Sampling Distributions & Variability

You’ll be able to

Define a sampling distribution as the distribution of a statistic over all samples
Explain what it means for a statistic to be an unbiased estimator
Describe how sample size affects the variability of a statistic

A statistic is itself a random variable

Take an SRS and compute a statistic (x-bar or p-hat). Take a different SRS and you get a different value — the statistic varies from sample to sample. The sampling distribution of a statistic is the distribution of its values over all possible samples of a given size from the population. It is not the distribution of the data and not one sample — it is the distribution of the statistic itself.

Unbiased and low-variability estimators

A statistic is an unbiased estimator if the mean of its sampling distribution equals the true parameter — it does not systematically over- or under-shoot. Bias is about the center of the sampling distribution; variability is about its spread. We want both: centered on the truth (unbiased) and tightly clustered (low variability). Larger samples reduce variability without affecting bias.

Bigger samples, less variability

The spread of a sampling distribution shrinks as the sample size n grows — larger samples give more consistent estimates. Crucially, it is the sample size, not the fraction of the population sampled, that governs this (as long as the population is much larger than the sample, at least 10 times). A sample of 1,000 is about equally precise whether the population is a hundred thousand or a hundred million.

Tip

Separate the two quality dimensions: bias = is the sampling distribution centered on the true parameter? Variability = is it tightly clustered? A good estimator hits the bullseye on average (unbiased) with a tight grouping (low variability). Increasing n tightens the grouping.

Worked example

A population proportion is p = 0.4. Explain why p-hat from an SRS of size 500 is a better estimator than p-hat from an SRS of size 50, in terms of bias and variability.

1.Both are unbiased: the mean of the sampling distribution of p-hat is p = 0.4 regardless of sample size, so neither systematically misses.
2.Variability differs: the standard deviation of p-hat is sqrt(p(1−p)/n), which decreases as n increases.
3.With n = 500 the standard deviation is smaller than with n = 50, so p-hat values cluster more tightly around 0.4.

Answer: Both sample sizes give an unbiased estimator (centered on 0.4), but n = 500 has smaller variability because the standard deviation sqrt(p(1−p)/n) shrinks as n grows. The larger sample produces estimates that are more consistently close to the true proportion.

Checkpoint

What does it mean to say the sample mean x-bar is an unbiased estimator of the population mean μ?

Watch out

The precision of an estimate depends on the sample size n, not on the fraction of the population sampled. A national poll of 1,500 people is about as precise as a state poll of 1,500 — provided the population is at least 10 times the sample. Do not assume you need a "percentage" of the population.

Checkpoint

A sampling distribution of x-bar is centered exactly at μ but has a wide spread. This estimator is best described as:

Answer the 2 checkpoints as you read.