Enter your sample mean, standard deviation, and sample size, then pick a confidence level. You'll get the confidence interval for the population mean, plus the margin of error and standard error.
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The interval is symmetric around your sample mean, and its half-width is the margin of error. Precision improves with the square root of the sample size, so quadrupling n only halves the width.
How the confidence interval calculator works
A sample mean is your best single guess at the population mean, but it's almost never exactly right; a different random sample would have given a slightly different average. A confidence interval turns that single guess into an honest range. The calculator finds the standard error (how much sample means typically wobble), multiplies it by a z-value that matches your chosen confidence level, and stretches that margin of error on either side of your sample mean.
The formula
CI = x̄ ± z × ( s ÷ √n )
Here x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and z is the critical value for your confidence level: 1.645 for 90%, 1.960 for 95%, and 2.576 for 99%. The quantity s ÷ √n is the standard error, and z × (s ÷ √n) is the margin of error.
Worked example
You test 36 students and find a mean score of 100 with a standard deviation of 15, and you want 95% confidence:
Standard error = 15 ÷ √36 = 15 ÷ 6 = 2.5
Margin of error = 1.960 × 2.5 = 4.9
CI = 100 ± 4.9 = 95.1 to 104.9. You can be 95% confident (in the repeated-sampling sense) that the true average score of the whole population lies in that range.
What "95% confident" does (and doesn't) mean
The most common misreading of a confidence interval is "there's a 95% chance the true mean is between 95.1 and 104.9." Strictly, that's wrong: the true mean is a fixed (if unknown) number, so it's either in the interval or it isn't. The 95% describes the method: build intervals this way from repeated random samples, and about 95 in 100 of them will capture the truth. Your interval is one draw from that process. Two practical notes: this calculator uses z-values, which is standard practice and accurate for n ≥ 30; for small samples, a t-distribution gives slightly wider intervals that account for the extra uncertainty in estimating s. And the interval only covers sampling error: a biased sample (surveying only your happiest customers, say) produces a beautifully precise interval around the wrong answer.
Frequently asked questions
What does a 95% confidence interval actually mean?
It means the procedure works 95% of the time: if you repeated your sampling over and over and built an interval each time, about 95% of those intervals would contain the true population mean. It does not mean there's a 95% probability the true mean is inside this particular interval; the true mean is a fixed number, not a random one.
How do I make my confidence interval narrower?
Three levers: collect a bigger sample (the margin of error shrinks with the square root of n, so quadrupling the sample halves the interval), accept a lower confidence level (90% is narrower than 99%), or reduce measurement noise so the standard deviation is smaller.
What z-values go with each confidence level?
The standard multipliers are 1.645 for 90% confidence, 1.960 for 95%, and 2.576 for 99%. These come from the standard normal distribution: for example, 95% of the distribution lies within 1.96 standard deviations of the mean.
When should I use a t-distribution instead of z?
When your sample is small (under about 30) and you're estimating the standard deviation from the sample itself, the t-distribution gives slightly wider, more honest intervals. Above n = 30 the two are nearly identical, so the z-based interval this calculator produces is a fine approximation.
What is the margin of error?
It's the half-width of the confidence interval, the amount you add to and subtract from the sample mean. It equals the z-value times the standard error (standard deviation divided by the square root of the sample size). A poll reported as "52% ± 3%" is quoting a margin of error of 3 percentage points.