A confidence interval (CI) is a range of values, calculated from sample data, that is likely to contain the true value of an unknown population parameter (such as a mean or a proportion). It is built around a point estimate as point estimate ± margin of error. A 95% confidence level means that if you repeated the same study many times, about 95% of the intervals you constructed would capture the true population value.
Suppose you want to find out how many children in a town leave school before they turn 18. Surveying every child would be expensive and, in practice, almost impossible, so you survey a smaller sample instead. Because a sample is only part of the whole population, each repeat of the survey would give a slightly different result. A confidence interval is how researchers express that uncertainty: rather than reporting a single number, they report a plausible range for the true value.
What Are Confidence Intervals?
A confidence interval is a range of values, bounded above and below a sample statistic such as the mean, that is likely to contain the true population parameter. It quantifies the uncertainty that comes from working with a sample rather than the whole population, and it is usually constructed using inferential methods based on the normal (z) or t-distribution.
Every confidence interval has the same structure:
where the margin of error = critical value × standard error.
The point estimate is your best single guess from the sample (for example, the sample mean). The margin of error is how far either side of that estimate you extend the range, and it depends on two things: how confident you want to be (the critical value) and how precise your estimate is (the standard error). A wider interval reflects more uncertainty; a narrower one reflects more precision.
How to Interpret a Confidence Interval (Correctly)
This is where confidence intervals are most often misunderstood, so it is worth being precise. The single most common mistake is to say:
“There is a 95% probability that the true population mean lies inside this particular interval.” — This statement is incorrect.— Common confidence-interval misconception
Why is it wrong? In classical (frequentist) statistics the true population parameter is a fixed, unknown constant — it is not random. Once you have computed a specific interval, say (102.95, 105.05), the true mean is either inside it or it is not. There is no probability attached to that one realised interval; the probability is 0 or 1, we just do not know which.
The 95% refers to the procedure, not to any single interval. The correct interpretation is:
- Right: “We are 95% confident that the interval (102.95, 105.05) contains the true mean” — understood as a statement about the method’s success rate.
- Wrong: “There is a 95% chance the true mean is between 102.95 and 105.05.”
- Wrong: “95% of the data values fall within this interval.” (A CI is about the parameter, not the spread of individual observations.)
- Wrong: “95% of future samples will fall in this interval.”
Why Do Researchers Use Confidence Intervals?
It is rarely feasible to study an entire population, so researchers work with a sample and use a confidence interval to show how well that sample estimate reflects the population. A CI does two valuable things at once: it gives a plausible range for the true value, and it communicates the precision of the estimate. A narrow interval signals a precise, trustworthy estimate; a wide one warns the reader that the estimate is shaky. Reporting an interval is therefore far more informative than reporting a single number with no sense of its reliability, which is why confidence intervals are central to inferential statistics and to interpreting statistical significance.
Confidence Level vs Interval Width
The confidence level (usually 90%, 95% or 99%) is the long-run proportion of intervals that would capture the true parameter. It is chosen by the researcher; 95% is the convention in most social-science and medical research. There is a direct trade-off between confidence and precision: the higher the confidence level, the wider the interval, because demanding more certainty forces you to cast a wider net.
| Confidence level | Two-tailed z critical value (z*) | Relative interval width |
|---|---|---|
| 80% | 1.282 | Narrowest |
| 90% | 1.645 | Narrow |
| 95% | 1.960 | Standard |
| 99% | 2.576 | Widest |
So a 99% interval is more likely to contain the true value than a 90% interval, but it is also less precise (wider). Choosing a confidence level is a judgement about how much you are willing to trade precision for reliability.
Factors that affect the width of a confidence interval
- Sample size (n): larger samples give a smaller standard error and therefore a narrower, more precise interval. Because the standard error shrinks with √n, quadrupling the sample size roughly halves the margin of error.
- Variability in the data: the more spread out the data (a larger standard deviation), the larger the standard error and the wider the interval.
- Confidence level: a higher confidence level (e.g. 99% vs 95%) uses a larger critical value and widens the interval.
- Proportion near 50% (for proportions): for a percentage estimate, uncertainty is greatest near 50/50. A 51%/49% split carries more sampling error than a 98%/2% split, regardless of sample size.
“The confidence level is the frequency of possible confidence intervals that contain the true value of the unknown population parameter.”— NIST/SEMATECH e-Handbook of Statistical Methods
How to Calculate a Confidence Interval
Which formula you use depends on whether the population standard deviation is known. When it is known (or the sample is large), use the z-distribution. When it is unknown and estimated from a small sample, use the t-distribution, which has heavier tails to account for the extra uncertainty. To decide between them and other tests, see our guide on which statistical test to use.
1. Using the normal (z) distribution — σ known
The formula for a confidence interval for a mean when the population standard deviation σ is known is:
- x̄ = the sample mean (the point estimate)
- z* = the critical value of the z-distribution for the chosen confidence level
- σ = the population standard deviation
- n = the sample size; √n is its square root
- σ / √n = the standard error of the mean
Step 1 – find the critical value. For 95% confidence, 2.5% sits in each tail. Looking up an area of 0.975 in the z-table gives z* = 1.96.
Step 2 – compute the standard error. σ / √n = 1.2 / √5 = 1.2 / 2.2361 = 0.5367.
Step 3 – compute the margin of error. z* × SE = 1.96 × 0.5367 = 1.05.
Step 4 – build the interval. Lower: 104 − 1.05 = 102.95. Upper: 104 + 1.05 = 105.05.
The 95% CI is (102.95, 105.05). Interpretation: if we repeated this procedure many times, about 95% of the intervals produced would contain the true mean weight.
2. Using the t-distribution — σ unknown, small sample
When the population standard deviation is unknown and estimated from a small sample, replace z* with the t critical value and σ with the sample standard deviation s:
Step 1 – degrees of freedom. df = n − 1 = 10 − 1 = 9.
Step 2 – tail area. (1 − 0.95) / 2 = 0.025 in each tail.
Step 3 – t critical value. For df = 9 and a tail area of 0.025, the t-table gives t* = 2.262.
Step 4 – standard error. s / √n = 20 / √10 = 20 / 3.1623 = 6.325.
Step 5 – margin of error. t* × SE = 2.262 × 6.325 = 14.31.
Step 6 – build the interval. Lower: 150 − 14.31 = 135.69. Upper: 150 + 14.31 = 164.31.
The 95% CI is (135.69, 164.31).
Steps to calculate any confidence interval
- Calculate the point estimate from your sample (e.g. the sample mean x̄).
- Choose a confidence level (commonly 95%) and find the matching critical value (z* or t*).
- Calculate the standard error of the estimate (σ/√n or s/√n).
- Multiply the critical value by the standard error to get the margin of error.
- Add and subtract the margin of error from the point estimate to get the upper and lower bounds.
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