Calculator Input
Formula Used
Z statistic for mean: z = (x̄ - μ₀) / (σ / √n)
One sample t statistic: t = (x̄ - μ₀) / (s / √n)
Pooled two sample t: t = [(x̄₁ - x̄₂) - Δ₀] / [sₚ√(1/n₁ + 1/n₂)]
Welch t statistic: t = [(x̄₁ - x̄₂) - Δ₀] / √(s₁²/n₁ + s₂²/n₂)
One proportion z: z = (p̂ - p₀) / √[p₀(1 - p₀) / n]
Two proportion z: z = [(p̂₁ - p̂₂) - Δ₀] / √[p̂(1 - p̂)(1/n₁ + 1/n₂)]
Chi square variance: χ² = (n - 1)s² / σ₀²
F statistic: F = s₁² / s₂²
Correlation t statistic: t = r√[(n - 2) / (1 - r²)]
How To Use This Calculator
Select the test type first. The form will show the fields needed for that test. Enter sample values, null values, deviations, variances, proportions, or correlation data. Choose an alternative hypothesis and alpha level for record keeping. Press the calculate button. The answer appears above the form. Use CSV or PDF buttons to save the result.
Example Data Table
| Case | Test | Main inputs | Expected statistic |
|---|---|---|---|
| Mean with known sigma | One sample z | x̄ = 52, μ₀ = 50, σ = 8, n = 40 | z = 1.5811 |
| Mean with sample deviation | One sample t | x̄ = 52, μ₀ = 50, s = 9, n = 40 | t = 1.4055 |
| Two independent means | Welch t | 84, 79, 10, 12, 35, 32 | t ≈ 1.84 |
| Variance claim | Chi square | n = 40, s² = 25, σ₀² = 16 | χ² = 60.9375 |
Test Statistic Guide
What The Calculator Does
A test statistic turns sample evidence into one standard number. That number can then be compared with a reference distribution. This calculator supports common tests used in introductory and advanced statistics work. It covers means, proportions, variances, and correlation. It also gives degrees of freedom where they are needed.
Why The Value Matters
The statistic measures distance from the null hypothesis. A value near zero usually means the sample result is close to the null value. A large positive or negative value means the sample result is farther away. The direction depends on the formula and the alternative hypothesis.
Choosing The Correct Test
Use a one sample z test when the population standard deviation is known. Use a one sample t test when the standard deviation comes from the sample. Use a pooled t test when two groups have similar variances. Use Welch t when equal variance is not safe. Welch is often the better default for two independent means.
Proportions And Counts
Use the one proportion z option for one success rate. Use the two proportion z option for two independent success rates. Enter successes and sample sizes carefully. The calculator converts these counts into sample proportions. It also uses the pooled proportion for the common two proportion null test.
Variance And Correlation Tests
The chi square option tests one variance against a claimed variance. The F option compares two variances. The correlation option tests whether a sample correlation differs from zero. These tests use different reference distributions, so their statistic symbols and degrees of freedom are not the same.
Reading The Output
The output shows the statistic, formula, standard error, and inputs. It does not replace a full hypothesis test. After finding the statistic, compare it with a critical value or use it to find a p-value. Always match the distribution, tail direction, and degrees of freedom.
Practical Notes
Check assumptions before using any result. Look for independence, random sampling, outliers, normality, and adequate sample size. Rounding can change the final statistic slightly. Keep more decimals during calculation. Round only the final answer when reporting the result.
FAQs
What is a test statistic?
A test statistic is a standardized value from sample data. It shows how far the sample result is from the null hypothesis under a selected test model.
Which test type should I choose?
Choose the test that matches your data. Use z for known population deviation or large proportion tests. Use t for means with sample deviation. Use chi square or F for variance questions.
Does this calculator give a p-value?
No. It focuses on finding the statistic value. After that, use the matching distribution table or p-value calculator with the correct degrees of freedom.
What does a large statistic mean?
A large absolute statistic means the sample result is farther from the null value. It may support stronger evidence against the null hypothesis, depending on the distribution and tail.
What is degrees of freedom?
Degrees of freedom describe how many independent pieces of information support the estimate. They are required for t, chi square, F, and correlation tests.
When should I use Welch t?
Use Welch t when comparing two means and equal variance is doubtful. It adjusts the degrees of freedom and is often safer than the pooled method.
Can I export the answer?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button for a printable calculation summary.
Why did I get an input error?
The calculator checks invalid values. Standard deviations, variances, and sample sizes must be positive. Proportions and correlations must stay within their allowed ranges.