# What is the critical value for my test?

Computes the critical values and rejection region of a left-, right-, or two-tailed test for the z, t, chi-square, and F distributions at a significance level.

- Page: https://www.acalculator.org/statistics/critical-value-calculator
- JSON spec: https://www.acalculator.org/statistics/critical-value-calculator.json
- Version: 4de8f84bf28f

## Default answer

Example with the default inputs (Distribution Z, Test type Two-tailed, Significance level (α) 0.05): The rejection region is (-∞, -1.9600) ∪ (1.9600, ∞) (Two-tailed, Z, α = 0.05).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| distribution | Distribution | The distribution of the test statistic: z (standard normal), Student’s t, chi-square, or F. |
| testType | Test type | Where the rejection region lies: both tails, the left tail, or the right tail. |
| significanceLevel | Significance level (α) | The probability of the rejection region when the null hypothesis is true, for example 0.05. |
| degreesOfFreedom | Degrees of freedom | Degrees of freedom of the t or chi-square distribution, or the numerator degrees of freedom of F. |
| degreesOfFreedom2 | Denominator degrees of freedom | The second (denominator) degrees of freedom of the F distribution. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| critical | Critical value | The boundary of the rejection region: the right one for right-tailed and two-tailed tests, the left one for left-tailed tests. |
| lower | Left critical value | For a two-tailed test, the boundary of the left part of the rejection region. |
| region | Rejection region | The values of the test statistic that reject the null hypothesis, as intervals. |

## Method

Two-tailed: the α/2 and 1 − α/2 quantiles; left-tailed: the α quantile; right-tailed: the 1 − α quantile of the chosen distribution.

## Assumptions

- The critical values invert the CDF numerically, not read from a table. They are good to about 10 significant digits or better (13 for most inputs).
- Degrees of freedom may be fractional (for example Welch’s t-test), from 1 up to 100,000.
- The rejection region excludes its boundary, and shows each critical value to 4 decimal places, at most 8 significant figures.

## Worked examples

1. distribution = z, testType = two-tailed, significanceLevel = 0.05 gives critical = 1.959964, lower = -1.959964, region = (-∞, -1.9600) ∪ (1.9600, ∞). Source: NIST/SEMATECH e-Handbook 1.3.6.7.1 (z 0.975 = 1.96).
2. distribution = z, testType = left-tailed, significanceLevel = 0.01 gives critical = -2.326348, region = (-∞, -2.3263). Source: NIST/SEMATECH e-Handbook 1.3.6.7.1 (2.326).
3. distribution = t, testType = two-tailed, significanceLevel = 0.05, degreesOfFreedom = 2 gives critical = 4.302653, lower = -4.302653. Source: NIST/SEMATECH e-Handbook 1.3.6.7.2 (t 0.975, 2 = 4.303).
4. distribution = t, testType = right-tailed, significanceLevel = 0.05, degreesOfFreedom = 10 gives critical = 1.812461. Source: NIST/SEMATECH e-Handbook 1.3.6.7.2 (t 0.95, 10 = 1.812).
5. distribution = chi-square, testType = right-tailed, significanceLevel = 0.05, degreesOfFreedom = 2 gives critical = 5.991465, region = (5.9915, ∞). Source: NIST/SEMATECH e-Handbook 1.3.6.7.4 (5.991).
6. distribution = chi-square, testType = two-tailed, significanceLevel = 0.05, degreesOfFreedom = 5 gives lower = 0.831212, critical = 12.832502, region = [0, 0.8312) ∪ (12.8325, ∞). Source: NIST/SEMATECH e-Handbook 1.3.6.7.4 (0.831 and 12.833).
7. distribution = f, testType = right-tailed, significanceLevel = 0.05, degreesOfFreedom = 2, degreesOfFreedom2 = 10 gives critical = 4.102821. Source: NIST/SEMATECH e-Handbook 1.3.6.7.3 (F 0.95; 2, 10 = 4.103).
8. distribution = f, testType = right-tailed, significanceLevel = 0.05, degreesOfFreedom = 5, degreesOfFreedom2 = 10 gives critical = 3.325835. Source: NIST/SEMATECH e-Handbook 1.3.6.7.3 (F 0.95; 5, 10 = 3.326).

## FAQ

### What is a critical value?

A critical value is a point on the scale of a test statistic that defines the boundary of the rejection region in hypothesis testing. It's the threshold value that determines whether to reject or fail to reject the null hypothesis. Critical values depend on the significance level (α), the type of test (one-tailed or two-tailed), and the distribution of the test statistic.

### How do I choose the right distribution for my test?

The choice of distribution depends on your test statistic and sample characteristics:

- **Z-distribution:** Use when you have a large sample size (n ≥ 30) and know the population standard deviation, or when testing proportions
- **t-distribution:** Use when you have a small sample size and don't know the population standard deviation
- **Chi-square distribution:** Use for tests of independence, goodness-of-fit tests, and tests of variance
- **F-distribution:** Use for ANOVA tests, comparing variances, and testing overall significance in regression

### What's the difference between one-tailed and two-tailed tests?

**One-tailed tests** are used when you're only interested in detecting an effect in one direction (either positive or negative). They have more statistical power but can only detect effects in the specified direction.

**Two-tailed tests** are used when you want to detect an effect in either direction. They're more conservative and require stronger evidence to reject the null hypothesis, but they can detect effects in both directions.

### How do I interpret the rejection region?

The rejection region shows the range of test statistic values that would lead you to reject the null hypothesis. For example, if the rejection region is (-∞, -1.96) ∪ (1.96, ∞) for a two-tailed Z-test with α = 0.05, you would reject the null hypothesis if your test statistic is less than -1.96 or greater than 1.96.

### What significance level should I use?

The most commonly used significance levels are:

- **α = 0.05 (5%):** Most common in social sciences and general research
- **α = 0.01 (1%):** More conservative, used when you want to be very confident in your results
- **α = 0.10 (10%):** Less conservative, used in exploratory research or when sample sizes are small

Choose based on the consequences of making a Type I error (rejecting a true null hypothesis) in your specific context.

### How do degrees of freedom affect critical values?

Degrees of freedom (df) represent the number of independent pieces of information in your data:

- **For t-tests:** df = n - 1 (where n is sample size)
- **For chi-square tests:** df = (rows - 1) × (columns - 1) for contingency tables, or k - 1 for goodness-of-fit tests
- **For F-tests:** df1 = numerator degrees of freedom, df2 = denominator degrees of freedom

As degrees of freedom increase, t-distributions approach the normal distribution, and critical values become more stable.

### When should I use a left-tailed vs right-tailed test?

The choice depends on your alternative hypothesis:

- **Left-tailed:** Use when H₁ states the parameter is less than some value (e.g., μ < μ₀)
- **Right-tailed:** Use when H₁ states the parameter is greater than some value (e.g., μ > μ₀)
- **Two-tailed:** Use when H₁ states the parameter is not equal to some value (e.g., μ ≠ μ₀)

### What are common critical values I should know?

Here are some commonly used critical values for Z-tests with α = 0.05:

- **Two-tailed test:** ±1.96
- **Right-tailed test:** 1.645
- **Left-tailed test:** -1.645

For α = 0.01, the two-tailed critical value is ±2.576.

### How accurate are these critical value calculations?

The calculator inverts each distribution's cumulative distribution function numerically, to at least 10 significant digits (about 13 for most inputs), for any significance level and degrees of freedom. Printed tables round to 3 decimals (for example 1.960 and 2.228), so they agree with the calculator to that many places.

### Can I use critical values for confidence intervals?

Yes! Critical values are directly related to confidence intervals. For a confidence level of (1 - α), you use the same critical values as for a two-tailed test with significance level α. For example, for a 95% confidence interval, you use the critical values from a two-tailed test with α = 0.05.

### What's the relationship between critical values and p-values?

Critical values and p-values are two different approaches to hypothesis testing:

- **Critical value approach:** Compare your test statistic to the critical value(s)
- **P-value approach:** Calculate the probability of observing a test statistic as extreme as yours under the null hypothesis

Both approaches lead to the same conclusion: reject H₀ if your test statistic falls in the rejection region OR if your p-value is less than α.