Example data table
| Distribution |
Tail |
Alpha |
df |
Statistic |
Expected use |
| z |
Two-tailed |
0.05 |
Not used |
2.12 |
Large sample mean test |
| t |
Right-tailed |
0.01 |
15 |
2.45 |
Small sample mean test |
| Chi-square |
Right-tailed |
0.05 |
10 |
19.7 |
Variance test |
| F |
Right-tailed |
0.05 |
5, 20 |
2.8 |
Variance ratio test |
Formula used
The calculator uses inverse cumulative distribution functions. Let Q(p) be the quantile at cumulative probability p.
Right-tailed test: critical value = Q(1 − α).
Left-tailed test: critical value = Q(α).
Two-tailed test: lower value = Q(α / 2), upper value = Q(1 − α / 2).
Confidence conversion: α = 1 − confidence level.
Decision rule: reject the null hypothesis when the statistic enters the rejection region.
How to use this calculator
- Select the distribution that matches your hypothesis test.
- Choose right-tailed, left-tailed, or two-tailed testing.
- Enter alpha directly, or choose confidence level mode.
- Add degrees of freedom for t, chi-square, or F tests.
- Enter a test statistic when you need a decision.
- Press the calculate button and review the rejection rule.
- Use the CSV or PDF buttons to save the result.
Critical values in hypothesis testing
A critical value is a boundary on a reference distribution. It marks where ordinary sampling variation becomes unlikely under the null hypothesis. This calculator finds that boundary for z, t, chi-square, and F tests. It also compares your test statistic with the rejection region.
Why the calculator is useful
Manual table lookup can be slow. Tables also limit decimal accuracy. Here, you can choose alpha, confidence level, tail direction, and distribution. The tool then returns lower and upper cutoffs where they apply. This is helpful for exams, reports, audits, quality checks, and research summaries.
Choosing the right distribution
Use the z distribution when the population standard deviation is known, or when a large sample supports normal approximation. Use the t distribution when estimating a mean with sample standard deviation. Use chi-square for variance tests and goodness-of-fit style checks. Use F for ratio tests, variance comparison, and analysis of variance settings.
Understanding tails
A right-tailed test rejects when the statistic is too large. A left-tailed test rejects when the statistic is too small. A two-tailed test rejects on either extreme. For two-tailed tests, alpha is split equally between both ends. This produces two limits for most distributions.
Interpreting the result
The result shows the active alpha, confidence level, critical value, p-value, and decision. If the statistic falls inside the rejection region, the calculator reports rejection of the null hypothesis. If not, it reports failure to reject. This wording matters. It does not prove the null is true.
Good statistical practice
Always define the null and alternative hypothesis before calculation. Select the distribution from the design of the study, not from the desired answer. Check degrees of freedom carefully. A small df can change the boundary greatly. Report alpha, tail choice, statistic, critical value, and p-value together. This makes your conclusion easier to review.
Common mistakes to avoid
Do not mix one-tailed and two-tailed rules after seeing data. Do not use z values for small samples without a justified model. Do not ignore denominator degrees of freedom in F tests. Keep rounding consistent. A rounded boundary can change a borderline decision, so use the displayed precision in final work. Save both exported files with your study notes.
FAQs
What is a critical value?
A critical value is a cutoff from a probability distribution. It separates the non-rejection region from the rejection region for a selected alpha level.
Which distribution should I choose?
Choose z for normal large sample tests, t for small sample mean tests, chi-square for variance tests, and F for variance ratio tests.
What does alpha mean?
Alpha is the significance level. It is the probability of rejecting the null hypothesis when the null hypothesis is actually true.
What is a two-tailed critical value?
A two-tailed test has two cutoffs. The calculator places half of alpha in the left tail and half in the right tail.
Can I use confidence level instead of alpha?
Yes. Select confidence level mode. The calculator converts it into alpha by using alpha equals one minus the confidence level.
Why do degrees of freedom matter?
Degrees of freedom shape t, chi-square, and F distributions. Changing them can change the critical value and final decision.
Does this calculator prove my hypothesis?
No. It only compares your statistic with a statistical boundary. Your study design and assumptions still control the strength of evidence.
What should I report with the result?
Report the distribution, tail type, alpha, degrees of freedom, test statistic, critical value, p-value, and final decision.