Example Data Table
| Distribution | Tail | Statistic | Alpha | Degrees | Expected rule |
|---|---|---|---|---|---|
| z | Two tailed | 2.10 | 0.05 | None | Compare with ±1.96 approximately |
| t | Right tailed | 2.30 | 0.05 | 15 | Compare with upper t limit |
| Chi square | Right tailed | 18.40 | 0.05 | 10 | Compare with upper chi square limit |
| F | Right tailed | 3.20 | 0.05 | 5, 20 | Compare with upper F limit |
Formula Used
Right tailed test: reject when test statistic ≥ Q(1 − α).
Left tailed test: reject when test statistic ≤ Q(α).
Two tailed test: reject when statistic ≤ Q(α / 2) or statistic ≥ Q(1 − α / 2).
Bonferroni adjustment: adjusted α = α / number of comparisons.
P value: right = 1 − CDF(x), left = CDF(x), two = 2 × min(CDF(x), 1 − CDF(x)).
The quantile function Q comes from the selected z, t, chi square, or F distribution.
How to Use This Calculator
Select the distribution that matches your test statistic.
Choose the correct tail from your alternative hypothesis.
Enter alpha directly, or use confidence level to derive alpha.
Add degrees of freedom for t, chi square, and F tests.
Use multiple comparisons when a Bonferroni adjustment is needed.
Press the submit button and read the result above the form.
Use the CSV or PDF button to save the output.
Understanding Critical Values
A critical value marks the boundary of a rejection region. It comes from a chosen distribution. The boundary depends on alpha, tail direction, and degrees of freedom. This calculator lets you compare a standardized test statistic with that boundary. It supports z, t, chi square, and F procedures.
Why Standardized Statistics Matter
A standardized statistic puts evidence on a common scale. A z score uses the standard normal curve. A t statistic uses a t curve, which changes with degrees of freedom. A chi square statistic is positive and often tests variance or goodness of fit. An F statistic compares two variance based quantities. Each scale needs its own critical rule.
Choosing Tail Direction
Tail direction follows the alternative hypothesis. A right tailed test looks for unusually large values. A left tailed test looks for unusually small values. A two tailed test splits alpha between both sides. The calculator reports the matching lower or upper limits. It also shows the reject or fail to reject decision.
Alpha, Confidence, and Error Risk
Alpha is the allowed Type I error rate. Common values are 0.10, 0.05, and 0.01. A smaller alpha makes rejection harder. Confidence level is related to alpha. For example, a 95 percent confidence level often matches alpha 0.05. This tool can use direct alpha or compute alpha from confidence.
Interpreting the Result
The decision is mathematical, not practical proof. A rejected null suggests the statistic falls inside the rejection region. A failed rejection means the result did not cross the chosen boundary. The p value estimate adds context. It shows how extreme the statistic is under the null model. Always check assumptions before reporting conclusions.
Good Reporting Practice
Report the distribution, degrees of freedom, alpha, tail type, statistic, critical value, p value, and decision. Add subject context. Mention whether alpha was adjusted for multiple comparisons. Use the export buttons to save a copy. The saved data helps audits, teaching notes, and repeated analyses.
Use With Care
The calculator is a guide for planned hypothesis tests. It cannot fix poor sampling, biased data, or weak design. Review assumptions first. Then use the numbers to support a clear statistical statement before final reporting safely.
FAQs
What is a critical value?
A critical value is a cutoff from a probability distribution. It separates ordinary test statistic values from values considered unusual under the null hypothesis.
Which distribution should I choose?
Use z for standard normal tests. Use t when degrees of freedom matter. Use chi square for variance or count tests. Use F for variance ratio tests.
What does alpha mean?
Alpha is the selected Type I error risk. It is the chance of rejecting a true null hypothesis under the test model.
How is confidence related to alpha?
Confidence is often one minus alpha. A 95 percent confidence level usually corresponds to alpha 0.05 for standard tests.
When do I use a two tailed test?
Use a two tailed test when the alternative hypothesis allows change in either direction. Alpha is split between both tails.
Why enter degrees of freedom?
Degrees of freedom shape t, chi square, and F distributions. Changing them changes the critical value and p value estimate.
What is the Bonferroni adjustment?
It divides alpha by the number of comparisons. This makes each individual test stricter when many tests are performed.
Can this replace statistical judgment?
No. It provides numerical boundaries and decisions. You still need good data, correct assumptions, and meaningful subject interpretation.