Aug 20

Sample statistics versus population parameters

by David Harper, CFA, FRM, CIPM

Learning objective

Distinguish between population and sample, and calculate the sample mean, variance, covariance, correlation, skewness, and kurtosis, respectively.

Statistics is using sample statistics to infer population parameters

In finance, we rarely collect the population. Generally we draw samples (e.g., market index returns over the last three years) on the sometimes dubious assumption that a true population metric exists (e.g., the equity risk premium). Often, we use a historical sample (historical volatility) to infer/estimate a current parameter (GARCH or EWMA to estimate current volatility).

The essence of statistics is to use sample statistics to infer population parameters; note statistics tend to be Roman and parameters tend to be Greek (“statistics are estimates of parameters”):

Sample mean

Sample variance

Divides by (n-1) instead of (n). Slightly larger.

Covariance and sample covariance

Similarly, sample covariance divides by (n-1) instead of (n) to return a slightly larger covariance.

Correlation and sample correlation

Sample correlation uses sample sample covariance. Since the (n-1) will cancel, in practice we often use population calculation instead.

Skewness and sample skew

Note sample skew has both a sample third moment (n-1) in the numerator and a sample standard deviation cubed (i.e., skew is a standardized third moment, not third moment per se):