T-Test Calculator

One-Sample T-TestTwo-Sample T-Test

Sample Data (comma or space separated)

Hypothesized Mean (μ₀)

Significance Level (α)0.10 (90% confidence)0.05 (95% confidence)0.01 (99% confidence)

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What is a T-Test?

A t-test is a statistical test used to compare means. It helps determine whether there’s a significant difference between group means or between a sample mean and a known value.

Types of T-Tests

One-Sample T-Test

Compares a sample mean to a known or hypothesized population mean.

Use when: Testing if your sample differs significantly from a standard or expected value.

Two-Sample T-Test (Independent)

Compares the means of two independent groups.

Use when: Comparing two separate groups (e.g., treatment vs. control).

Paired T-Test

Compares means from the same group at different times or under different conditions.

Use when: Comparing before/after measurements on the same subjects.

T-Test Formulas

One-Sample T-Test

t = (x̄ - μ₀) / (s / √n)

Two-Sample T-Test (Welch’s)

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Understanding the Results

• T-statistic: Measures the size of the difference relative to variation
• Degrees of freedom: Related to sample size(s)
• P-value: Probability of observing results if null hypothesis is true
• Significance: If p-value < α (e.g., 0.05), the result is statistically significant

Assumptions of T-Tests

1. Data is approximately normally distributed
2. Random sampling from the population
3. Independence of observations
4. For two-sample: Similar variances (unless using Welch’s t-test)

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One-Sample T-Test: When and How

A one-sample t-test compares a sample mean to a known or hypothesized population value. Use it when you want to test whether your data differs from a specific standard.

Example: A manufacturer claims light bulbs last 1,000 hours. You test 25 bulbs and find a mean of 980 hours with s = 40. Is the mean significantly different from 1,000?

1. H₀: μ = 1,000 hours
2. H₁: μ ≠ 1,000 hours
3. t = (980 − 1000) / (40 / √25) = −20 / 8 = −2.50
4. df = 25 − 1 = 24
5. From the t-table: t-critical at α=0.05 (two-tailed) = 2.064
6. Since |−2.50| > 2.064, reject H₀ — the bulbs last significantly less than 1,000 hours.

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Paired T-Test: Before-and-After Comparisons

A paired t-test (dependent samples t-test) compares two related measurements on the same subjects — for example, before and after a treatment.

Example: 12 students took a math test, then completed a tutoring program, and took the test again.

Student	Before	After	Difference (d)
1	72	78	+6
2	65	70	+5
3	80	82	+2
…	…	…	…

1. Calculate the mean difference: d̄ = 4.2
2. Calculate the SD of differences: s_d = 2.1
3. t = d̄ / (s_d / √n) = 4.2 / (2.1 / √12) = 6.93
4. df = 12 − 1 = 11
5. This is highly significant — the tutoring program worked.

When to use: Pre/post measurements, matched pairs, same subjects under two conditions.

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Independent Two-Sample T-Test: Comparing Two Groups

An independent (two-sample) t-test compares means from two separate, unrelated groups.

Example: A pharmaceutical company tests a new drug against a placebo.

• Drug group (n₁ = 30): mean = 8.2, s₁ = 1.5
• Placebo group (n₂ = 30): mean = 6.8, s₂ = 1.8

1. H₀: μ₁ = μ₂ (no difference between groups)
2. H₁: μ₁ ≠ μ₂
3. t = (8.2 − 6.8) / √(1.5²/30 + 1.8²/30) = 1.4 / 0.429 = 3.26
4. Degrees of freedom (Welch’s): df ≈ 56
5. p-value < 0.01 — the drug shows a significant effect.

When to use: Treatment vs. control, comparing two independent groups, A/B testing results.

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Which T-Test Should You Use?

Scenario	Test	df
Compare sample to known value	One-sample	n − 1
Before/after on same subjects	Paired	n − 1
Two independent groups	Two-sample (Welch’s)	Complex formula
Two groups, equal variances assumed	Two-sample (pooled)	n₁ + n₂ − 2

Not sure? Start with How to Choose the Right Statistical Test.

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Frequently Asked Questions

What is a t-test used for?

A t-test is used to compare means and determine if there’s a statistically significant difference between them. Use a one-sample t-test to compare a sample mean to a known value, two-sample to compare two independent groups, and paired to compare before/after measurements.

How do I calculate the t-value?

For a one-sample t-test: t = (sample mean - hypothesized mean) / (sample SD / √n). For two-sample: t = (mean₁ - mean₂) / √(s₁²/n₁ + s₂²/n₂). This calculator does the computation automatically.

What is a good t-value?

There’s no single “good” t-value—it depends on degrees of freedom and significance level. Generally, |t| > 2 suggests significance at α=0.05 for moderate sample sizes. Compare your t-value to the critical value from the t-table.

What does the p-value mean in a t-test?

The p-value is the probability of getting results as extreme as yours if there’s actually no difference (null hypothesis is true). p < 0.05 means there’s less than 5% chance the difference is due to random sampling—typically considered statistically significant.

When should I use a paired t-test vs independent t-test?

Use a paired t-test when the same subjects are measured twice (before/after, matched pairs). Use an independent t-test when comparing two separate groups with different subjects. The paired test is more powerful because it controls for individual differences.

When should I use t-test vs z-test?

Use t-test when population standard deviation is unknown (most real-world situations). Use z-test when population σ is known or sample size is very large (n > 100). The t-test is more conservative and appropriate for smaller samples.

What is the difference between one-tailed and two-tailed t-test?

A two-tailed test checks if means are different (either direction). A one-tailed test checks if one mean is specifically greater or less than the other. Use two-tailed unless you have a strong directional hypothesis before collecting data.

What is Welch’s t-test?

Welch’s t-test is a two-sample t-test that does not assume equal variances between groups. It adjusts degrees of freedom based on sample sizes and variances, making it more reliable than the pooled t-test when group sizes or variances differ.

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Related Tools

• t-Table - Look up critical t-values
• Z-Score Calculator - Standardize values
• Mean Calculator - Calculate sample means
• Standard Deviation Calculator - Calculate standard deviations
• Confidence Interval Calculator - Build confidence intervals
• Degrees of Freedom Reference - df formulas for every test