Your sample mean is 72, but what is the likely value in the whole population? A single number hides the uncertainty that comes from sampling. A confidence interval shows it: a range around the estimate that tells readers how precise your result is. APA style recommends reporting one with most estimates, yet the confidence interval is also one of the most misread statistics in theses. This guide explains what it means, how to compute it, and how to report it.
This article is part of our guide How to Write a Master’s Thesis by the Mastermind PhD team and relates to our Statistical Analysis service. Sources last checked: November 2026.
What is a confidence interval? A confidence interval (CI) is a range calculated from sample data that, under the method used, would contain the true population parameter in a stated percentage of repeated samples, usually 95%. It is written as the estimate followed by the lower and upper limits, for example M = 72.0, 95% CI [67.9, 76.1].
How do you calculate a 95% confidence interval for a mean?
The formula is:
CI = mean ± t × (SD / √n)
where t is the critical value from the t distribution with n − 1 degrees of freedom, SD the sample standard deviation and n the sample size. The part SD / √n is the standard error.
Worked example (illustrative data): 25 students, mean score 72, SD 10.
| Quantity | Value |
|---|---|
| Standard error | 10 / √25 = 2.0 |
| Critical t (24 df, 95%) | 2.064 |
| Margin of error | 2.064 × 2.0 = 4.13 |
| 95% confidence interval | [67.87, 76.13] |
In SPSS, Analyze > Descriptive Statistics > Explore prints the 95% confidence interval for the mean; t-test and regression output include intervals for differences and coefficients.
What does a 95% confidence interval really mean?
The 95% describes the method, not one particular interval. If you repeated the study many times and computed an interval each time, about 95% of those intervals would contain the true population mean. Any single interval either contains it or does not. In practice, Cumming (2014) suggests reading the interval as the range of values reasonably compatible with the data: values near the center are more plausible than values near the limits.
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What are the 5 common misreadings?
Hoekstra and colleagues (2014) found that researchers and students endorsed false statements about a confidence interval at high rates. The common ones:
- “There is a 95% probability that the true value lies in this interval.” Not in the frequentist sense; the probability refers to the procedure.
- “95% of the sample data fall within the interval.” No; the interval is about the parameter, not individual scores.
- “A wider interval means a bigger effect.” No; width reflects precision, driven mainly by sample size and variability.
- “Overlapping intervals mean no significant difference.” Not necessarily; two 95% CIs can overlap a little while the difference is still significant. Test the difference, or report the CI of the difference.
- “The interval for a replication will be the same.” Intervals vary from sample to sample; that variability is exactly what they describe.
How is a confidence interval related to the p-value?
For many tests they agree: if the 95% confidence interval for a difference excludes zero (or a ratio excludes 1), the two-sided p-value is below .05. The interval adds what the p-value lacks: the size and precision of the effect. Greenland and colleagues (2016) recommend interpreting intervals as estimates rather than as significance tests in disguise.
| What you want to know | p-value | Confidence interval |
|---|---|---|
| Is the result compatible with zero effect? | Yes | Yes |
| How large is the effect? | No | Yes |
| How precise is the estimate? | No | Yes |
How do you report a confidence interval in APA style?
APA 7 uses square brackets and repeats the confidence level the first time: “Participants in the intervention group scored higher, mean difference = 5.4, 95% CI [2.1, 8.7], t(58) = 3.29, p = .002.” After the first mention, you may write the interval as [2.1, 8.7] if the level stays the same. Use the same number of decimals as the estimate, and include intervals for effect sizes where possible.
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Frequently asked questions
Why 95% and not 99%?
95% is a convention matching the .05 significance level. A 99% confidence interval is wider; use it if your field or multiple testing calls for it, and state the level.
How can I make my confidence interval narrower?
Increase the sample size, reduce measurement error, or use a more precise design. Lowering the confidence level also narrows it but reduces coverage.
Can a confidence interval include negative values?
Yes, for differences or coefficients. For a difference, an interval that includes zero means the data are compatible with no difference as well as with the effects in the interval.
Do I need a confidence interval for every result?
APA recommends intervals for point estimates and effect sizes. In a thesis, report them at least for the main hypotheses.
What is the difference between a confidence interval and a standard error?
The standard error measures the sampling variability of an estimate. The interval uses it, multiplied by a critical value, to build a range.
Sources
- Cumming, G. (2014). The new statistics: Why and how. Psychological Science, 25(1), 7-29. https://doi.org/10.1177/0956797613504966
- Hoekstra, R., Morey, R. D., Rouder, J. N., & Wagenmakers, E.-J. (2014). Robust misinterpretation of confidence intervals. Psychonomic Bulletin & Review, 21(5), 1157-1164. https://doi.org/10.3758/s13423-013-0572-3
- Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31(4), 337-350. https://doi.org/10.1007/s10654-016-0149-3
- American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://apastyle.apa.org/
- Altman, D. G., Machin, D., Bryant, T. N., & Gardner, M. J. (Eds.). (2000). Statistics with confidence (2nd ed.). BMJ Books.
Sources last checked: November 2026. This article is general educational guidance and does not replace your university regulations or your supervisor’s instructions. How we prepare our content.