Populations and samples
A population is the whole group of interest; a sample is the part we actually measure. A number describing a population is a parameter (such as the population mean μ). A number computed from a sample is a statistic (the sample mean x̄). We use statistics to estimate parameters.
Sampling well
A simple random sample (SRS) gives every possible sample of the given size an equal chance of being chosen. It avoids systematic bias. Common sources of bias:
- Convenience samples (surveying only people who are easy to reach, such as gym users about exercise habits).
- Voluntary response (online polls where only those with strong views reply).
- Undercoverage (parts of the population left out of the list).
- Non-response and badly worded questions.
A bigger sample reduces random error, but it does not fix bias. A huge biased sample is still biased.
Sampling distributions
If you took many samples of the same size and calculated x̄ each time, the values would vary. The distribution of those x̄ values is the sampling distribution of the sample mean. It has:
- Mean μ (the sample mean is an unbiased estimate of μ).
- Standard deviation, called the standard error, SE = σ / √n.
Larger samples give a smaller SE. Because of the square root, quadrupling n cuts SE in half.
The Central Limit Theorem (CLT)
For a large enough sample size (a common rule of thumb is n ≥ 30), the sampling distribution of x̄ is approximately normal, whatever the shape of the population. If the population itself is normal, x̄ is normal for any n.
Worked example
A population has mean μ = 100 and SD σ = 20. A random sample of n = 64 is taken.
- SE = 20 / √64 = 20 / 8 = 2.5.
- By the CLT, x̄ is approximately normal with mean 100 and SD 2.5.
- What is P(x̄ > 105)? z = (105 − 100) / 2.5 = 2.
- The area above z = 2 is about 0.023, so there is about a 2.3 percent chance that a sample mean this high occurs.
Contrast with a single value: P(X > 105) for one individual is z = 5/20 = 0.25, about 40 percent. Sample means vary much less than individual values.
Another quick check: σ = 15 and n = 25 give SE = 15 / 5 = 3.
Proportions
For a sample proportion p̂, the standard error is √(p(1 − p)/n). The same ideas apply: bigger n, smaller SE.
Common mistakes
- Using σ instead of σ/√n when working with a sample mean.
- Thinking the CLT says the data become normal; it says the distribution of the mean does.
- Believing a larger sample cures a biased sampling method.