t-statistic Calculator
Calculate t-statistic (t-value) for hypothesis testing and Student's t-test
Calculate t-statistic
Average value of your sample data
Hypothesized or known population mean
Number of observations in your sample
Standard deviation of your sample
t-statistic Formula
Basic Formula
x̄: Sample mean
μ: Population mean
s: Sample standard deviation
n: Sample size
Standard Error
Basketball Example
Scenario
Basketball player's average: 15 points
Population average: 10 points
Games played: 36
Standard deviation: 6 points
Calculation
t = (15 - 10) / (6 / √36)
t = 5 / (6 / 6)
t = 5.0
Strong evidence of above-average performance!
Interpretation Guide
|t| > 2: Strong evidence against null hypothesis
1 < |t| ≤ 2: Moderate evidence against null hypothesis
|t| ≤ 1: Weak evidence against null hypothesis
Positive t: Sample mean > Population mean
Negative t: Sample mean < Population mean
Understanding the t-statistic
What is the t-statistic?
The t-statistic (or t-value) is a measure that describes how many standard errors a sample mean is away from a population mean. It's central to Student's t-test and is used to determine whether to support or reject the null hypothesis.
When to Use t-statistic vs z-score
- •t-statistic: Small sample size (<30) or unknown population σ
- •z-score: Large sample size (≥30) and known population σ
- •t-distribution has heavier tails than normal distribution
- •As sample size increases, t-distribution approaches normal
Hypothesis Testing Steps
Step 1: State Hypotheses
H₀: μ = μ₀ (null hypothesis)
H₁: μ ≠ μ₀ (alternative hypothesis)
Step 2: Calculate t-statistic
Use the formula: t = (x̄ - μ) / (s / √n)
Step 3: Compare with Critical Value
If |t| > critical value, reject null hypothesis
Applications of t-statistic
One-Sample t-test
Compare sample mean to hypothesized population mean. Used when testing if a sample comes from a population with a specific mean.
Two-Sample t-test
Compare means of two independent samples. Tests whether two groups have significantly different means.
Paired t-test
Compare paired observations (before/after, matched pairs). Tests whether the mean difference between pairs is zero.
Historical Background
The t-statistic was developed by William Sealy Gosset in 1908 while working at the Guinness Brewery in Dublin. Due to company policy preventing employees from publishing, he published under the pseudonym "Student." This is why the t-distribution is also known as "Student's t-distribution." Gosset developed this test to solve practical problems related to quality control in brewing with small sample sizes.