T-Test Calculator

T-Test Calculator - Student's T-Test

Calculate t-tests for statistical significance with our free online calculator. Support for one-sample, independent two-sample, and paired t-tests with p-values and step-by-step explanations.

Calculate T-Test

Test Type:

• One-Sample T-Test
• Independent Two-Sample T-Test
• Paired T-Test

Input Parameters:

• Sample Mean(s): [Input]
• Population Mean (for one-sample): [Input]
• Sample Standard Deviation(s): [Input]
• Sample Size(s): [Input]
• Significance Level (α): [Dropdown: 0.01, 0.05, 0.10]

Results:

• T-Statistic: [Result]
• P-Value: [Result]
• Degrees of Freedom: [Result]
• Critical Value: [Result]
• Conclusion: [Reject/Fail to reject H₀]

---

What is a T-Test?

A T-Test is a statistical hypothesis test that compares the means of one or two groups to determine if they're significantly different from each other. It's used when the population standard deviation is unknown and the sample size is small (typically n < 30).

Basic Concept

T-Test asks: "Is the difference between means statistically significant, or could it be due to random chance?"

When to Use T-Test

Why T-Tests Matter

1. Scientific Research: Validate experimental results
2. Quality Control: Compare production batches
3. Medical Studies: Test treatment effectiveness
4. Business Analytics: A/B testing
5. Education: Evaluate teaching methods

Types of T-Tests

1. One-Sample T-Test

Purpose: Compare sample mean to a known or hypothesized population mean

Formula:

t = (x̄ - μ) / (s / √n)

Where:

• x̄ = sample mean
• μ = hypothesized population mean
• s = sample standard deviation
• n = sample size

Degrees of Freedom:

df = n - 1

Example: Test if average height differs from 170cm

x̄ = 175, μ = 170, s = 10, n = 25
t = (175 - 170) / (10 / √25)
t = 5 / 2 = 2.5
df = 25 - 1 = 24

Hypotheses:

• H₀: μ = 170 (no difference)
• H₁: μ ≠ 170 (two-tailed) or μ > 170 (one-tailed)

2. Independent Two-Sample T-Test

Purpose: Compare means of two independent groups

Formula:

t = (x̄₁ - x̄₂) / SE

Standard Error:

SE = √(s₁²/n₁ + s₂²/n₂)

Degrees of Freedom (Welch's):

df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]

Equal Variances (Pooled):

s_p = √[((n₁-1)s₁² + (n₂-1)s₂²) / (n₁ + n₂ - 2)]
t = (x̄₁ - x̄₂) / [s_p · √(1/n₁ + 1/n₂)]
df = n₁ + n₂ - 2

Example: Compare test scores of two classes

Class 1: x̄₁ = 85, s₁ = 8, n₁ = 20
Class 2: x̄₂ = 80, s₂ = 10, n₂ = 25
SE = √(64/20 + 100/25) = √(3.2 + 4) = √7.2 = 2.683
t = (85 - 80) / 2.683 = 5 / 2.683 = 1.864

3. Paired T-Test

Purpose: Compare means of related samples (before/after, matched pairs)

Formula:

t = d̄ / (s_d / √n)

Where:

• d̄ = mean of differences
• s_d = SD of differences
• n = number of pairs

Degrees of Freedom:

df = n - 1

Example: Test weight loss program effectiveness

Before: [80, 85, 90, 88, 92]
After:  [78, 82, 87, 85, 89]
Differences: [2, 3, 3, 3, 3]
d̄ = 2.8, s_d = 0.447, n = 5
t = 2.8 / (0.447 / √5) = 2.8 / 0.2 = 14
df = 5 - 1 = 4

---