A confidence interval is a range of values, calculated from your sample data, that likely contains the true value you're trying to estimate in the whole population — a 95% confidence interval means that if you repeated your study many times, about 95% of the intervals you calculated would capture the true value. If you've ever reported a mean or a difference between groups and wondered "but how sure am I about this number?", the confidence interval is the answer your reviewer or supervisor is looking for.
Key Takeaways
- A confidence interval (CI) gives a range of plausible values for a population parameter, not a single point estimate — it communicates uncertainty.
- A 95% confidence level is the most common in psychology, medicine, and social science; it means the method captures the true value 95% of the time over repeated sampling.
- A wider interval means more uncertainty; a narrower interval means more precision. Larger samples produce narrower intervals.
- If a 95% CI for a difference between two groups does not include zero, the result is statistically significant at p < .05.
- APA 7 requires confidence intervals for most estimates, written as 95% CI [lower, upper].
What does a confidence interval actually tell you?
A confidence interval tells you how precise your estimate is. When you measure something in a sample — say, the average anxiety score of 40 students — you only have an estimate of the true average for all students. Your sample mean might be 22.5, but the real population mean could be a bit higher or lower. The confidence interval draws a boundary around your estimate and says, "the true value is plausibly somewhere in here."
Think of it like a weather forecast. A point estimate is "it will be 20°C tomorrow." A confidence interval is "it will be between 17°C and 23°C." The second version is more honest because it admits the natural uncertainty in prediction.
The confidence level (usually 95%) describes the reliability of the method, not the probability that one specific interval is correct. This is the single most misunderstood point in statistics, so it's worth stating plainly.
What does a 95% confidence interval mean, exactly?
A 95% confidence interval means that if you repeated your study 100 times with fresh random samples, roughly 95 of the resulting intervals would contain the true population value. It does not mean there's a 95% probability that the true value sits inside your particular interval — once you've calculated it, the true value either is or isn't inside it.
In everyday research writing, though, people interpret it loosely as "I'm 95% confident the true value falls in this range," and that's an acceptable shorthand as long as you understand the technical meaning underneath.
How do you calculate a confidence interval?
The general formula for a confidence interval around a mean is:
CI = sample estimate ± (critical value × standard error)
The three ingredients:
- Sample estimate — your mean, difference, or proportion.
- Critical value — a number from the t or z distribution tied to your confidence level (for 95% with a large sample, it's about 1.96).
- Standard error — how much your estimate would wobble from sample to sample; it shrinks as your sample size grows.
Because the standard error gets smaller with bigger samples, larger studies produce narrower, more precise confidence intervals. This is why sample size matters so much.
A worked example with real numbers
Suppose you measure reaction times in a sample of 30 participants and find a mean of 450 ms with a standard deviation of 60 ms.
Step 1 — Calculate the standard error:
SE = SD / √n = 60 / √30 = 60 / 5.48 = 10.95 ms
Step 2 — Find the critical value. With 29 degrees of freedom (n − 1), the t critical value for 95% is about 2.045.
Step 3 — Build the margin of error:
2.045 × 10.95 = 22.4 ms
Step 4 — Add and subtract from the mean:
450 − 22.4 = 427.6
450 + 22.4 = 472.4
So your result is M = 450 ms, 95% CI [427.6, 472.4]. You can now say the true average reaction time in the population is plausibly between roughly 428 and 472 ms. If you'd tested 300 people instead of 30, that interval would shrink to something like [443, 457] — much tighter, because larger samples reduce uncertainty.
Running this by hand is fine for one mean, but once you're comparing groups or estimating effect sizes, the arithmetic gets error-prone. Tools like StatRyx calculate the CI, choose the correct distribution, and format it for you automatically.
How do you know if a confidence interval is "significant"?
For a difference between two groups or a correlation, check whether the confidence interval includes zero.
- If a 95% CI for a mean difference is [2.1, 8.4], it does not include zero → the difference is statistically significant at p < .05.
- If the 95% CI is [−1.3, 5.7], it does include zero → the result is not significant; a "no difference" outcome is still plausible.
This is why confidence intervals are more informative than a bare p-value: they tell you not only whether an effect exists, but how big it might be and how precisely you've pinned it down.
Confidence interval vs p-value: what's the difference?
| Feature | Confidence Interval | p-value |
|---|---|---|
| What it shows | Range of plausible values for the effect | Probability of your data if the null were true |
| Effect size info | Yes — you see magnitude and direction | No — only a threshold decision |
| Precision info | Yes — width shows uncertainty | No |
| Answers "is it significant?" | Yes (does it include zero?) | Yes (is it below .05?) |
| APA 7 preference | Strongly encouraged for all estimates | Reported, but not sufficient alone |
APA 7 explicitly favours confidence intervals alongside effect sizes, because a p-value on its own hides how large or precise the effect actually is.
How do you report a confidence interval in APA 7 format?
APA 7 formats confidence intervals inside square brackets, with the confidence level stated once and no repeated units inside the brackets. A few correct examples:
- Single mean: M = 450, 95% CI [427.6, 472.4]
- Mean difference: The intervention group scored higher, M~diff~ = 5.2, 95% CI [2.1, 8.3].
- Correlation: r = .42, 95% CI [.18, .61].
- Regression coefficient: b = 1.35, 95% CI [0.60, 2.10].
Note the APA conventions: statistics like M, r, and b are italicised, and p values drop the leading zero (p = .019, not 0.019). Confidence bounds keep their leading zeros because they're not probabilities.
Which confidence level should I use — 90%, 95%, or 99%?
Use 95% by default — it's the standard across psychology, medicine, and the social sciences. A 99% interval is wider (more cautious, used when the cost of being wrong is high, such as clinical trials), while a 90% interval is narrower but riskier. Don't pick a level after seeing your data; choose it in advance to keep your analysis honest.
If you're not sure which interval or test your data calls for, our related guide on choosing the right statistical test walks through it — or StatRyx will detect it from your dataset automatically.
Stop calculating this by hand — run it free in StatRyx → Try StatRyx
Frequently Asked Questions
What does a 95% confidence interval actually mean?
A 95% confidence interval means that if you repeated your study many times with new random samples, about 95% of the intervals you calculated would cont