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Contrast ways of sampling and explain sampling error

CFA® Level I · Quantitative Methods · Estimation and Hypothesis Testing · QM.5.c

The idea

Sampling error is the difference between a sample statistic and the population parameter it estimates, and it shrinks with sample size. Sampling bias is a different problem: a systematically unrepresentative sample stays wrong however large it gets. Simple random sampling gives every member an equal chance; stratified sampling divides the population into groups and samples within each, which improves precision when the groups differ. In finance the biases that bite hardest are survivorship — measuring only the funds that still exist — and look-ahead, using information that was not available at the time.

Formulas

Standard error of the sample mean

standard error = σ ÷ √n

σ
the standard deviation of the population, or the sample's when the population's is unknown
n
the sample size

How far a sample mean typically sits from the true mean. The square root is the reason precision is expensive: four times the sample halves the error, and a hundred times cuts it to a tenth.

A practice question

From the question bank, with the reason each wrong option is wrong.

What is sampling error?

  1. The difference between a sample statistic and the population parameter — correct
  2. A mistake made when recording observationsWhy not: That is measurement error, which a careful process can eliminate. Sampling error is unavoidable and shrinks only with sample size.
  3. The bias introduced by choosing an unrepresentative sampleWhy not: That is sampling bias, and the distinction is the point: error shrinks as the sample grows, bias does not.

Sampling error is the natural variation between a sample and the population, and it falls as the sample grows. Bias is systematic and does not, which is why the two must not be confused.

A harder one

A study of fund performance uses only funds still operating today. Which bias does this introduce?

  1. Look-ahead biasWhy not: That is using information in a test that was not available at the time. Here the problem is which funds are in the sample, not when the data were known.
  2. Survivorship bias — correct
  3. Data-mining biasWhy not: That comes from searching many specifications until one works. This sample was not searched over; it was filtered by survival.

Funds that closed were typically the poor performers, so excluding them flatters the average. The sample is unrepresentative in a way more data cannot fix.

The hardest tier

When does stratified sampling produce a more precise estimate than simple random sampling of the same size?

  1. Always, since it uses more informationWhy not: If the strata are alike, forcing proportional representation changes nothing — the gain comes entirely from the subgroups differing.
  2. Never, since randomness is what guarantees representativenessWhy not: Stratified sampling is still random within each stratum. It removes one source of variation without giving up randomisation.
  3. When the population contains distinct subgroups that differ from one another — correct

Sampling within each stratum guarantees every subgroup is represented, typically in proportion to its size, which removes the variation between subgroups from the estimate. The gain is largest when the subgroups differ and each is fairly uniform inside; if they are alike there is nothing to gain.

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