When you calculate a sample mean — say, the average height in a sample of 40 people — you rarely care about those exact 40 people. You want to say something about the whole population. But your sample mean is just an estimate; a different sample would give a slightly different number. A confidence interval quantifies that uncertainty by giving a range of plausible values for the true population figure.
What "95% confidence" really means
Suppose you compute a 95% confidence interval for a mean and get [68.2, 71.8]. The tempting interpretation — "there's a 95% probability the true mean is between 68.2 and 71.8" — is technically not quite right, and examiners love to catch it.
The correct interpretation: if you repeated your study many times and built a 95% interval each time, about 95% of those intervals would contain the true population value.
In everyday practice, people do treat the interval as "the range where the true value plausibly lies," and for interpretation purposes that's a reasonable working summary — just know the precise meaning is about the long-run reliability of the method, not the probability of one specific interval.
Reading an interval
Three things matter when you look at a confidence interval:
- Where it's centered — that's your best estimate (the sample mean or proportion).
- How wide it is — a narrow interval means a precise estimate; a wide one means high uncertainty.
- What it does or doesn't contain — this is where intervals connect to hypothesis tests.
The link to significance
Confidence intervals and p-values are two sides of the same coin:
- A 95% CI for a difference between two groups that does not contain 0 corresponds to a significant result at p < 0.05.
- A 95% CI for a correlation that doesn't contain 0 means the relationship is significant.
- If the interval does straddle 0, the effect isn't statistically significant.
The advantage of the interval is that it shows the range of plausible effects, not just a yes/no verdict. An interval of [0.1, 0.3] and an interval of [0.1, 9.5] can both be "significant," but they tell very different stories about how much you actually know.
What makes an interval narrow?
Three levers control the width:
| Factor | Effect on width |
|---|---|
| Larger sample size (n) | Narrower — more data, more precision |
| Lower confidence level (90% vs 99%) | Narrower, but less certain |
| Smaller variability in the data | Narrower |
The sample-size lever is the one you usually control. Because precision improves with the square root of n, cutting your margin of error in half requires roughly four times the sample. You can work out exactly how many participants you need with the sample size calculator.
Worked example. A poll finds 52% support with a 95% CI of [49%, 55%]. Because the interval includes 50%, you cannot confidently say the true support is above half — the race is a statistical tie, despite the headline "52%."
Confidence interval vs margin of error
You'll often see results reported as "52% ± 3%." The ±3% is the margin of error — literally half the width of the confidence interval. So "52% ± 3%" and "95% CI [49%, 55%]" are the same statement expressed two ways.
Calculate one now
Confidence Interval Calculator →For a population mean or a proportion — enter your data, pick a confidence level, get the interval.Related reading: Interpreting p-values · Effect size explained · Choosing a statistical test.