T-Test Calculator

Hypothesis Testing and Statistical Inference

Select the type of t-test and enter your data to determine statistical significance.

Examples

Use these examples to see how the T-Test Calculator works with different data sets.

One-Sample T-Test Example

oneSample

A researcher wants to know if the average height of a plant species is different from the known average of 15 cm.

Type: one-sample

Data 1: 14.5, 15.2, 14.8, 15.5, 16.0, 14.9, 15.1

Pop. Mean: 15

Independent Samples T-Test Example

independentSamples

Comparing the test scores of two different groups of students who were taught with different methods.

Type: independent-samples

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

Data 2: 75, 82, 70, 79, 85, 73

Paired Samples T-Test Example

pairedSamples

Measuring the blood pressure of the same group of patients before and after taking a new medication.

Type: paired-samples

Data 1: 140, 135, 142, 148, 130

Data 2: 132, 130, 135, 140, 125

Right-Tailed T-Test Example

oneSampleRightTailed

A company claims their batteries last more than 40 hours. A sample is tested to verify this claim.

Type: one-sample

Data 1: 42, 45, 39, 41, 43, 44, 38

Pop. Mean: 40

Other Titles
Understanding the T-Test: A Comprehensive Guide
A deep dive into hypothesis testing, statistical significance, and the practical applications of Student's t-test for comparing means.

What is a T-Test?

  • Core Concept of Hypothesis Testing
  • The Role of the Null Hypothesis (H₀)
  • Types of T-Tests Explained
A T-Test is a type of inferential statistic used to determine if there is a significant difference between the means of two groups, which may be related in certain features. It is one of the most widely used statistical hypothesis tests. The t-test is based on Student's t-distribution, which is a probability distribution that is used to estimate population parameters when the sample size is small and/or when the population standard deviation is unknown.
Core Concept of Hypothesis Testing
At its core, a t-test helps you decide if you should accept or reject a 'null hypothesis'. The null hypothesis (H₀) is a general statement that there is no relationship or no difference between two measured phenomena. For a t-test, it typically states that the two population means are equal. The alternative hypothesis (H₁) is what you might believe to be true or hope to prove true; it states that there is a difference.
The Role of the Null Hypothesis (H₀)
The entire process revolves around testing the validity of the null hypothesis. The t-test calculates a t-value, which is then compared to a critical value from the t-distribution table (or used to calculate a p-value). If the t-value is large enough (or the p-value is small enough), it suggests that the observed difference between the samples is unlikely to be due to random chance, leading you to reject the null hypothesis in favor of the alternative.
Types of T-Tests Explained
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 Samples T-Test: Compares the means of two independent or unrelated groups. 3) Paired Samples T-Test: Compares the means of two related groups, such as the same subjects tested at two different points in time.

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

  • Selecting the Correct T-Test Type
  • Entering Your Data Correctly
  • Interpreting the Results
Our calculator simplifies the process, but understanding each step is crucial for accurate results.
1. Selecting the Correct T-Test Type
Choose 'One-Sample' if you are comparing a single group's average to a known value. Select 'Independent Samples' if you are comparing two separate, unrelated groups. Use 'Paired Samples' if you are comparing the same group under two different conditions (e.g., before and after a treatment).
2. Entering Your Data Correctly
Input your sample data as comma-separated numbers. The calculator will automatically parse the numbers and ignore any text or invalid entries. For a one-sample test, you must also provide the population mean you are testing against. For two-sample tests, ensure the data for each group is in the correct input box.
3. Interpreting the Results
The key outputs are the t-value and the p-value. The t-value measures the size of the difference relative to the variation in your sample data. The p-value is the probability of observing a result as extreme as you did, assuming the null hypothesis is true. A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject it. A large p-value (> 0.05) indicates weak evidence, so you fail to reject it.

Real-World Applications of the T-Test

  • A/B Testing in Marketing
  • Medical Research and Clinical Trials
  • Quality Control in Manufacturing
A/B Testing in Marketing
Marketers use independent samples t-tests to determine if a new website design (Version B) leads to a significantly higher conversion rate compared to the old design (Version A). The two groups are the users who see Version A and the users who see Version B.
Medical Research and Clinical Trials
Paired samples t-tests are crucial in medical research. For example, researchers might measure patients' cholesterol levels before and after a new drug regimen to see if the drug had a statistically significant effect.
Quality Control in Manufacturing
A one-sample t-test can be used in quality control. A factory might produce bolts that need to have a diameter of 2 cm. They can take a sample of bolts, measure their diameters, and use a one-sample t-test to determine if the average diameter of their sample is significantly different from the required 2 cm.

Common Misconceptions and Correct Methods

  • Correlation vs. Causation
  • The Meaning of 'Significance'
  • Assumptions of the T-Test
Correlation vs. Causation
A common mistake is to assume that because a t-test shows a significant difference, one variable caused the change in the other. A t-test can only show a statistical relationship; it cannot prove causation. Other factors could be at play.
The Meaning of 'Significance'
Statistical significance does not necessarily mean the result is practically important or meaningful. A very large sample size might yield a statistically significant result for a very small, trivial difference between means. Always consider the effect size and the context of your research.
Assumptions of the T-Test
For the results of a t-test to be valid, several assumptions must be met: the data should be continuous or ordinal, the data should be a simple random sample from the population, the data should be approximately normally distributed, and there should be homogeneity of variances (the variances of the groups are equal).

Mathematical Derivation and Formulas

  • One-Sample T-Test Formula
  • Independent Samples T-Test Formula
  • Paired Samples T-Test Formula
One-Sample T-Test Formula
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.
Independent Samples T-Test Formula
The formula is: t = (x̄₁ - x̄₂) / √((s₁²/n₁) + (s₂²/n₂)), where x̄₁ and x̄₂ are the means of the two samples, s₁² and s₂² are their variances, and n₁ and n₂ are their sizes. A pooled variance is often used when variances are assumed to be equal.
Paired Samples T-Test Formula
This test is essentially a one-sample t-test on the differences between paired values. 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.