Hypothesis tests show whether a difference in data is larger than chance alone would explain. Run the most common tests from summary statistics and get the test statistic, p-value and decision at your chosen significance level.
How it works
Run the most common significance tests from summary statistics: one-sample and two-sample (Welch) t-tests, one- and two-proportion z-tests, and the chi-square test of independence.
One-sample t = (x̄ − μ₀) ÷ (s ÷ √n), df = n − 1
Welch t = (x̄₁ − x̄₂) ÷ √(s₁²/n₁ + s₂²/n₂), Welch–Satterthwaite df
One-proportion z = (p̂ − p₀) ÷ √(p₀(1 − p₀) ÷ n)
Two-proportion z = (p̂₁ − p̂₂) ÷ √(p̄(1 − p̄)(1/n₁ + 1/n₂))
χ² = Σ (Observed − Expected)² ÷ Expected, df = (rows − 1)(columns − 1)Common uses
- Comparing performance before and after a change
- A/B tests and survey comparisons
- Statistics coursework and research
Worked example
The calculator above opens with this example. Change any input to see the results update instantly, or select Reset to return to these figures.
| Test | Two-sample t-test (Welch, means) |
| Sample 1 mean | 48.2 |
| Sample 1 standard deviation | 9.4 |
| Sample 1 size | 42 |
| Sample 2 mean | 43.6 |
| Sample 2 standard deviation | 10.1 |
| Sample 2 size | 38 |
| Alternative hypothesis | Two-tailed (≠) |
| Significance level (α) | 5% |
| Decision | Reject H₀ |
| p-value | 0.0389 |
| Test statistic (t) | 2.1022 |
| Degrees of freedom | 75.74 |
| Difference in means | 4.60 |
| Standard error | 2.1882 |
The result is statistically significant: data this extreme would be unlikely if the null hypothesis were true. Statistical significance does not measure practical importance; check the size of the effect as well.
Interpreting the results
- Choose the alternative hypothesis before looking at the data; switching to a one-tailed test after the fact inflates false positives.
- Welch’s t-test does not assume equal variances and is a safe default for comparing two means.
- Chi-square results are unreliable when expected counts fall below 5; combine categories or use an exact test.
Frequently asked questions
What does a p-value mean?
The p-value is the probability of observing results at least as extreme as the data if the null hypothesis were true. A small p-value, typically below 0.05, suggests the data are inconsistent with the null hypothesis.
When should I use Welch’s t-test?
Use Welch’s t-test to compare two means when the groups may have different variances or sizes. It does not require the equal-variance assumption of Student’s t-test and performs well in most practical situations.
What does the chi-square test of independence show?
It tests whether two categorical variables, such as payer type and service line, are related. A significant result means the distribution of one variable differs across categories of the other.
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