T-Statistic Calculator

Hypothesis Testing and Statistical Inference

Choose the test type and enter your data to calculate the t-statistic, p-value, and degrees of freedom.

Practical Examples

Explore different scenarios to understand how the t-statistic calculator works.

One-Sample T-Test Example

oneSample

A researcher wants to know if the average weight of a new species of turtles is different from the known average of 300 grams.

Data 1: 310, 312, 298, 305, 308, 301, 299, 304, 307, 296

Pop. Mean: 300

Two-Sample T-Test Example

twoSample

Comparing the test scores of two groups of students who used different study methods.

Data 1: 85, 90, 78, 88, 92, 95, 80

Data 2: 78, 82, 75, 80, 84, 88, 79

Paired T-Test Example

paired

Measuring the effectiveness of a new medication by comparing blood pressure readings before and after treatment for the same group of patients.

Data 1: 140, 135, 150, 155, 142

Data 2: 132, 130, 142, 145, 138

A/B Testing Scenario

twoSample

An e-commerce site tests two different button colors (A and B) to see which one leads to a higher average purchase value.

Data 1: 55, 60, 58, 62, 57, 53, 59, 61

Data 2: 50, 52, 48, 55, 51, 49, 53, 54

Other Titles
Understanding the T-Statistic: A Comprehensive Guide
A deep dive into the t-statistic, its applications in hypothesis testing, and the interpretation of its results. This guide will walk you through the core concepts needed to use this calculator effectively.

What is the T-Statistic?

  • Definition and Purpose
  • The T-Distribution vs. The Normal Distribution
  • Types of T-Tests
The t-statistic, also known as the t-value, is a measure used in hypothesis testing to determine whether there is a significant difference between the means of two groups, or between a sample mean and a hypothesized population mean. It quantifies the difference relative to the variation in your sample data. In simpler terms, a large t-statistic indicates that the difference between the groups is large compared to the difference within the groups, suggesting that the observed difference is not due to random chance.
The T-Distribution
The t-statistic follows a Student's t-distribution, which is similar to the normal distribution (bell-shaped and symmetric) but has heavier tails. This means it is more prone to producing values that fall far from its mean. The shape of the t-distribution depends on the 'degrees of freedom' (df), which is related to the sample size. As the sample size increases, the t-distribution gets closer to the normal distribution.
Types of T-Tests
There are three main types of t-tests: 1. One-Sample T-Test: Compares the mean of a single sample to a known or hypothesized population mean. 2. Independent Two-Sample T-Test: Compares the means of two independent groups to determine if they come from the same population. 3. Paired T-Test: Compares the means of two related groups (e.g., the same subjects before and after a treatment) to see if there is a significant change.

Step-by-Step Guide to Using the T-Statistic Calculator

  • Selecting the Right Test
  • Entering Your Data Correctly
  • Interpreting the Results
1. Select the Test Type
Begin by choosing the appropriate t-test from the dropdown menu: 'One-Sample', 'Two-Sample', or 'Paired'.
2. Input Your Data
Enter your sample data as comma-separated numbers. For a one-sample test, you also need to provide the hypothesized population mean. For two-sample and paired tests, you'll need to enter data for both samples.
3. Set Hypothesis Parameters
Choose your hypothesis type (two-tailed, left-tailed, or right-tailed) and set the significance level (α), which is typically 0.05.
4. Interpret the Output
The calculator provides the t-statistic, p-value, and degrees of freedom. The most important value is the p-value. If the p-value is less than or equal to your significance level (p ≤ α), you reject the null hypothesis and conclude that your results are statistically significant. Otherwise, you fail to reject the null hypothesis.

Real-World Applications of the T-Statistic

  • Medical Research
  • A/B Testing in Marketing
  • Quality Control
Medical Research
Researchers use paired t-tests to determine if a new drug is effective by comparing patient metrics (like blood pressure or cholesterol levels) before and after treatment.
A/B Testing
A marketing team might use a two-sample t-test to compare the conversion rates of two different website designs (A and B) to see which one performs better.
Quality Control
A factory can use a one-sample t-test to check if the average weight of a product from a recent batch matches the required standard weight.

Mathematical Formulas and Derivations

  • One-Sample T-Test Formula
  • Two-Sample T-Test (Welch's T-Test) Formula
  • Paired T-Test Formula
One-Sample T-Test
The formula is: t = (x̄ - μ₀) / (s / √n), where x̄ is the sample mean, μ₀ is the population mean, s is the sample standard deviation, and n is the sample size. Degrees of freedom (df) = n - 1.
Independent Two-Sample T-Test (Welch's)
This test does not assume equal variances. The formula is: t = (x̄₁ - x̄₂) / √((s₁²/n₁) + (s₂²/n₂)). The degrees of freedom calculation is more complex (known as the Welch-Satterthwaite equation), which the calculator handles for you.
Paired T-Test
This test is essentially a one-sample t-test on the differences between paired data points. The formula is: t = d̄ / (sd / √n), where d̄ is the mean of the differences, sd is the standard deviation of the differences, and n is the number of pairs. Degrees of freedom (df) = n - 1.