Test Result
Calculator Inputs
Detailed Statistics
| Statistic | Group 1 | Group 2 |
|---|
Step-by-Step Calculation
T Distribution View
The vertical line marks the calculated t statistic. Shaded interpretation is described by the selected tail and alpha.
Formula Used
Welch’s test uses a standard error that allows unequal variances.
Its degrees of freedom use the Welch–Satterthwaite approximation.
The pooled test assumes equal population variances.
How to Use This Calculator
Choose raw observations when individual measurements are available. Paste values for each independent group. The calculator computes sample statistics automatically. Choose summary statistics when only sample size, mean, and standard deviation are known. Select Welch’s test when equal population variances are uncertain. Select the pooled test only when equal variance is defensible.
Next, choose the alternative hypothesis. A two-tailed test detects differences in either direction. A right-tailed test evaluates whether Group 1 has a larger population mean. A left-tailed test evaluates whether Group 1 has a smaller population mean. Set alpha before calculating. Alpha represents the chosen Type I error threshold.
After calculation, review the t statistic, degrees of freedom, and p-value. The tool also displays a confidence interval for the mean difference. Cohen’s d and Hedges’ g provide standardized effect-size estimates. Statistical significance does not automatically indicate practical importance, so effect size and context remain important.
Understanding Independent T Tests
An independent t test compares the means of two unrelated groups. Examples include treatment and control groups, two classrooms, or separate production lines. Observations in one group should not be paired with observations in the other group. When the same people are measured twice, a paired t test is usually more appropriate.
Welch’s independent t test is widely useful because it does not require equal population variances. It adjusts the standard error and degrees of freedom using each sample variance. The pooled Student test combines both sample variances into one estimate. That approach can be efficient when the equal-variance assumption is reasonable.
The p-value measures compatibility between observed data and the null hypothesis under the selected test model. A small p-value indicates that the observed mean difference would be unusual if the null hypothesis were true. The confidence interval adds useful scale information. An interval excluding zero corresponds to statistical significance for a matching two-sided alpha level.
Sample size affects precision strongly. Larger samples usually reduce standard errors. Extreme outliers can distort means and standard deviations. For small samples, severe non-normality may also affect results. Inspect data quality before interpreting conclusions. Confidence intervals and effect sizes should accompany significance testing whenever possible. This combination gives a fuller statistical picture for decisions.
Example Table
| Example | n₁ | Mean₁ | SD₁ | n₂ | Mean₂ | SD₂ | Recommended method |
|---|---|---|---|---|---|---|---|
| Class scores | 20 | 78.4 | 8.1 | 22 | 73.5 | 10.2 | Welch |
| Machine output | 30 | 51.2 | 4.4 | 30 | 49.8 | 4.1 | Pooled if equal variance is justified |
| Treatment response | 14 | 23.1 | 5.7 | 11 | 18.6 | 8.9 | Welch |
Frequently Asked Questions
What does an independent t test measure?
It tests whether two independent population means differ, using sample data and estimated sampling variability.
When should I use Welch’s test?
Use Welch’s test when equal population variances are uncertain. It also works well with unequal sample sizes.
When is the pooled test appropriate?
The pooled version is appropriate when independent samples reasonably satisfy an equal-variance assumption.
What does p < 0.05 mean?
It means the observed result is statistically significant at alpha 0.05 under the selected test model. It does not measure effect importance.
Can sample sizes be different?
Yes. Both Welch and pooled independent t tests can handle unequal sample sizes.
What is Cohen’s d?
Cohen’s d standardizes the mean difference using a pooled standard deviation, helping describe effect magnitude on a common scale.
Why calculate Hedges’ g?
Hedges’ g applies a small-sample correction to Cohen’s d and can reduce bias when sample sizes are limited.
What assumptions matter?
Groups should be independent, observations should be reasonably representative, and severe outliers should be investigated. The pooled test additionally assumes equal variances.
Can I export my results?
Yes. Use the CSV button for structured data or the PDF button for a formatted statistical report.