Confidence Intervals: Formula, Interpretation and Worked Examples
Reviewed by Prof. Dr. Alexander Holtermann · Updated 16 October 2026 · 5 min read
A 95% confidence interval gives a range of plausible values for a population parameter. It shows both the size of an estimate and its precision.
Key takeaways
- A confidence interval (CI) = estimate ± critical value × standard error. For a mean:
M ± t_crit · SD / √n. - Correct interpretation: if you repeated the study many times, 95% of the intervals computed this way would contain the true value. It is not a 95% probability that this particular interval contains it.
- A 95% CI for a difference that does not contain 0 corresponds to p < .05 in a two-sided test of that difference.
- Wider intervals mean less precision. Width shrinks with larger samples (roughly with 1/√n), lower variability and a lower confidence level.
- APA 7 recommends reporting confidence intervals for estimates and effect sizes, written as 95% CI [lower, upper].
What is a confidence interval?
When you compute a mean from a sample, you get a point estimate: one number. But another sample would give a slightly different number. A confidence interval turns the point estimate into an interval estimate that reflects this sampling uncertainty.
The general structure is always the same:
CI = estimate ± critical value × standard error
The critical value comes from the sampling distribution (z for proportions and large samples, t for means), and the standard error measures how much the estimate varies from sample to sample. The term “critical value × SE” is called the margin of error.
| Confidence level | z (large samples) | t with df = 9 (n = 10) |
|---|---|---|
| 90% | 1.645 | 1.833 |
| 95% | 1.960 | 2.262 |
| 99% | 2.576 | 3.250 |
Worked example: CI for a mean
Ten students report their weekly study hours: 21, 25, 19, 23, 26, 22, 24, 20, 27, 23. M = 23.00, SD = 2.58, n = 10, df = 9.
| Step | Calculation | Result |
|---|---|---|
| Standard error | 2.58 / √10 | 0.82 |
| Critical t (95%, df = 9) | from the t-distribution | 2.262 |
| Margin of error | 2.262 × 0.82 | 1.85 |
| 95% CI | 23.00 ± 1.85 | [21.15, 24.85] |
| 99% CI | 23.00 ± 3.250 × 0.82 | [20.35, 25.65] |
Worked example: CI for a proportion
In a survey of 300 students, 120 (40%) say they work more than 10 hours per week. For proportions with large samples:
CI = p ± z · √(p(1 − p) / n)
SE = √(0.40 · 0.60 / 300) = 0.028. Margin of error = 1.96 · 0.028 = 0.055.
95% CI: 40% ± 5.5 percentage points, i.e. [34.5%, 45.5%].
More examples: differences and correlations
Confidence intervals exist for almost every statistic. For effects such as differences and correlations, check whether the interval contains the “no effect” value: 0 for differences and correlations, 1 for odds ratios.
| Estimate | Point estimate | 95% CI | Contains “no effect”? |
|---|---|---|---|
| Difference in exam points, study plan vs. none (Welch) | 2.50 | [0.92, 4.08] | No (0 outside), p = .004 |
| Mean drop in exam anxiety, before vs. after training | 3.83 | [2.15, 5.51] | No, p = .002 |
| Correlation of study hours and exam score (n = 10, Fisher z) | .84 | [.44, .96] | No, p = .002 |
| Odds ratio, review session and passing | 3.00 | [1.38, 6.50] | No (1 outside), p = .005 |
How to interpret a confidence interval correctly
The confidence level refers to the method, not to one specific interval. Once computed, a particular interval either contains the true value or it does not.
- Correct: “If we repeated this study many times, 95% of the intervals constructed this way would capture the population mean.”
- Acceptable shorthand: “The data are compatible with population values between 21.15 and 24.85 hours.”
- Wrong: “There is a 95% probability that the true mean lies between 21.15 and 24.85.”
- Wrong: “95% of students study between 21.15 and 24.85 hours.” The CI is about the mean, not about individuals; individual values spread much more widely.
What makes an interval narrow or wide?
Three things determine the width: variability, sample size and confidence level. You control the last two.
- Sample size: the SE falls with √n. To halve the margin of error you need about four times as many participants.
- Variability: a more homogeneous sample or a more reliable measure gives a smaller SD and a narrower interval.
- Confidence level: 99% intervals are wider than 95% intervals, which are wider than 90% intervals.
- Overlap of two CIs: if the 95% CIs of two group means do not overlap, the difference is significant. If they overlap slightly, the difference may still be significant; compute the CI of the difference instead of judging by eye.
How to report confidence intervals in APA style
APA 7 format: the confidence level, “CI”, and the limits in square brackets separated by a comma. After the first mention you may drop “95%” if you use the same level throughout.
Frequently asked questions
What does a 95% confidence interval mean?
It is a range computed by a method that captures the true population value in 95% of repeated samples. It shows the plausible values for the parameter given your data.
How do I calculate a 95% confidence interval for a mean?
Compute the standard error SD / √n, multiply it by the critical t-value for n − 1 degrees of freedom (about 2 for larger samples) and add and subtract the result from the mean.
What is the relationship between confidence intervals and p-values?
For the same test, a 95% CI that excludes the null value (0 for differences, 1 for ratios) corresponds to p < .05 two-sided. The CI additionally shows the size and precision of the effect.
Why is my confidence interval so wide?
Usually because the sample is small or the data vary a lot. Quadrupling the sample size roughly halves the width.
What is the difference between a confidence interval and the margin of error?
The margin of error is the half-width of the interval, the part after the ±. The confidence interval is the full range from estimate minus margin to estimate plus margin.
Sources and further reading
- Agresti, A., & Finlay, B. (2018). Statistical Methods for the Social Sciences (5th ed.). Pearson.
- American Psychological Association. (2020). Publication Manual of the American Psychological Association (7th ed.).
- Howell, D. C. (2013). Statistical Methods for Psychology (8th ed.). Wadsworth Cengage Learning.
Your university’s examination regulations and style guide always take precedence.
Related articles
- p-value and significanceThe p-value is the probability of a result at least as extreme as yours if the null hypothesis were true. p < .05 is usually called significant.
- Standard deviation and varianceThe standard deviation is the typical distance of values from the mean. Variance is its square. Sample formula: SD = √(Σ(x − M)² / (n − 1)).
- Sample sizeFor a survey with 95% confidence and a ±5% margin of error you need about 385 responses. For group comparisons, a power analysis sets the size.
- Effect sizeAn effect size tells you how large a difference or relationship is, independent of sample size. Cohen’s d of 0.2, 0.5 and 0.8 is small, medium, large.
- Reporting statistics in APAAPA style reports test statistic, degrees of freedom, exact p-value and effect size, e.g. t(13.46) = 3.42, p = .004, d = 1.71. Examples for every test.
