T-Value Calculator
Calculate the t-statistic (t-value) for hypothesis testing. The t-value measures how many standard errors the sample mean is from the hypothesized population mean. Used in t-tests to determine statistical significance when the population standard deviation is unknown and sample sizes are small.
T-Value Formulas
One-sample: t = (x̄ - μ₀) / (s / √n)
Two-sample: t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)
Paired: t = d̄ / (s_d / √n)
T-Distribution Critical Values Table
| df | α=0.10 (90%) | α=0.05 (95%) | α=0.01 (99%) |
|---|---|---|---|
| 1 | 6.314 | 12.706 | 63.657 |
| 5 | 2.015 | 2.571 | 4.032 |
| 10 | 1.812 | 2.228 | 3.169 |
| 15 | 1.753 | 2.131 | 2.947 |
| 20 | 1.725 | 2.086 | 2.845 |
| 25 | 1.708 | 2.060 | 2.787 |
| 30 | 1.697 | 2.042 | 2.750 |
| 50 | 1.676 | 2.009 | 2.678 |
| 100 | 1.660 | 1.984 | 2.626 |
| ∞ | 1.645 | 1.960 | 2.576 |
Frequently Asked Questions
What does the t-value tell you?
The t-value measures how far your sample statistic is from the null hypothesis value, in units of standard error. A larger absolute t-value means stronger evidence against the null hypothesis.
When do I use a t-test vs z-test?
Use a t-test when the population standard deviation is unknown (you estimate it from the sample) or when sample size is small (n < 30). Use a z-test when the population standard deviation is known and n ≥ 30.
What are degrees of freedom?
Degrees of freedom (df) represent the number of independent values that can vary. For a one-sample t-test, df = n - 1. More degrees of freedom means the t-distribution approaches the normal distribution.
How do I determine statistical significance?
Compare your calculated t-value to the critical value from the t-table at your chosen significance level (α). If |t| > t_critical, reject the null hypothesis. Alternatively, calculate the p-value and compare to α.
What is a paired t-test?
A paired t-test compares two related measurements (before/after, matched pairs). Instead of comparing two independent groups, it tests whether the mean difference between paired observations is significantly different from zero.
T-Test vs Z-Test: When to Use Each
The t-test and z-test both assess whether a sample mean differs significantly from a hypothesized value, but they apply in different situations. Use a t-test when the population standard deviation (σ) is unknown and must be estimated from the sample, or when sample size is small (n < 30). Use a z-test when σ is known and n ≥ 30. In practice, the t-test is far more common because population parameters are rarely known. As degrees of freedom increase beyond 30, the t-distribution converges to the normal distribution, making t-test and z-test results nearly identical.
Interpreting Your T-Value
After calculating the t-statistic, determine significance by comparing it to critical values:
- Calculate degrees of freedom (df = n - 1 for one-sample)
- Choose significance level α (commonly 0.05 for 95% confidence)
- Look up the critical t-value in the table above for your df and α
- If |t_calculated| > t_critical, reject the null hypothesis — the result is statistically significant
- If |t_calculated| ≤ t_critical, fail to reject the null hypothesis
Real-World Applications of T-Tests
- Medical research: Testing if a new drug lowers blood pressure more than a placebo (two-sample t-test)
- A/B testing: Determining if a website redesign increases conversion rates (two-sample t-test)
- Quality control: Checking if manufactured parts meet specification tolerances (one-sample t-test)
- Education: Comparing test scores before and after a teaching intervention (paired t-test)
- Psychology: Measuring if therapy reduces anxiety scores (paired t-test, pre/post design)
- Agriculture: Comparing crop yields between two fertilizer treatments (two-sample t-test)
Assumptions of the T-Test
For valid results, t-tests require: (1) data is continuous (interval or ratio scale), (2) sample is randomly selected from the population, (3) data is approximately normally distributed (less critical for n > 30 due to the Central Limit Theorem), and (4) for two-sample tests, equal variances (use Welch's t-test if variances are unequal). Violations of normality can be addressed using non-parametric alternatives like the Mann-Whitney U test or Wilcoxon signed-rank test.