Mathematics
Confidence Intervals and Their Interpretation
Quick fact
A 95% confidence interval does NOT mean there is a 95% chance the true parameter lies in a given interval. The '95%' refers to the method: if you repeat the sampling and computation many times, about 95% of the resulting intervals will capture the true parameter.
Why this is interesting
You've probably seen a poll say 'the result is accurate within ±3% with 95% confidence.' But what does that actually mean? Surprisingly, it's not a 95% probability that the true value lies in that range.
Read the full explanation
Understanding Confidence Intervals and Their Interpretation
Imagine you want to know the average height of all students in a large university. Measuring everyone is impractical, so you take a random sample of, say, 100 students and calculate their average height. That number is a point estimate. But you know it's probably not exactly the true average of the entire population. To express uncertainty, you create an interval around the estimate, such as 'average is 5'9" plus or minus 1 inch.' This interval is a confidence interval. The width of the interval depends on several factors: how much the data varies (standard deviation), how large your sample is, and the level of confidence you want (e.g., 95%). The key idea is that if you could repeat your sampling process many times, about 95% of the intervals you calculate would actually contain the true population average.
A deeper explanation
The logic of confidence intervals rests on the sampling distribution of the estimate, often the sample mean. For a mean, the Central Limit Theorem tells us that the sampling distribution of the sample mean is approximately normal for sufficiently large samples, with a standard error equal to the population standard deviation divided by the square root of the sample size. In practice, we estimate this standard error from the sample. The confidence interval is then constructed as: sample mean ± (critical value) × (standard error). The critical value comes from the standard normal distribution (z) or the t-distribution, depending on sample size and whether the population standard deviation is known. For a 95% confidence level, the critical z-value is roughly 1.96. This interval calculation is designed so that, in the long run, 95% of all such intervals cover the true parameter. The common misconception is to assign a probability to a specific interval after it's computed, which is invalid because the true parameter is fixed and the interval is a random result. Correct interpretation: 'We are 95% confident that the interval contains the true mean,' meaning that the method is reliable. Common pitfalls include ignoring that the interval reflects only sampling error, not all biases (e.g., measurement error, nonresponse).