations from the sample mean over n observations. Because the data are random, a sample moment is itself a random variable that changes from sample to sample, and it is used as an estimator of the population moment. The second raw sample moment is unbiased for E(X²). The central versions subtract an estimated mean, and so need the n − 1 adjustment. Do not confuse the fixed parameter with the random statistic that estimates it.
Formulas
Sample variance from the first two raw sample moments
s² = (m₂ − m₁²) × n ÷ (n − 1)
- m₁
- the first raw sample moment, the average of x
- m₂
- the second raw sample moment, the average of x²
- n
- the number of observations
The n ÷ (n − 1) factor is needed because the mean was estimated from the same data. If the true mean is known to be zero, m₂ alone is unbiased.
A practice question
Which statement best distinguishes a sample moment from the corresponding population moment?
- A sample moment is a fixed number, while a population moment is recomputed each time a new sample is drawn from the distribution.Why not: Reverses the two: the population moment is the fixed constant; the sample moment is random.
- A population moment is a fixed number set by the distribution; a sample moment is computed from the data and varies by sample. - correct
- A sample moment divides by n while a population moment divides by n - 1 to correct for estimating the mean.Why not: Confuses the moment definitions with the bias correction for the sample variance; a population moment is an expectation, with no divisor.
- A sample moment is always biased, whereas a population moment is always unbiased.Why not: Bias is a property of an estimator. Population moments are parameters, and the sample mean, for instance, is an unbiased sample moment.
A population moment such as E[X] or E[X²] is an expectation over the whole distribution. A sample moment replaces the expectation with an average over the data, so it is itself a random variable and an estimator of the population moment.
A harder one
From 40 daily returns an analyst computes the first raw sample moment as 0.8 and the second raw sample moment as 5.0 (in % and %²). What is the sample variance with the n - 1 divisor, closest to?
- 4.36 %²Why not: Gets the central second moment (divisor n) but does not apply the n/(n - 1) adjustment.
- 4.25 %²Why not: Applies the adjustment in the wrong direction, multiplying by (n - 1)/n.
- 5.13 %²Why not: Applies n/(n - 1) to the raw second moment without first subtracting the squared mean.
- 4.47 %² - correct
Central second moment = 5.0 - 0.8² = 4.36. With the n - 1 divisor the variance is 4.36 x 40/39 = 4.472.
The hardest tier
An analyst knows that a P&L series is demeaned by construction, so its population mean is exactly zero. From 60 observations, the sum of squared values is 180 (%²) and the sample mean is 0.5%. Which is the best estimate of the variance, using the second raw sample moment?
- 3.05 %²Why not: Applies the n - 1 divisor although no mean was estimated: with a known mean there is no degree of freedom to correct for.
- 3.00 %² - correct
- 2.80 %²Why not: Subtracts the noisy sample mean and uses n - 1, discarding the knowledge that the true mean is zero.
- 2.75 %²Why not: Subtracts the sample mean and divides by n, which is biased downward; the known zero mean should be used instead.
With a known mean of zero, the variance equals E[X²], and the unbiased estimator is the second raw sample moment: 180/60 = 3.00. The n - 1 divisor is only needed when the mean is estimated from the same data.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Sample Moments
- Use sample data to estimate a mean, a variance and a standard deviation.
- Distinguish an estimator, which is a rule, from an estimate, which is a value it produces.
- Explain what the bias of an estimator is and what it indicates about the estimator.
- Interpret the claim that the estimator of the mean is BLUE.
- Define the consistency of an estimator and describe why the idea is useful.
- Apply the Central Limit Theorem (CLT) and the Law of Large Numbers (LLN) to a sample mean, saying what each result implies.
- Use data to estimate the skewness and kurtosis of a random variable and interpret the results.
- Use sample data to estimate quantiles, the median among them.
- Apply the CLT after estimating the means of two variables from sample data.
- From sample data, calculate estimates of how two random variables covary and how they correlate.
- Relate coskewness and cokurtosis back to the skewness and kurtosis of a single variable.