Standard Error of the Mean
As sample size grows, the sample mean becomes a more precise estimate of the true population mean, with variability controlled by the standard error.
Sample Means Vary
Every random sample drawn from a population yields a slightly different sample mean. The distribution formed by collecting these means across all possible samples of size n is called the sampling distribution. Its expected value equals the population mean, showing that the sample mean is an unbiased estimator.
\mathbb{E}[\bar{X}] = \muQuantifying the Error
The standard deviation of the sampling distribution is called the standard error. It quantifies how much sample means typically deviate from the population parameter. Because the denominator contains the square root of n, larger samples produce tighter, more accurate distributions.
\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}Central Limit Effect
The Central Limit Theorem guarantees that for sufficiently large sample sizes, the sampling distribution of the mean approximates a normal distribution. This holds true regardless of the underlying population shape, allowing precise probability calculations using z-scores.
Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}Flashcard
Standard error of the sample mean
Check your understanding
If a researcher quadruples the sample size from n = 25 to n = 100, how does the standard error of the sample mean change?