Advanced T Test Calculator
Example Data Table
| Observation |
Sample A |
Sample B |
Paired Difference |
| 1 | 12 | 10 | 2 |
| 2 | 15 | 11 | 4 |
| 3 | 14 | 13 | 1 |
| 4 | 13 | 12 | 1 |
| 5 | 16 | 14 | 2 |
| 6 | 17 | 15 | 2 |
Formula Used
One Sample Test
t = (x̄ - μ₀) / (s / √n)
df = n - 1
Paired Sample Test
t = (d̄ - μ₀) / (sd / √n)
df = n - 1
Equal Variance Two Sample Test
sp² = [((n₁ - 1)s₁²) + ((n₂ - 1)s₂²)] / (n₁ + n₂ - 2)
t = [(x̄₁ - x̄₂) - μ₀] / √[sp²(1/n₁ + 1/n₂)]
df = n₁ + n₂ - 2
Welch Two Sample Test
t = [(x̄₁ - x̄₂) - μ₀] / √[(s₁²/n₁) + (s₂²/n₂)]
df uses the Welch Satterthwaite adjustment.
How to Use This Calculator
Select the correct test type. Use one sample for one mean. Use paired for matched observations. Use equal variance for independent groups with similar spread. Use Welch when independent group spreads differ.
Choose raw data when you have observations. Separate values with commas, spaces, semicolons, or new lines. Choose summary mode when you know n, mean, and standard deviation.
Enter the null value. For two groups, this is usually zero. Pick the tail direction. Then click calculate. Use CSV or PDF buttons after entering all values.
T Test Overview
A t test compares a sample mean with a target mean, or compares two related or independent means. It is useful when population standard deviation is unknown. The test uses sample variation to judge whether an observed difference is larger than random sampling noise.
When This Calculator Helps
This calculator supports one sample, paired sample, equal variance two sample, and Welch two sample methods. You can enter raw values or summary statistics. Raw values are useful when observations are available. Summary mode is faster when you already know the mean, standard deviation, and sample size.
Important Inputs
Choose the test type first. Then select the alternative hypothesis. A two tailed test checks for any difference. A left tailed test checks whether the estimated mean difference is lower. A right tailed test checks whether it is higher. The confidence level controls the interval width. Higher confidence gives a wider interval.
Understanding Results
The t statistic measures the estimated difference in standard error units. A larger absolute value gives stronger evidence against the null hypothesis. The p value estimates how unusual the result would be if the null hypothesis were true. The degrees of freedom define the reference t distribution. Welch tests use adjusted degrees of freedom because group variances may differ.
Effect Size and Interval
The calculator also reports confidence intervals and common effect size values. Cohen d describes the practical size of the difference. For paired data, the effect is based on difference scores. For independent groups, the effect can use pooled or unpooled spread. Always read effect size with context, sample design, and measurement scale.
Assumptions and Caution
A t test works best with independent observations and roughly normal data, especially for small samples. Large samples are usually more robust. Paired tests require matched observations. Independent tests require separate groups. Equal variance tests assume similar group variance. Welch tests are safer when spread differs. Results should support judgment, not replace domain knowledge.
Reporting the Test
Report the test type, t value, degrees of freedom, p value, confidence interval, and effect size. Mention the alternative hypothesis and confidence level. Keep raw data notes, rounding settings, and exclusion rules with the exported file for transparent review and audits.
FAQs
What is a t test?
A t test checks whether a mean difference is large compared with sample variation. It is often used when the population standard deviation is unknown.
Which t test should I choose?
Use one sample for one mean. Use paired for matched values. Use Welch for separate groups with unequal spread. Use equal variance only when spreads are similar.
What does the p value mean?
The p value shows how unusual your result is under the null hypothesis. Smaller values give stronger evidence against the null hypothesis.
What is the null value?
The null value is the mean difference assumed by the null hypothesis. It is usually zero for paired and two sample comparisons.
Can I enter raw data?
Yes. Enter values separated by commas, spaces, semicolons, or line breaks. The calculator computes mean, standard deviation, and sample size.
Can I use summary statistics?
Yes. Choose summary mode. Then enter n, mean, and standard deviation. For paired summary tests, use difference mean and difference standard deviation.
What is degrees of freedom?
Degrees of freedom define the t distribution shape. Smaller samples have lower degrees of freedom and wider confidence intervals.
What is effect size?
Effect size describes practical difference size. It helps compare results beyond p values, but it should be interpreted with subject knowledge.