t-Statistic Calculator

Compute the t-statistic from your sample data to use in hypothesis testing.

📖 What is the t-Statistic?

The t-statistic is the standardised test statistic used in Student's t-test. It measures the size of the difference relative to the variation in the sample data. Specifically, it tells you how many standard errors the observed sample mean is from the null hypothesis mean.

The formula t = (x̄ − μ₀) / (s/√n) has three components: the numerator (x̄ − μ₀) is the raw difference between the sample mean and the null hypothesis value; the denominator s/√n is the standard error of the mean (SEM) - a measure of how precisely the sample mean estimates the population mean. A t-statistic of 2.5 means the sample mean is 2.5 standard errors above the null hypothesis value.

The t-statistic is compared to the t-distribution (not the normal distribution) because we are using the estimated standard deviation s rather than the known population σ. The t-distribution has heavier tails than the normal distribution, reflecting the extra uncertainty from estimating σ. As sample size increases, the t-distribution approaches the standard normal distribution.

After computing the t-statistic, the next step is to find the p-value or compare to the critical value to make a conclusion about the hypothesis test. Use the p-Value Calculator or t-Test Calculator for the complete test.

📐 Formula

t = (x̄ − μ₀) / SE, where SE = s / √n

x̄ = sample mean | μ₀ = null hypothesis mean

s = sample standard deviation | n = sample size

SE (standard error) = s / √n = standard deviation of the sampling distribution of x̄

Degrees of freedom: df = n − 1 (one-sample and paired)

Two-sample pooled: t = (x̄₁ − x̄₂) / [sp√(1/n₁+1/n₂)], df = n₁+n₂−2

Welch's: t = (x̄₁−x̄₂)/√(s₁²/n₁+s₂²/n₂), df from Welch-Satterthwaite equation

📖 How to Use This Calculator

Select the test type: one-sample (compare to reference mean), two-sample pooled, Welch's (unequal variances), or paired.

Enter the sample statistics. For one-sample, input x̄, s, n, and μ₀. For two-sample, enter statistics for both groups. For paired, enter mean and SD of differences.

Click Calculate t-Statistic. The t-value, standard error, and degrees of freedom are shown.

Use the t-statistic and df in the p-Value Calculator or t-Test Calculator to complete the hypothesis test.

📝 Example Calculations

Example 1 - One-Sample

x̄ = 52.3, s = 8.4, n = 25, μ₀ = 50. SE = 8.4/√25 = 1.68. t = (52.3−50)/1.68 = 1.369, df = 24.

t = 1.369 (df = 24)

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Example 2 - Two-Sample Pooled

Group 1: x̄=68, s=10, n=30. Group 2: x̄=62, s=11, n=28. sp = √[(29×100+27×121)/56] = √108.9 = 10.44. SE = 10.44×√(1/30+1/28) = 2.76. t = (68−62)/2.76 = 2.17, df = 56.

t = 2.17 (df = 56)

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Example 3 - Paired

d̄ = 4.2, s_d = 6.1, n = 20 pairs. SE = 6.1/√20 = 1.364. t = 4.2/1.364 = 3.08, df = 19.

t = 3.08 (df = 19)

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Example 4 - Welch's

Group 1: x̄=105, s=15, n=18. Group 2: x̄=95, s=25, n=12. SE = √(225/18+625/12) = √(12.5+52.08) = 8.04. t = (105−95)/8.04 = 1.24.

t = 1.24

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❓ Frequently Asked Questions

What is a t-statistic?+

The t-statistic (also called the t-value) measures how many standard errors the sample mean (or difference in means) is away from the null hypothesis value. A t-statistic of 0 means the sample mean equals the null value. A large |t| (positive or negative) indicates the sample mean is far from the null, providing evidence to reject H₀.

How do you calculate the t-statistic?+

For a one-sample test: t = (x̄ − μ₀) / (s/√n), where x̄ is the sample mean, μ₀ is the null hypothesis mean, s is the sample standard deviation, and n is the sample size. The denominator s/√n is the standard error of the mean (SEM).

What does the sign of the t-statistic mean?+

A positive t-statistic means the sample mean is above the null hypothesis mean. A negative t-statistic means it is below. For a two-tailed test, only the absolute value |t| matters. For a one-tailed test, the sign determines which tail to use.

What is the standard error of the mean?+

The standard error (SE) = s/√n is the standard deviation of the sampling distribution of the mean. It measures how much the sample mean is expected to vary across repeated samples. A smaller SE (from a larger sample or less variable data) makes it easier to detect true differences.

How large does the t-statistic need to be to reject H₀?+

Compare |t| to the critical value t_(α/2, df). For df = 20 at α = 0.05 (two-tailed), the critical value is 2.086 - reject H₀ if |t| > 2.086. For large df (> 30), the critical value approaches 1.96 (the normal distribution value). Use the Critical Value Calculator to find the exact threshold.

What is the difference between t-statistic and t-score?+

In hypothesis testing, 't-statistic' usually refers to the test statistic used to test a hypothesis about a population mean. 'T-score' sometimes refers to a standardised score on a scale with mean 50 and SD 10 (used in educational and psychological testing). The t-Test Calculator uses the test statistic definition.

t-Statistic Calculator

📖 What is the t-Statistic?

📐 Formula

📖 How to Use This Calculator

📝 Example Calculations

❓ Frequently Asked Questions

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📌 Quick Tips