Sampling

Sampling is the process of choosing a smaller group (called a sample) from a larger population so you can study the sample and draw conclusions about the entire population without surveying every single member.

Sampling is a statistical method in which a subset of individuals, items, or observations is selected from a larger population for the purpose of estimating population parameters. The validity of inferences drawn from a sample depends on how the sample is selected; well-designed sampling methods, such as simple random sampling, aim to produce samples that are representative of the population and minimize bias. The size and selection method of the sample directly affect the precision and reliability of the resulting estimates.

Key Formula

n = \frac{z^2 \cdot p(1 - p)}{E^2}

Where:

$n$ = the required sample size
$z$ = the z-score corresponding to the desired confidence level
$p$ = the estimated proportion of the population with the characteristic of interest
$E$ = the desired margin of error

Worked Example

Problem: You want to estimate the proportion of students at a large university who walk to campus. You'd like a 95% confidence level with a margin of error of 4%. You have no prior estimate for the proportion. How large a sample do you need?

Step 1: Since there is no prior estimate, use

p = 0.5

because this maximizes the required sample size and is the most conservative choice.

p = 0.5

Step 2: For a 95% confidence level, the corresponding z-score is 1.96.

z = 1.96

Step 3: The desired margin of error is 4%, so

E = 0.04

. Substitute all values into the sample size formula.

n = \frac{(1.96)^2 \cdot 0.5(1 - 0.5)}{(0.04)^2} = \frac{3.8416 \cdot 0.25}{0.0016}

Step 4: Calculate the result and round up to the next whole number, since you can't survey a fraction of a person.

n = \frac{0.9604}{0.0016} = 600.25 \approx 601

Answer: You need a sample of at least 601 students to estimate the proportion with 95% confidence and a 4% margin of error.

Why It Matters

Sampling is central to nearly every field that relies on data — from medical trials determining whether a drug works, to political polls predicting election outcomes, to quality control in manufacturing. Surveying or testing an entire population is usually impossible or prohibitively expensive, so researchers depend on well-chosen samples to make trustworthy generalizations. In AP Statistics, understanding sampling methods is essential because the way a sample is collected determines whether your conclusions are valid.

Common Mistakes

Mistake: Using a convenience sample and treating it as representative of the population.

Correction: Convenience samples (e.g., surveying only your friends) introduce selection bias. Use a random sampling method so every member of the population has a known chance of being selected.

Mistake: Confusing sample size with population size when judging reliability.

Correction: The precision of an estimate depends primarily on the absolute size of the sample, not on what fraction of the population it represents. A random sample of 1,000 people can give reliable results whether the population is 100,000 or 100 million.

Related Terms

Population — The full group a sample is drawn from
Mean — Sample mean estimates the population mean
Probability — Underlies random selection in sampling methods