3.1 Introducing Statistics: Why Is My Sample Not Producing the Same Statistic as Others' Samples?

Sample statistics vary

Every time you take a new random sample from a population, you likely get a different sample statistic (like a sample proportion). This natural variation between samples is called sampling variability. It's not an error — it's expected because samples are random subsets of a population, not the whole population.

Why variability matters

Because statistics vary from sample to sample, a single sample statistic is just an estimate of the true, fixed population parameter. Understanding this variability helps us judge how much we can trust a sample result and sets up the need for sampling distributions to describe the pattern of that variation.

Bigger samples, less variability

As sample size increases, sampling variability tends to decrease — larger samples give estimates that cluster more closely around the true parameter. This is why researchers care about sample size: it affects how precisely a statistic estimates a parameter, though it doesn't eliminate variability entirely.

Key terms

Sampling variability

The natural variation in a statistic's value from one random sample to another, due to chance.

Parameter

A fixed numerical value describing a characteristic of an entire population.

Statistic

A numerical value calculated from sample data, used to estimate a population parameter.

Effect of sample size on variability

As sample size increases, sampling variability generally decreases, giving more precise estimates.

Why do samples differ from the population?

Because a sample is only a subset of individuals chosen at random, it won't perfectly represent the whole population every time.

Try an AP-style question

Several students each take a random sample of 50 from the same school and compute p̂. Why do their answers differ?

1. Samples of size 50 are always biased by their design
2. The true proportion p changed between the different samples
3. Different random samples naturally contain different individuals
4. Some of the students must have made calculation errors