Calculator Form
Example Data Table
| Test | Inputs | Formula | Expected statistic |
|---|---|---|---|
| One sample z mean | x̄ = 52.4, μ₀ = 50, σ = 8, n = 36 | (52.4 - 50) / (8 / √36) | 1.8 |
| One sample t mean | x̄ = 18.6, μ₀ = 20, s = 4.2, n = 25 | (18.6 - 20) / (4.2 / √25) | -1.666667 |
| One proportion z | p̂ = 0.58, p₀ = 0.50, n = 100 | (0.58 - 0.50) / √(0.50 × 0.50 / 100) | 1.6 |
| Goodness of fit | Observed: 22, 18, 25, 35. Expected: 25, 25, 25, 25 | Σ[(O - E)² / E] | 6.32 |
Formula Used
One sample z mean: z = (x̄ - μ₀) / (σ / √n)
One sample t mean: t = (x̄ - μ₀) / (s / √n)
One proportion z: z = (p̂ - p₀) / √[p₀(1 - p₀) / n]
Two independent means: t = [(x̄₁ - x̄₂) - Δ₀] / √(s₁²/n₁ + s₂²/n₂)
Two proportions: z = (p̂₁ - p̂₂) / √[p̂(1 - p̂)(1/n₁ + 1/n₂)]
Paired t: t = d̄ / (sᵈ / √n)
Variance chi square: χ² = (n - 1)s² / σ₀²
Goodness of fit: χ² = Σ[(O - E)² / E]
How to Use This Calculator
Choose the test type that matches your hypothesis problem.
Select the tail type used by your alternative hypothesis.
Enter only the fields needed for the selected test.
Add a critical value when you want a decision note.
Press the calculate button to view the statistic above the form.
Use the CSV or PDF button to save your result.
Understanding the Test Statistic
A test statistic turns sample evidence into one measured number. It shows how far an observed result sits from a claimed population value. The distance is scaled by standard error, so different studies can be compared fairly. A large absolute value usually means the sample is less consistent with the null claim. A small value usually means the sample is close to the expected value.
Why This Calculator Helps
Manual work can be slow when many test types are possible. Means, proportions, paired differences, variances, and goodness of fit tests all use different inputs. This calculator keeps those choices in one place. It also displays the substituted formula, standard error, degrees of freedom, and decision note. That helps users check work before writing a report.
Selecting the Right Test
Use a one sample z test when the population standard deviation is known. Use a one sample t test when it is unknown and a sample standard deviation is used. Use a proportion test for success counts or sample proportions. Use two sample tests when two independent groups are compared. Use a paired t test when each observation has a matched before and after value. Use a chi square variance test for one population variance. Use goodness of fit when observed counts are compared with expected counts.
Interpreting Results
The statistic alone does not prove a claim. It must be compared with a critical value or p value. This tool can compare the statistic with your entered critical value. Choose a two tailed, left tailed, or right tailed test to match the research question. The decision note is only as reliable as the chosen test and entered assumptions.
Good Practice
Always check sample size, independence, measurement scale, and expected count rules. Round only at the final step when possible. Keep raw values in your notes. If assumptions are weak, use caution. A clean statistic supports clear reasoning, but sound study design gives the result meaning. Use the export buttons to save records. Review formulas before submission. Update entries when your teacher or analyst provides a different null value. Document units, tails, and rounding choices so another reader can repeat the calculation without guessing later or confusion errors.
FAQs
What is a test statistic?
A test statistic is a standardized value. It compares sample evidence with a null hypothesis. It helps decide whether the observed result is unusual under the assumed claim.
Which test should I choose for a mean?
Use a z test when the population standard deviation is known. Use a t test when only the sample standard deviation is known.
Can this calculator make a hypothesis decision?
Yes. Enter a critical value and choose the correct tail type. The calculator will compare the statistic with that critical value.
Does this tool calculate p values?
No. This tool focuses on computing test statistics and critical value decisions. Use a distribution table or software for exact p values.
What does degrees of freedom mean?
Degrees of freedom show how many values can vary after constraints are applied. They are used mainly in t and chi square tests.
Why is standard error important?
Standard error scales the difference between the sample result and null value. Smaller standard error usually produces a larger test statistic.
Can I use counts for proportions?
Yes. For the two proportion test, enter successes and sample sizes. The calculator converts those counts into sample proportions.
What should I check before using results?
Check independence, sample size, expected counts, and the correct test choice. Wrong assumptions can make a correct calculation misleading.