A confidence interval gives you a range instead of a single number — which is usually a more honest way to express an estimate, since a single sample statistic is almost never exactly right. Here's the confidence interval formula broken into steps you can follow for any intro stats problem.
What a confidence interval actually tells you
A 95% confidence interval means: if you repeated this sampling process many times and built an interval each time, about 95% of those intervals would contain the true population value. It's a statement about the reliability of the method, not a 95% probability that this specific interval contains the true value.
The step-by-step method
- Find your sample mean (x̄) and sample standard deviation (s).
- Calculate the standard error: SE = s / √n
- Find the critical value for your confidence level. Use a t-value (not z) for most intro-course problems, especially with smaller samples — look it up using your sample size minus 1 (degrees of freedom) and your confidence level.
- Calculate the margin of error: critical value × standard error
- Build the interval: sample mean ± margin of error
Worked example: 95% confidence interval
A sample of 25 students has a mean exam score of 75, with a sample standard deviation of 10. Find the 95% confidence interval for the true average score.
Standard error: SE = 10 / √25 = 10 / 5 = 2
Degrees of freedom: n − 1 = 24. The t-critical value for 95% confidence with 24 df is approximately 2.064.
Margin of error: 2.064 × 2 = 4.13
95% confidence interval: 75 ± 4.13 → (70.87, 79.13)
Interpretation: we're 95% confident the true average score for the full population falls between about 70.9 and 79.1.
Common mistakes
Using z instead of t for small samples. The t-distribution accounts for the extra uncertainty that comes with a smaller sample. Most intro courses expect t unless you're specifically told the population standard deviation is known.
Misinterpreting what "95% confidence" means. It describes the reliability of the method over repeated sampling, not a 95% chance that this specific interval contains the true value.
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