🧪 One-Sample T-Test Calculator
Test whether a sample mean differs from a hypothesized population mean when population standard deviation is unknown.
📊 Statistics📐 Formula included🧠 Practice questions✓ Free
Please enter valid values.
Formula and Explanation
Test whether a sample mean differs from a hypothesized population mean when population standard deviation is unknown.
| Concept | Formula or rule |
|---|---|
| One-Sample T-Test Calculator | t = (x̄ − μ₀) / (s/√n) |
How to Use This Calculator
- Enter the sample, counts, probabilities, or summary statistics requested above.
- Keep paired lists in the same order and separate values with commas.
- Select the correct test direction or sample/population option when available.
- Press Calculate and interpret the result together with the supporting statistics.
Worked Example
Example: x̄=52, μ₀=50, s=8, n=30 gives t≈1.369.
✓ Compare the p-value with your chosen α.
Where This Is Used
School & coursework
Check homework, understand formulas, and verify step-by-step statistics exercises.
Research
Explore estimates, uncertainty, distributions, and relationships before using full statistical software.
Business & data analysis
Summarize variation, compare groups, and communicate data-driven findings clearly.
Common Mistakes to Avoid
⚠️
Using a z-test automatically
A t-test is appropriate when σ is unknown.
A t-test is appropriate when σ is unknown.
⚠️
Ignoring dependence
Observations should be independent or modeled appropriately.
Observations should be independent or modeled appropriately.
⚠️
Equating statistical and practical significance
A small p-value does not show that the effect is important.
A small p-value does not show that the effect is important.
Practice Questions
1. What is df when n=20?
Answer: 19
2. If sample mean equals μ0, what is t?
Answer: 0
3. Does larger n usually increase power?
Answer: Yes.
Frequently Asked Questions
What are the degrees of freedom?▾
n−1.
What does a two-sided test ask?▾
Whether the mean differs in either direction.
What assumption matters for small samples?▾
The population should be approximately normal without severe outliers.