T-Test Calculator
Use the T-Test Calculator to compare the means of two independent samples. This tool performs Welch's t-test (unequal variances) and returns the t statistic, degrees of freedom, and a two-tailed p-value. Enter each sample as a comma-separated list of numbers. Results update automatically as you edit inputs.
When to use a t-test
A t-test is used when you want to compare the means of two groups to determine whether they are statistically different from each other. Welch's t-test is recommended when the two samples may have different variances and/or sizes. Researchers frequently apply this method in various fields such as psychology, medicine, and economics when analyzing quantitative data.
This calculator performs Welch's t-test by default. It computes sample means and variances, the t statistic, Welch-Satterthwaite degrees of freedom, and a two-tailed p-value. T-tests are crucial for hypothesis testing and help in drawing significant conclusions.
How the calculation works
Given two samples x and y:
- Compute sample means: mean(x), mean(y)
- Compute sample variances (sample variance with n-1 denominator)
- t statistic: (mean(x) - mean(y)) / sqrt(var(x)/n_x + var(y)/n_y)
- Degrees of freedom (Welch-Satterthwaite):df = ( (s1^2 / n1 + s2^2 / n2)^2 ) / ( (s1^4 / (n1^2 * (n1 - 1))) + (s2^4 / (n2^2 * (n2 - 1))) )
- Two-tailed p-value is computed from the Student's t distribution using the calculated t and df.
The calculator uses numerical integration to evaluate the cumulative distribution of the t statistic and returns the two-tailed p-value. Statistical software often uses similar methods, but this calculator offers an accessible online option.
Example of a t-test calculation
Suppose you have two samples:
- Sample 1: 10, 12, 14, 16, 18
- Sample 2: 15, 17, 19, 21, 23
Paste each sample into the corresponding input field. The calculator will display means, variances, sample sizes, the t statistic, degrees of freedom, and the two-tailed p-value. Use the chart toggle to visualize the sample means. Understanding these results provides valuable insight into the comparative performance of the two samples.
Interpreting the results from the t-test
A small p-value (typically < 0.05) indicates evidence against the null hypothesis that the two population means are equal. A large p-value suggests insufficient evidence to reject the null hypothesis. Importantly, results should be contextualized within the scope of study.
Note: Statistical significance does not imply practical significance. Consider effect sizes and confidence intervals when making decisions. These additional metrics can help interpret the relevance of your results in practical terms.
Frequently Asked Questions about t-tests
Can I use this for paired data?
No — this calculator performs an independent two-sample t-test (Welch's). For paired samples, calculate the differences and perform a one-sample t-test on the differences.
What if my samples have different sizes?
Welch's t-test accounts for unequal variances and sample sizes; it is appropriate for samples with different n.
Why might the calculator return no result?
Ensure each sample has at least two numeric values. Values must be comma-separated numbers (e.g., "1, 2, 3"). If the inputs are valid and you still encounter issues, refresh the page and try again.
Can I use this calculator for one-sample tests?
This calculator is designed for independent two-sample t-tests. For one-sample tests, consider the specifics of your data and statistical needs.
References & further reading
- Welch, B.L. (1947). The generalisation of Student's problem when several different population variances are involved. Biometrika.
- Student. (1908). The probable error of a mean. Biometrika.
- Moore, D.S., & McCabe, G.P. (2006). Introduction to the Practice of Statistics. New York: W.H. Freeman.
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Thousand Oaks, CA: Sage Publications.