acalculator

What is the critical value for my test?

Find the critical values and rejection region of a z, t, chi-square, or F test at your significance level.

Your numbers

Distribution
Test type
Common values: 0.10, 0.05, 0.01.
Critical value
1.96

The rejection region is (-∞, -1.9600) ∪ (1.9600, ∞) (Two-tailed, Z, α = 0.05).

Left critical value
−1.96
Rejection region
(-∞, -1.9600) ∪ (1.9600, ∞)

Critical value: 1.96. The rejection region is (-∞, -1.9600) ∪ (1.9600, ∞) (Two-tailed, Z, α = 0.05).

Where is the rejection region?

How to calculate

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.

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).

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.

  • 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.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Distribution Z, Test type Two-tailed, Significance level (α) 0.05 gives Critical value 1.959964, Left critical value -1.959964, Rejection region (-∞, -1.9600) ∪ (1.9600, ∞).Source: NIST/SEMATECH e-Handbook 1.3.6.7.1 (z 0.975 = 1.96)
  2. Distribution Z, Test type Left-tailed, Significance level (α) 0.01 gives Critical value -2.326348, Rejection region (-∞, -2.3263).Source: NIST/SEMATECH e-Handbook 1.3.6.7.1 (2.326)
  3. Distribution t, Test type Two-tailed, Significance level (α) 0.05, Degrees of freedom 2 gives Critical value 4.302653, Left critical value -4.302653.Source: NIST/SEMATECH e-Handbook 1.3.6.7.2 (t 0.975, 2 = 4.303)
  4. Distribution t, Test type Right-tailed, Significance level (α) 0.05, Degrees of freedom 10 gives Critical value 1.812461.Source: NIST/SEMATECH e-Handbook 1.3.6.7.2 (t 0.95, 10 = 1.812)
  5. Distribution Chi-square, Test type Right-tailed, Significance level (α) 0.05, Degrees of freedom 2 gives Critical value 5.991465, Rejection region (5.9915, ∞).Source: NIST/SEMATECH e-Handbook 1.3.6.7.4 (5.991)
  6. Distribution Chi-square, Test type Two-tailed, Significance level (α) 0.05, Degrees of freedom 5 gives Left critical value 0.831212, Critical value 12.832502, Rejection region [0, 0.8312) ∪ (12.8325, ∞).Source: NIST/SEMATECH e-Handbook 1.3.6.7.4 (0.831 and 12.833)
  7. Distribution F, Test type Right-tailed, Significance level (α) 0.05, Degrees of freedom 2, Denominator degrees of freedom 10 gives Critical value 4.102821.Source: NIST/SEMATECH e-Handbook 1.3.6.7.3 (F 0.95; 2, 10 = 4.103)
  8. Distribution F, Test type Right-tailed, Significance level (α) 0.05, Degrees of freedom 5, Denominator degrees of freedom 10 gives Critical value 3.325835.Source: NIST/SEMATECH e-Handbook 1.3.6.7.3 (F 0.95; 5, 10 = 3.326)

How it works

A critical value is a quantile of the test statistic's distribution under the null hypothesis. With significance level α:

  • Two-tailed: the left critical value is the α/2 quantile and the right critical value is the 1 − α/2 quantile. The rejection region is below the left value or above the right value.
  • Left-tailed: the critical value is the α quantile. The rejection region is below it.
  • Right-tailed: the critical value is the 1 − α quantile. The rejection region is above it.

The p quantile of a distribution is the x with P(X ≤ x) = p, found by inverting the cumulative distribution function (CDF). The distributions:

  • Z: the standard normal distribution N(0, 1).
  • t: Student's t with ν degrees of freedom.
  • Chi-square: χ² with k degrees of freedom. Its values are never negative.
  • F: F with d1 (numerator) and d2 (denominator) degrees of freedom. Its values are never negative.

For Z and t the distribution is symmetric, so the two-tailed left value is minus the right value. Chi-square and F are not symmetric, so the two values differ.

The page shows each critical value to 4 decimal places. The rejection region is written as intervals. Each end is rounded to 4 decimal places, but to at most 8 significant figures: with d digits before the decimal point, it keeps min(4, 8 − d) decimal places, and a number with more than 8 digits before the point is rounded to 8 significant figures and written in full (37149.187, 520549970000000). Numbers below 1 count as d = 1. Rounding is half away from zero on the exact binary value. For example (-∞, -1.9600) ∪ (1.9600, ∞) for a two-tailed test, (-∞, -2.3263) for a left-tailed test, and (5.9915, ∞) for a right-tailed test. For chi-square and F, whose values start at 0, the left part starts at 0 and includes it: [0, 0.8312) ∪ (12.8325, ∞). (The old page wrote (0, …).)

Assumptions

  • The critical values invert each CDF numerically; they are not read from a table. The CDFs come from the incomplete gamma and beta functions, which lose some precision for large degrees of freedom, so the values are good to about 10 significant digits in the worst case (13 or more for most inputs). That is why large region ends stop at 8 significant figures.
  • The significance level is between 0.0000000001 and 0.5.
  • Degrees of freedom may be fractional (for example Welch's t-test), from 1 up to 100,000.
  • The boundary itself is not in the rejection region.

Worked examples by hand

Two-tailed Z test, α = 0.05. The right value is the 0.975 quantile of N(0, 1). The standard normal table gives 1.96 (1.959964 to 6 decimals), so the region is (-∞, -1.9600) ∪ (1.9600, ∞). The old page showed ±0.2665 here, which is wrong.

Left-tailed Z test, α = 0.01. The 0.01 quantile of N(0, 1) is −2.3263.

Two-tailed t test, α = 0.05, 2 degrees of freedom. For ν = 2 the t CDF has a closed form, and its inverse is t = (2p − 1) ÷ √(2p(1 − p)). At p = 0.975: 0.95 ÷ √(2 × 0.975 × 0.025) = 0.95 ÷ √0.04875 = 4.3027. The t table gives 4.303.

Right-tailed chi-square test, α = 0.05, 2 degrees of freedom. For k = 2, P(X > x) = e^(−x/2), so x = −2 ln α = −2 ln 0.05 = 5.9915. The chi-square table gives 5.991.

Two-tailed chi-square test, α = 0.05, 5 degrees of freedom. The 0.025 and 0.975 quantiles are 0.8312 and 12.8325 (table: 0.831 and 12.833).

Right-tailed F test, α = 0.05, d1 = 2, d2 = 10. For d1 = 2, P(F > x) = (1 + 2x ÷ d2)^(−d2/2), so x = (d2 ÷ 2)(α^(−2/d2) − 1) = 5 × (0.05^(−0.2) − 1) = 5 × 0.820564 = 4.1028. The F table gives 4.103.

Other questions people ask

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.