What is a confidence interval?
A confidence interval is your estimate shown honestly, as a range instead of a single number: the estimate plus and minus its margin of error. "62 percent of children can read, ±4.8 points" means the data supports anywhere from 57.2 to 66.8 percent. The confidence level (usually 95 percent) says how often the method that built this range captures the true value across repeated studies.
A single number from a sample is always slightly wrong; the only question is by how much. The confidence interval is the tool that answers it. Reporting "62 percent" alone invites readers to treat a sample estimate as an exact fact. Reporting "57 to 67 percent" tells them precisely how much weight the number can carry, which is what separates an honest evaluation from a confident-sounding one.
The formula, for a proportion
A worked example
You test 385 randomly sampled children and 239 of them (62 percent) read at grade level.
- Standard error = √(0.62 × 0.38 / 385) ≈ 0.0247
- Margin of error = 1.96 × 0.0247 ≈ 0.048, that is ±4.8 percentage points
- 95 percent confidence interval: 57.2 to 66.8 percent
What 95 percent confidence really promises
It is a promise about the method, not about this one interval. If you ran the whole survey 100 times, drawing a fresh random sample each time, about 95 of the 100 intervals you built would contain the true value, and about 5 would miss. You never know which kind you are holding; what you know is the reliability of the machine that produced it.
Two common misreadings to avoid:
- "There is a 95 percent chance the truth is in this interval." Close enough for everyday conversation, but strictly the truth is fixed and it is the interval that varies from sample to sample. The 95 percent describes the procedure's hit rate.
- "Values in the middle are more likely than values at the edges, and values just outside are impossible." The interval is a plausibility range, not a hard wall; a true value just outside it is unlucky, not impossible.
Frequently asked questions
What does a 95 percent confidence interval mean?
That the method delivers: across many repeated studies, about 95 percent of intervals built this way would contain the true value. It quantifies the reliability of the procedure, not a probability attached to this single interval.
How do I calculate one for a survey proportion?
p ± 1.96 × √(p(1-p)/n) at 95 percent confidence. From 385 children with 62 percent readers: 62 ± 4.8, so 57.2 to 66.8 percent. If your sample is clustered by school, widen the interval by the square root of the design effect.
Is a wide confidence interval a failure?
No. It is the honest size of your evidence. A wide interval warns decision-makers not to over-read the headline number. The failure would be reporting the point estimate without it.
Watch intervals catch, and miss, the truth
Sampling Shala's Learn mode runs 100 surveys in a row so you can see 95 percent confidence physically happen.
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