A reliable method
For independent observations, the standard deviation of a sample mean is population SD divided by square root of n. For a sample proportion it is square root of p(1−p)/n.
The common trap
A larger sample usually narrows a sampling distribution, but it does not automatically remove bias. A statistic is unbiased when its sampling-distribution center equals the parameter.
Make the connection
To cut standard error in half, multiply sample size by four. Independence and appropriate large-sample conditions must be checked before using a normal approximation.
Data distributions and sampling distributions
The distribution of 36 measured weights describes individual objects. Repeating the sampling process and plotting one mean from each sample describes sample means. If individual SD is 12, independent sample means with n=36 have SD 2. The center can stay the same while the spread changes substantially.
Precision does not repair bias
Increasing n from 25 to 100 halves SE when the other assumptions stay fixed. By contrast, surveying four times as many volunteers can preserve the same selection bias. An estimator may have low variability around the wrong center; a narrow distribution is not automatically an accurate one. Compare both center and spread with the target.
Transfer check
Explain why 1000 simulated averages near a target provide evidence but not proof of unbiasedness. The theoretical property concerns expected value over the full sampling distribution. Then describe how you would distinguish the roles of a larger sample size and a larger number of simulation repetitions.
In independent random samples of size 100 from a population with p = 1/2, what is the standard deviation of the sample proportion?
- Use square root of p(1 − p)/n.
- The numerator p(1 − p) is 1/4 and n is 100.
- The standard deviation is 1/20.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Statistics 2026–27 course framework