What Is Indicated Critical Value?
A critical value is the boundary of the rejection region. If your test statistic lands beyond it, you reject the null hypothesis at the significance level you chose.
Pick the distribution that matches your test, enter alpha or a confidence level, and choose a left, right or two-tailed test. For t, chi-square and F add the degrees of freedom.
The graph shades the rejection region so you can see how much of the distribution it holds.
How to Use the Indicated Critical Value Calculator
- Choose the distribution that matches your test.
- Enter α or the confidence level and pick the tail.
- Add degrees of freedom for t, chi-square or F, then read the cut-off.
Indicated Critical Value Formula
Right-tailed
Two-tailed
Indicated Critical Value Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| α | Significance level | |
| Tail | Left, right or two-tailed | Set by the alternative hypothesis |
| df | Degrees of freedom | Not needed for z |
| c | Critical value | Reject H₀ beyond it |
Indicated Critical Value Worked Example
With the example values loaded in the calculator above, the steps are:
- Distribution: z. Significance level α = 0.05.
- Tail: two. That puts 0.025 in each tail.
- Invert the cumulative distribution to find the cut-offs: ± 1.96.
- Reject H₀ if the statistic falls in the region z < -1.96 or z > 1.96.
Result: Critical value = ± 1.96; Rejection region = z < -1.96 or z > 1.96.
Worked example: z at α = 0.05
For a two-tailed z test, each tail holds 0.025, so the critical values are ±1.96.
For a right-tailed test the whole 0.05 sits in one tail, so the critical value is 1.645.
Which distribution to use
Use z for means with a known standard deviation or for large-sample proportions. Use t for means with an estimated standard deviation. Use chi-square for variances and goodness-of-fit tests, and F for comparing variances and in ANOVA.
Where Indicated Critical Value Is Used
Hypothesis tests
Find the cut-off for a z, t, chi-square or F test.
Exam practice
Look up a critical value without a table.
Checking a decision
Confirm that a test statistic is inside or outside the rejection region.
Common Indicated Critical Value Mistakes to Avoid
- Decide the tail before collecting the data.
- For chi-square and F the distribution is skewed, so the two critical values of a two-tailed test are not symmetric.
Indicated Critical Value Terms and Definitions
- Rejection region
- The set of test statistic values for which the null hypothesis is rejected.
- Significance level
- The probability of rejecting a true null hypothesis, α.
Indicated Critical Value FAQs
What is a critical value?
The boundary of the rejection region. A statistic beyond it is significant at level α.
What is the z critical value for 95% confidence?
1.96, for a two-tailed test with α = 0.05.
How do I pick the tail?
From the alternative hypothesis. A "not equal" claim is two-tailed, "greater than" is right-tailed and "less than" is left-tailed.
Do critical values and p-values agree?
Yes. A statistic beyond the critical value has a p-value below α.
Why are chi-square critical values not symmetric?
The chi-square and F distributions are skewed, so each tail needs its own cut-off.