Critical Value Calculator (2024)

Use this calculator for critical values to easily convert a significance level to its corresponding Z value, T score, F-score, or Chi-square value. Outputs the critical region as well. The tool supports one-tailed and two-tailed significance tests / probability values.

Quick navigation:

  1. Using the critical value calculator
  2. What is a critical value?
  3. T critical value calculation
  4. Z critical value calculation
  5. F critical value calculation

Using the critical value calculator

If you want to perform a statistical test of significance (a.k.a. significance test, statistical significance test), determining the value of the test statistic corresponding to the desired significance level is necessary. You need to know the desired error probability (p-value threshold, common values are 0.05, 0.01, 0.001) corresponding to the significance level of the test. If you know the significance level in percentages, simply subtract it from 100%. For example, 95% significance results in a probability of 100%-95% = 5% = 0.05.

Then you need to know the shape of the error distribution of the statistic of interest (not to be mistaken with the distribution of the underlying data!). Our critical value calculator supports statistics which are either:

  • Z-distributed (normally distributed, e.g. absolute difference of means)
  • T-distributed (Student's T distribution, usually appropriate for small sample sizes, equivalent to the normal for sample sizes over 30)
  • X2-distributed (Chi square distribution, often used in goodness-of-fit tests, but also for tests of hom*ogeneity or independence)
  • F-distributed (Fisher-Snedecor distribution), usually used in analysis of variance (ANOVA)

Then, for distributions other than the normal one (Z), you need to know the degrees of freedom. For the F statistic there are two separate degrees of freedom - one for the numerator and one for the denominator.

Finally, to determine a critical region, one needs to know whether they are testing a point null versus a composite alternative (on both sides) or a composite null versus (covering one side of the distribution) a composite alternative (covering the other). Basically, it comes down to whether the inference is going to contain claims regarding the direction of the effect or not. Should one want to claim anything about the direction of the effect, the corresponding null hypothesis is direction as well (one-sided hypothesis).

Depending on the type of test - one-tailed or two-tailed, the calculator will output the critical value or values and the corresponding critical region. For one-sided tests it will output both possible regions, whereas for a two-sided test it will output the union of the two critical regions on the opposite sides of the distribution.

What is a critical value?

A critical value (or values) is a point on the support of an error distribution which bounds a critical region from above or below. If the statistics falls below or above a critical value (depending on the type of hypothesis, but it has to fall inside the critical region) then a test is declared statistically significant at the corresponding significance level. For example, in a two-tailed Z test with critical values -1.96 and 1.96 (corresponding to 0.05 significance level) the critical regions are from -∞ to -1.96 and from 1.96 to +∞. Therefore, if the statistic falls below -1.96 or above 1.96, the null hypothesis test is statistically significant.

You can think of the critical value as a cutoff point beyond which events are considered rare enough to count as evidence against the specified null hypothesis. It is a value achieved by a distance function with probability equal to or greater than the significance level under the specified null hypothesis. In an error-probabilistic framework, a proper distance function based on a test statistic takes the generic form [1]:

Critical Value Calculator (1)

X (read "X bar") is the arithmetic mean of the population baseline or the control, μ0 is the observed mean / treatment group mean, while σx is the standard error of the mean (SEM, or standard deviation of the error of the mean).

Here is how it looks in practice when the error is normally distributed (Z distribution) with a one-tailed null and alternative hypotheses and a significance level α set to 0.05:

Critical Value Calculator (2)

And here is the same significance level when applied to a point null and a two-tailed alternative hypothesis:

Critical Value Calculator (3)

The distance function would vary depending on the distribution of the error: Z, T, F, or Chi-square (X2). The calculation of a particular critical value based on a supplied probability and error distribution is simply a matter of calculating the inverse cumulative probability density function (inverse CPDF) of the respective distribution. This can be a difficult task, most notably for the T distribution [2].

T critical value calculation

The T-distribution is often preferred in the social sciences, psychiatry, economics, and other sciences where low sample sizes are a common occurrence. Certain clinical studies also fall under this umbrella. This stems from the fact that for sample sizes over 30 it is practically equivalent to the normal distribution which is easier to work with. It was proposed by William Gosset, a.k.a. Student, in 1908 [3], which is why it is also referred to as "Student's T distribution".

To find the critical t value, one needs to compute the inverse cumulative PDF of the T distribution. To do that, the significance level and the degrees of freedom need to be known. The degrees of freedom represent the number of values in the final calculation of a statistic that are free to vary whilst the statistic remains fixed at a certain value.

It should be noted that there is not, in fact, a single T-distribution, but there are infinitely many T-distributions, each with a different level of degrees of freedom. Below are some key values of the T-distribution with 1 degree of freedom, assuming a one-tailed T test is to be performed. These are often used as critical values to define rejection regions in hypothesis testing.

Probability to T critical value table
Probability valueDegrees of FreedomT critical value

Z critical value calculation

The Z-score is a statistic showing how many standard deviations away from the normal, usually the mean, a given observation is. It is often called just a standard score, z-value, normal score, and standardized variable. A Z critical value is just a particular cutoff in the error distribution of a normally-distributed statistic.

Z critical values are computed by using the inverse cumulative probability density function of the standard normal distribution with a mean (μ) of zero and standard deviation (σ) of one. Below are some commonly encountered probability values (significance levels) and their corresponding Z values for the critical region, assuming a one-tailed hypothesis.

Probability to Z critical value table
Probability valueZ critical value

The critical region defined by each of these would span from the Z value to plus infinity for the right-tailed case, and from minus infinity to minus the Z critical value in the left-tailed case. Our calculator for critical value will both find the critical z value(s) and output the corresponding critical regions for you.

Chi Square (Χ2) critical value calculation

Chi square distributed errors are commonly encountered in goodness-of-fit tests and hom*ogeneity tests, but also in tests for independence in contingency tables. Since the distribution is based on the squares of scores, it only contains positive values. Calculating the inverse cumulative PDF of the distribution is required in order to convert a desired probability (significance) to a chi square critical value.

Just like the T and F distributions, there is a different chi square distribution corresponding to different degrees of freedom. Hence, to calculate a Χ2 critical value one needs to supply the degrees of freedom for the statistic of interest.

F critical value calculation

F distributed errors are commonly encountered in analysis of variance (ANOVA), which is very common in the social sciences. The distribution, also referred to as the Fisher-Snedecor distribution, only contains positive values, similar to the Χ2 one. Similar to the T distribution, there is no single F-distribution to speak of. A different F distribution is defined for each pair of degrees of freedom - one for the numerator and one for the denominator.

Calculating the inverse cumulative PDF of the F distribution specified by the two degrees of freedom is required in order to convert a desired probability (significance) to a critical value. There is no simple solution to find a critical value of f and while there are tables, using a calculator is the preferred approach nowadays.


1 Mayo D.G., Spanos A. (2010) – "Error Statistics", in P. S. Bandyopadhyay & M. R. Forster (Eds.), Philosophy of Statistics, (7, 152–198). Handbook of the Philosophy of Science. The Netherlands: Elsevier.

2 Shaw T.W. (2006) – "Sampling Student's T distribution – use of the inverse cumulative distribution function", Journal of Computational Finance 9(4):37-73, DOI:10.21314/JCF.2006.150

3 "Student" [William Sealy Gosset] (1908) - "The probable error of a mean", Biometrika 6(1):1–25. DOI:10.1093/biomet/6.1.1

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Critical Value Calculator (2024)


What is the critical value of 0.05 in a two-tailed test? ›

For example if �� = 0.05 and it is an upper tailed test, the critical value is 1.645. For a lower tailed test it is -1.645. But if it is two tailed test then the critical values are -1.96 and 1.96.

How to find critical t-value with confidence level and sample size calculator? ›

Step 1: Express the confidence level as a number (decimal) with 0 < c < 1 . Step 2: Obtain the significance level, denoted , by α = 1 − c . Step 3: Use the -table or a calculator to obtain the -score (critical value) t α / 2 where (i) the is from Step 2 and (ii) the degrees of freedom equals , where is the sample size.

What is the critical value of 95%? ›

For a 95% confidence interval Zcritical = 1.96.

What is your critical value? ›

A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval, or which defines the threshold of statistical significance in a statistical test.

How to find critical value for 2 sample t test? ›

For our two-tailed t-test, the critical value is t1-α/2,ν = 1.9673, where α = 0.05 and ν = 326. If we were to perform an upper, one-tailed test, the critical value would be t1-α,ν = 1.6495.

How to calculate the T critical value? ›

How to find the critical value of t?
  1. Solution: Step 1: Identify the values.
  2. Degree of freedom = 30. Step 2: Look for the significance level in the top row of the t distribution table below (one tail) and degree of freedom (df) on the left side of the table. ...
  3. T critical value (one-tailed) = 1.6978.

What is the critical value of 0.025 two tailed? ›

Since this is a two-tailed test, of of the values would be in the left tail, and the other 2.5% would be in the right tail. Looking up the -score associated with 0.025 on a reference table, we find 1.96. Therefore, +1.96 is the critical value of the right tail and -1.96 is the critical value of the left tail.

What is an example of a critical value? ›

If the z -score is used then reading straight from the tables gives the critical values. For example, the critical values for a 5 % significance test are: For a one-tailed test, the critical value is 1.645 . So the critical region is Z<−1.645 for a left-tailed test and Z>1.645 for a right-tailed test.

How to calculate critical value in Excel? ›

You can use the T. INV() function to find the critical value of t for one-tailed tests in Excel, and you can use the T. INV. 2T() function for two-tailed tests.

How to find critical value from confidence level? ›

Example question: Find a critical value for a 90% confidence level (Two-Tailed Test). Step 1: Subtract the confidence level from 100% to find the α level: 100% – 90% = 10%. Step 2: Convert Step 1 to a decimal: 10% = 0.10. Step 3: Divide Step 2 by 2 (this is called “α/2”).

What is the critical value of 99.5 confidence interval? ›

Short Answer. The critical value z α 2 for 99.5% level of confidence is 2.81.

What is the critical value of 85? ›

Answer and Explanation:

1.44 is correct for an 85% confidence level.

How to calculate the critical value of z? ›

Step 1: Determine the confidence level, denoted , where is a number (decimal) between 0 and 100. Step 2: Obtain the confidence level, denoted by evaluating α = 1 − C 100 . Step 3: Use the -table (or a calculator) to obtain the -score z α / 2 .

What is the Z critical value calculator? ›

The Z critical value calculator calculates the critical value for the z statistic (normal distribution), choose the normal distribution and enter the mean and standard deviation. For a standard normal distribution enter μ = 0 and &sigma = 1.

How do you find the critical value of 80%? ›

We see that in the z table, a probability of 0.9 lies between 1.28 and 1.29. Thus, the value of the critical value is approximately 1.285. Hence, the critical value of a z score for 80% confidence interval is 1.285.

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