The Interval That Doesn't Mean What You Think
Confidence intervals are one of the most widely used and most widely misunderstood tools in statistics. When a poll reports "52% ± 3%," most people interpret this as a 95% probability that the true value is between 49% and 55%. This interpretation, while intuitive, is technically incorrect — and the correct interpretation reveals a deep philosophical distinction between frequentist and Bayesian probability.
How Confidence Intervals Work
A confidence interval is constructed by taking the sample mean and adding or subtracting a margin of error. The margin depends on three things: the z-score corresponding to your chosen confidence level, the population standard deviation, and the sample size. The key insight is that the interval is random (it depends on the sample), while the true parameter is fixed. Different samples produce different intervals, and the confidence level tells you what fraction of those intervals capture the truth.
The Square Root Law
The margin of error shrinks with the square root of the sample size — one of the most important relationships in statistics. To halve the margin of error, you need four times the sample size. This explains why opinion polls use about 1,000 respondents: going from 1,000 to 10,000 would only reduce the margin from ±3% to ±1%, rarely worth the tenfold increase in cost. This simulator lets you see this diminishing-returns curve in real time.
Coverage and Calibration
The ultimate test of a confidence interval procedure is its coverage rate — does a 95% interval actually contain the true value 95% of the time? This simulator draws many intervals from repeated samples, letting you verify that roughly 95 out of 100 intervals capture the true mean. The ones that miss are not "wrong" — they are the expected 5% that any honest procedure must produce.