Margin of Error

This formula gives the sampling uncertainty in a sample mean when the population standard deviation is known or well-estimated. It works because the mean of repeated samples varies with standard error \(\sigma/\sqrt{n}\), and multiplying by the critical value \(z\) scales that typical spread to the chosen confidence level, producing the half-width of the interval around the sample mean. In other words, it tells you how far above and below the estimate you should extend to capture the true mean with the desired confidence. When \(n\) increases, the denominator grows and the uncertainty in the mean shrinks; when you choose a larger confidence level, \(z\) increases and the interval widens.

Use this when building a confidence interval for a population mean in settings like quality control, survey analysis, or lab measurements, especially when the population standard deviation is known or the normal approximation is appropriate. It is the go-to expression for the margin of error attached to a mean estimate.

• Using the sample standard deviation \(s\) here without checking whether a z-interval is appropriate
• Forgetting to divide by \(\sqrt{n}\) and using \(n\) instead
• Interpreting the margin of error as the standard deviation of individual observations rather than the uncertainty in the mean