Critical Value Calculator

Q: What is a critical value in statistics?

It is the cutoff on the test distribution that separates the rejection region from the rest. If the test statistic is beyond it, you reject the null hypothesis.

Q: How do you find a t critical value?

Use the significance level, the number of tails and the degrees of freedom. For a two-tailed test at α = 0.05 with df = 10, the t critical value is 2.228.

Q: Why is the two-tailed critical value larger?

The significance level is split between two tails, so each tail holds only α ÷ 2. You must go further from the center to leave that smaller area.

Q: Do chi-square and F tests have two tails?

They are usually right-tailed, because large statistics show a poor fit or a real difference. A two-tailed version is used for some variance tests, and then there are separate lower and upper critical values.

StatisticsFreeNo sign-up

A critical value marks where the rejection region of a test starts. For a two-tailed z test at α = 0.05 it is ±1.96; for a t test with 10 degrees of freedom it is ±2.228. Pick the distribution, the significance level and the tails.

FreeNo sign-up

Distribution

Tails

Significance level α

0.05 for 95% confidence, 0.01 for 99%.

Degrees of freedom

For F, this is the numerator df.

Critical value

±2.2281

For a two-tailed t test at α = 0.05, the critical value is ±2.2281.

Inputs
Distribution	t (df = 10)
α	0.05
Tails	two-tailed
Critical value
Reject H₀ if	t < -2.2281 or t > 2.2281
Confidence level	95%

Show steps

Split α between the two tails: 0.05 ÷ 2 = 0.025 in each tail.
Find the value with area 0.975 to its left: 2.2281.

How to use the calculator

Choose the distribution your test uses: z for large samples or known σ, t for means with an estimated standard deviation, chi-square for variance and goodness-of-fit tests, F for ANOVA and comparing variances.
Choose the tails from your alternative hypothesis: ≠ is two-tailed, > is right-tailed, < is left-tailed.
Enter α and the degrees of freedom, then press Calculate.

Common critical values

Confidence	α	z one-tailed	z two-tailed
90%	0.10	1.2816	±1.6449
95%	0.05	1.6449	±1.9600
98%	0.02	2.0537	±2.3263
99%	0.01	2.3263	±2.5758

df	t two-tailed α = 0.10	α = 0.05	α = 0.01
5	2.015	2.571	4.032
10	1.812	2.228	3.169
20	1.725	2.086	2.845
30	1.697	2.042	2.750

As the degrees of freedom grow, t critical values move toward the z values.

Critical value vs p-value

Both lead to the same decision. With critical values you compare your test statistic with the cutoff: reject H₀ if it lands in the rejection region. With p-values you compare p with α: reject H₀ if p ≤ α. A z statistic of 2.3 in a two-tailed test at α = 0.05 is beyond 1.96, and its p-value of about 0.021 is below 0.05.

Degrees of freedom

One-sample t test: df = n − 1.
Two-sample t test with equal variances: df = n₁ + n₂ − 2.
Chi-square goodness of fit: df = categories − 1; test of independence: (rows − 1)(columns − 1).
One-way ANOVA F test: numerator df = groups − 1, denominator df = total observations − groups.

Critical value practice questions

Test yourself with 6 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What is a critical value in statistics?

It is the cutoff on the test distribution that separates the rejection region from the rest. If the test statistic is beyond it, you reject the null hypothesis.

How do you find a t critical value?

Use the significance level, the number of tails and the degrees of freedom. For a two-tailed test at α = 0.05 with df = 10, the t critical value is 2.228.

Why is the two-tailed critical value larger?

The significance level is split between two tails, so each tail holds only α ÷ 2. You must go further from the center to leave that smaller area.

Do chi-square and F tests have two tails?

They are usually right-tailed, because large statistics show a poor fit or a real difference. A two-tailed version is used for some variance tests, and then there are separate lower and upper critical values.

Sources

Related calculators

Browse all calculators

More statistics calculators

Descriptive Statistics CalculatorPaste a data set to get mean, median, mode, range, variance, standard deviation, quartiles and IQR, with a box plot.Open →Standard Deviation CalculatorEnter a data set for the population and sample standard deviation, variance, and mean, with steps and a dot plot.Open →Probability CalculatorFind the probability of a single event, or combine two independent events (and / or / neither), with the steps.Open →Permutations & Combinations CalculatorEnter n and r to get nPr and nCr with the factorial formulas worked out.Open →Binomial Probability CalculatorFind P(X = k) and the cumulative probabilities for n trials with success probability p, plus the mean, variance and SD.Open →Pascal's Triangle CalculatorBuild Pascal's triangle, find any row or entry C(n, k), expand (x + y)^n and see the Sierpinski pattern.Open →

Browse all 517 calculators

Effortless Math Team

37 minutes ago

Effortless Math Team

Related to This Article

What people say about "Critical Value Calculator: z, t, Chi-Square and F"?

No one replied yet.

Critical Value Calculator

How to use the calculator

Common critical values

Critical value vs p-value

Degrees of freedom

Critical value practice questions

Quiz complete