A test statistic is located on its null reference distribution. The selected tail rule defines results at least as extreme.
H₀ referenceobserved statistic
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Statistics result
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Two-Sample t-Test. The observed effect is scaled by its null-model uncertainty and located on a reference distribution selected before looking at the result.
1State H₀, the effect, and direction2Calculate the statistic under H₀3Report p-value with effect and interval
Test statistic and reference model
t=(x̄₁−x̄₂)/√(s₁²/n₁+s₂²/n₂).
Welch’s method avoids assuming equal variances but still relies on independent, defensibly sampled observations.
Responsible interpretation
A p-value is not the probability that the null is true, the chance the result occurred ‘by luck,’ or a measure of effect size, importance, replication, bias, or causation.
Required assumption: Do not use for paired measurements; use the paired t procedure on within-pair differences instead.
Quick guide
How to use this calculator
Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
Read the result together with its assumptions and interpretation. Statistical output summarizes uncertainty under a model; it does not repair biased data or establish causation.
Calculation method
How the two-sample t-test calculator works
t=(x̄₁−x̄₂)/√(s₁²/n₁+s₂²/n₂).
Welch’s method avoids assuming equal variances but still relies on independent, defensibly sampled observations.
Worked example
Two-Sample t-Test example
The statistic compares the observed mean difference with its estimated standard error.
t=(x̄₁−x̄₂)/√(s₁²/n₁+s₂²/n₂).
Supported inputs
Precision and limits
Model and design
Do not use for paired measurements; use the paired t procedure on within-pair differences instead.
Numerical scope
Inputs use double-precision numerical methods with guarded domains. Datasets accept up to 10,000 finite plain-decimal values. Extremely large parameters or probabilities deep in a numerical tail may require specialist statistical software.
Interpretation
Welch’s method avoids assuming equal variances but still relies on independent, defensibly sampled observations.
Decision boundary
The calculator does not validate how data were collected, diagnose dependence or bias, choose a scientifically meaningful effect, or replace review by a qualified statistician for consequential research, medical, regulatory, safety, or policy decisions.
Privacy
Entered values and calculated results stay in this browser and are not sent to an analytics service.
