When do I use t instead of z?

Use z when the population standard deviation σ is known (or the sample is very large), and t with n−1 degrees of freedom when σ has to be estimated by the sample standard deviation s. The two converge as n grows and differ little past about n = 30.

What does a p-value actually mean?

The probability of seeing a result at least as extreme as this one if the null hypothesis were true. A small p means the data sit badly with the null, and p < α is the rule for rejecting it — but p is not the probability that the null is true.

What does “95% confidence” mean?

If you repeated the sampling many times, about 95% of the intervals produced this way would contain the true parameter. For one particular interval the parameter is either in it or not, so it is wrong to say the parameter has a 95% chance of being inside.

Why is the Wilson interval better than Wald?

The Wald interval can run past 0 or 1 and under-covers badly when the sample is small or the proportion is near an endpoint. The Wilson interval always stays inside [0, 1] and its coverage stays close to the nominal level, so it is the one to trust.

Why does the two-sample test default to Welch?

The pooled-variance t test assumes both groups have the same variance, which often is not true. Welch’s test drops that assumption and performs virtually identically when the variances are equal, so it is the safer default.

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🔬Confidence Interval Calculator - Means, Proportions, t-test and z-test Online

Q: What does “95% confidence” mean?

If you repeated the sampling many times, about 95% of the intervals produced this way would contain the true parameter. For one particular interval the parameter is either in it or not, so it is wrong to say the parameter has a 95% chance of being inside.

Q: Why is the Wilson interval better than Wald?

The Wald interval can run past 0 or 1 and under-covers badly when the sample is small or the proportion is near an endpoint. The Wilson interval always stays inside [0, 1] and its coverage stays close to the nominal level, so it is the one to trust.

Q: Why does the two-sample test default to Welch?

The pooled-variance t test assumes both groups have the same variance, which often is not true. Welch’s test drops that assumption and performs virtually identically when the variances are equal, so it is the safer default.

Enter the sample mean, standard deviation and size to get a confidence interval or a test result straight away. Means use the t distribution when the population standard deviation is unknown, proportions come with both the Wald and the sturdier Wilson interval, and every test reports the statistic, degrees of freedom, p-value, critical value and decision.

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How to use Confidence Intervals & Tests

1Choose what you need: interval for a mean, interval for a proportion, one-sample test or two-sample test
2Enter the sample statistics (mean, standard deviation, size — or successes and trials)
3Pick the confidence level or significance level α, and the tail for a test
4Read the interval, or the p-value and the conclusion

FAQ

When do I use t instead of z?: Use z when the population standard deviation σ is known (or the sample is very large), and t with n−1 degrees of freedom when σ has to be estimated by the sample standard deviation s. The two converge as n grows and differ little past about n = 30.
What does a p-value actually mean?: The probability of seeing a result at least as extreme as this one if the null hypothesis were true. A small p means the data sit badly with the null, and p < α is the rule for rejecting it — but p is not the probability that the null is true.
What does “95% confidence” mean?: If you repeated the sampling many times, about 95% of the intervals produced this way would contain the true parameter. For one particular interval the parameter is either in it or not, so it is wrong to say the parameter has a 95% chance of being inside.
Why is the Wilson interval better than Wald?: The Wald interval can run past 0 or 1 and under-covers badly when the sample is small or the proportion is near an endpoint. The Wilson interval always stays inside [0, 1] and its coverage stays close to the nominal level, so it is the one to trust.
Why does the two-sample test default to Welch?: The pooled-variance t test assumes both groups have the same variance, which often is not true. Welch’s test drops that assumption and performs virtually identically when the variances are equal, so it is the safer default.