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the critical value of t

Student's T Critical Values

The values in the table are the areas critical values for the given areas in the right tail or in both tails.

Critical Value Calculator

What is a critical value, critical value definition, how to calculate critical values, how to use this critical value calculator, z critical values, t critical values, chi-square critical values (χ²), f critical values.

Welcome to the critical value calculator! Here you can quickly determine the critical value(s) for two-tailed tests, as well as for one-tailed tests. It works for most common distributions in statistical testing: the standard normal distribution N(0,1) (that is, when you have a Z-score), t-Student, chi-square, and F-distribution .

What is a critical value? And what is the critical value formula? Scroll down - we provide you with the critical value definition and explain how to calculate critical values in order to use them to construct rejection regions (also known as critical regions).

In hypothesis testing, critical values are one of the two approaches which allow you to decide whether to retain or reject the null hypothesis. The other approach is to calculate the p-value (for example, using the p-value calculator ).

The critical value approach consists of checking if the value of the test statistic generated by your sample belongs to the so-called rejection region , or critical region , which is the region where the test statistic is highly improbable to lie . A critical value is a cut-off value (or two cut-off values in case of a two-tailed test) that constitutes the boundary of the rejection region(s). In other words, critical values divide the scale of your test statistic into the rejection region and non-rejection region.

Once you have found the rejection region, check if the value of test statistic generated by your sample belongs to it :

But, how to calculate critical values? First of all, you need to set a significance level , α \alpha α , which quantifies the probability of rejecting the null hypothesis when it is actually correct. The choice of α is arbitrary; in practice, we most often use a value of 0.05 or 0.01. Critical values also depend on the alternative hypothesis you choose for your test , elucidated in the next section .

To determine critical values, you need to know the distribution of your test statistic under the assumption that the null hypothesis holds. Critical values are then the points on the distribution which have the same probability as your test statistic , equal to the significance level α \alpha α . These values are assumed to be at least as extreme at those critical values .

The alternative hypothesis determines what "at least as extreme" means. In particular, if the test is one-sided, then there will be just one critical value; if it is two-sided, then there will be two of them: one to the left and the other to the right of the median value of the distribution.

Critical values can be conveniently depicted as the points with the property that the area under the density curve of the test statistic from those points to the tails is equal to α \alpha α :

left-tailed test: the area under the density curve from the critical value to the left is equal to α \alpha α ;

right-tailed test: the area under the density curve from the critical value to the right is equal to α \alpha α ; and

two-tailed test: the area under the density curve from the left critical value to the left is equal to α 2 \frac{\alpha}{2} 2 α ​ and the area under the curve from the right critical value to the right is equal to α 2 \frac{\alpha}{2} 2 α ​ as well; thus, total area equals α \alpha α .

Critical values for symmetric distribution

As you can see, finding the critical values for a two-tailed test with significance α \alpha α boils down to finding both one-tailed critical values with a significance level of α 2 \frac{\alpha}{2} 2 α ​ .

The formulae for the critical values involve the quantile function , Q Q Q , which is the inverse of the cumulative distribution function ( c d f \mathrm{cdf} cdf ) for the test statistic distribution (calculated under the assumption that H 0 holds!): Q = c d f − 1 Q = \mathrm{cdf}^{-1} Q = cdf − 1

Once we have agreed upon the value of α \alpha α , the critical value formulae are the following:

left-tailed test : ( − ∞ , Q ( α ) ] (-\infty, Q(\alpha)] ( − ∞ , Q ( α )]

right-tailed test : [ Q ( 1 − α ) , ∞ ) [Q(1-\alpha), \infty) [ Q ( 1 − α ) , ∞ )

two-tailed test : ( − ∞ , Q ( α 2 ) ]   ∪   [ Q ( 1 − α 2 ) , ∞ ) (-\infty, Q(\frac{\alpha}{2})] \ \cup \ [Q(1 - \frac{\alpha}{2}), \infty) ( − ∞ , Q ( 2 α ​ )]   ∪   [ Q ( 1 − 2 α ​ ) , ∞ )

In the case of a distribution symmetric about 0 , the critical values for the two-tailed test are symmetric as well: Q ( 1 − α 2 ) = − Q ( α 2 ) Q(1 - \frac{\alpha}{2}) = -Q(\frac{\alpha}{2}) Q ( 1 − 2 α ​ ) = − Q ( 2 α ​ )

Unfortunately, the probability distributions that are the most widespread in hypothesis testing have somewhat complicated c d f \mathrm{cdf} cdf formulae. To find critical values by hand, you would need to use specialized software or statistical tables. In these cases, the best option is, of course, our critical value calculator! 😁

Now that you have found our critical value calculator, you no longer need to worry how to find critical value for all those complicated distributions! Here are the steps you need to follow:

Tell us the distribution of your test statistic under the null hypothesis: is it a standard normal N(0,1), t-Student, chi-squared, or Snedecor's F? If you are not sure, check the sections below devoted to those distributions, and try to localize the test you need to perform.

Choose the alternative hypothesis : two-tailed, right-tailed, or left-tailed.

If needed, specify the degrees of freedom of the test statistic's distribution. If you are not sure, check the description of the test you are performing. You can learn more about the meaning of this quantity in statistics from the degrees of freedom calculator .

Set the significance level, α \alpha α . We pre-set it to the most common value, 0.05, by default, but you can, of course, adjust it to your needs.

The critical value calculator will then display not only your critical value(s) but also the rejection region(s).

Go to the advanced mode of the critical value calculator if you need to increase the precision with which the critical values are computed.

Use the Z (standard normal) option if your test statistic follows (at least approximately) the standard normal distribution N(0,1) .

In the formulae below, u u u denotes the quantile function of the standard normal distribution N(0,1):

left-tailed Z critical value: u ( α ) u(\alpha) u ( α )

right-tailed Z critical value: u ( 1 − α ) u(1-\alpha) u ( 1 − α )

two-tailed Z critical value: ± u ( 1 − α 2 ) \pm u(1- \frac{\alpha}{2}) ± u ( 1 − 2 α ​ )

Check out Z-test calculator to learn more about the most common Z-test used on the population mean. There are also Z-tests for the difference between two population means, in particular, one between two proportions.

Use the t-Student option if your test statistic follows the t-Student distribution . This distribution is similar to N(0,1) , but its tails are fatter - the exact shape depends on the number of degrees of freedom . If this number is large (>30), which generically happens for large samples, then the t-Student distribution is practically indistinguishable from N(0,1). Check our t-statistic calculator to compute the related test statistic.

t-Student distribution densities

In the formulae below, Q t , d Q_{\text{t}, d} Q t , d ​ is the quantile function of the t-Student distribution with d d d degrees of freedom:

left-tailed t critical value: Q t , d ( α ) Q_{\text{t}, d}(\alpha) Q t , d ​ ( α )

right-tailed t critical value: Q t , d ( 1 − α ) Q_{\text{t}, d}(1 - \alpha) Q t , d ​ ( 1 − α )

two-tailed t critical values: ± Q t , d ( 1 − α 2 ) \pm Q_{\text{t}, d}(1 - \frac{\alpha}{2}) ± Q t , d ​ ( 1 − 2 α ​ )

Visit the t-test calculator to learn more about various t-tests: the one for a **population mean with an unknown population standard deviation, those for the difference between the means of two populations (with either equal or unequal population standard deviations), as well as about the t-test for paired samples .

Use the χ² (chi-square) option when performing a test in which the test statistic follows the χ²-distribution . You need to determine the number of degrees of freedom of the χ²-distribution of your test statistic - below, we list them for the most commonly used χ²-tests.

Here we give the formulae for chi square critical values; Q χ 2 , d Q_{\chi^2, d} Q χ 2 , d ​ is the quantile function of the χ²-distribution with d d d degrees of freedom:

Left-tailed χ² critical value: Q χ 2 , d ( α ) Q_{\chi^2, d}(\alpha) Q χ 2 , d ​ ( α )

Right-tailed χ² critical value: Q χ 2 , d ( 1 − α ) Q_{\chi^2, d}(1 - \alpha) Q χ 2 , d ​ ( 1 − α )

Two-tailed χ² critical values: Q χ 2 , d ( α 2 ) Q_{\chi^2, d}(\frac{\alpha}{2}) Q χ 2 , d ​ ( 2 α ​ ) and Q χ 2 , d ( 1 − α 2 ) Q_{\chi^2, d}(1 - \frac{\alpha}{2}) Q χ 2 , d ​ ( 1 − 2 α ​ )

Several different tests lead to a χ²-score:

Goodness-of-fit test : does the empirical distribution agree with the expected distribution?

This test is right-tailed . Its test statistic follows the χ²-distribution with k − 1 k - 1 k − 1 degrees of freedom, where k k k is the number of classes into which the sample is divided.

Independence test : is there a statistically significant relationship between two variables?

This test is also right-tailed , and its test statistic is computed from the contingency table. There are ( r − 1 ) ( c − 1 ) (r - 1)(c - 1) ( r − 1 ) ( c − 1 ) degrees of freedom, where r r r is the number of rows, and c c c is the number of columns in the contingency table.

Test for the variance of normally distributed data : does this variance have some pre-determined value?

This test can be one- or two-tailed! Its test statistic has the χ²-distribution with n − 1 n - 1 n − 1 degrees of freedom, where n n n is the sample size.

Finally, choose F (Fisher-Snedecor) if your test statistic follows the F-distribution . This distribution has a pair of degrees of freedom .

Let us see how those degrees of freedom arise. Assume that you have two independent random variables, X X X and Y Y Y , that follow χ²-distributions with d 1 d_1 d 1 ​ and d 2 d_2 d 2 ​ degrees of freedom, respectively. If you now consider the ratio ( X d 1 ) ÷ ( Y d 2 ) (\frac{X}{d_1})\div(\frac{Y}{d_2}) ( d 1 ​ X ​ ) ÷ ( d 2 ​ Y ​ ) , it turns out it follows the F-distribution with ( d 1 , d 2 ) (d_1, d_2) ( d 1 ​ , d 2 ​ ) degrees of freedom. That's the reason why we call d 1 d_1 d 1 ​ and d 2 d_2 d 2 ​ the numerator and denominator degrees of freedom , respectively.

In the formulae below, Q F , d 1 , d 2 Q_{\text{F}, d_1, d_2} Q F , d 1 ​ , d 2 ​ ​ stands for the quantile function of the F-distribution with ( d 1 , d 2 ) (d_1, d_2) ( d 1 ​ , d 2 ​ ) degrees of freedom:

Left-tailed F critical value: Q F , d 1 , d 2 ( α ) Q_{\text{F}, d_1, d_2}(\alpha) Q F , d 1 ​ , d 2 ​ ​ ( α )

Right-tailed F critical value: Q F , d 1 , d 2 ( 1 − α ) Q_{\text{F}, d_1, d_2}(1 - \alpha) Q F , d 1 ​ , d 2 ​ ​ ( 1 − α )

Two-tailed F critical values: Q F , d 1 , d 2 ( α 2 ) Q_{\text{F}, d_1, d_2}(\frac{\alpha}{2}) Q F , d 1 ​ , d 2 ​ ​ ( 2 α ​ ) and Q F , d 1 , d 2 ( 1 − α 2 ) Q_{\text{F}, d_1, d_2}(1 -\frac{\alpha}{2}) Q F , d 1 ​ , d 2 ​ ​ ( 1 − 2 α ​ )

Here we list the most important tests that produce F-scores: each of them is right-tailed .

ANOVA : tests the equality of means in three or more groups that come from normally distributed populations with equal variances. There are ( k − 1 , n − k ) (k - 1, n - k) ( k − 1 , n − k ) degrees of freedom, where k k k is the number of groups, and n n n is the total sample size (across every group).

Overall significance in regression analysis . The test statistic has ( k − 1 , n − k ) (k - 1, n - k) ( k − 1 , n − k ) degrees of freedom, where n n n is the sample size, and k k k is the number of variables (including the intercept).

Compare two nested regression models . The test statistic follows the F-distribution with ( k 2 − k 1 , n − k 2 ) (k_2 - k_1, n - k_2) ( k 2 ​ − k 1 ​ , n − k 2 ​ ) degrees of freedom, where k 1 k_1 k 1 ​ and k 2 k_2 k 2 ​ are the number of variables in the smaller and bigger models, respectively, and n n n is the sample size.

The equality of variances in two normally distributed populations . There are ( n − 1 , m − 1 ) (n - 1, m - 1) ( n − 1 , m − 1 ) degrees of freedom, where n n n and m m m are the respective sample sizes.

What is a Z critical value?

A Z critical value is the value that defines the critical region in hypothesis testing when the test statistic follows the standard normal distribution . If the value of the test statistic falls into the critical region, you should reject the null hypothesis and accept the alternative hypothesis.

How do I calculate Z critical value?

To find a Z critical value for a given confidence level α :

Is a t critical value the same as Z critical value?

In theory, no . In practice, very often, yes . The t-Student distribution is similar to the standard normal distribution, but it is not the same . However, if the number of degrees of freedom (which is, roughly speaking, the size of your sample) is large enough (>30), then the two distributions are practically indistinguishable , and so the t critical value has practically the same value as the Z critical value.

What is the Z critical value for 95% confidence?

The Z critical value for a 95% confidence interval is:

Standard deviation

Wilcoxon rank-sum test.

Mastering Data Analytics

Related Articles

How to Calculate Critical t-Value in R ?

A critical-T value is a “cut-off point” on the t distribution. A t-distribution is a probability distribution that is used to calculate population parameters when the sample size is small and when the population variance is unknown. T values are used to analyze whether to support or reject a null hypothesis. 

After conducting a t-test, we get its statistics as result. In order to determine the significance of the result, we compare the t- score obtained by a critical t value.  If the absolute value of the t-score is greater than the t critical value, then the results of the test are statistically significant.

Function used:

In order to find the T Critical value we make use of qt() function provided in R Programming Language. 

Syntax: qt(p=conf_value, df= df_value, lower.tail=True/False) Parameters: p:- Confidence level df: degrees of freedom lower.tail: If TRUE, the probability to the left of p in the t distribution is returned. If FALSE, the probability to the right is returned. By default  it’s value is TRUE.

There are three methods for calculating critical t value, all of them are discussed below:

Method 1: Right tailed test

A right-tailed test is a test in which the hypothesis statement contains a greater than (>) symbol i.e. the inequality points to the right. Sometimes it is also referred to as the upper test.

Here we are assuming a confidence value of 96% i.e.. p= .04 and degree of freedom 4 i.e.. df=4 . We are also using the format() function to reduce the decimal value to three decimal places. For the Right tail test, we are setting the value of lower.tail as FALSE .

The t critical value is 2.333. Thus, if the test score is greater than this value, the results of the test are statistically significant.

Method 2: Left tailed test

A left-tailed test is a test in which the hypothesis statement contains a less than (<) symbol i.e… the inequality points to the left. Sometimes it is also referred to as the lower test.

Here we are assuming a confidence value of 95% i.e.. p= .05 and degree of freedom 4 i.e.. df=4 .  We are also using the format() function to reduce the decimal value to three decimal places. For the Left tail test, we are setting the value of lower.tail as TRUE .

The t critical value is -2.132. Thus, if the test score is less than this value, the results of the test are statistically significant.

Method 3: Two-tailed test

A two-tailed test is a test in which the hypothesis statement contains both a greater than (>) symbol and a less-than symbol(<) i.e. the inequality points between a certain range. 

In a two-tailed test, we simply need to pass in half of our confidence level in “p” parameter. Here we are assuming confidence value of 96% i.e.. p= .04 and degree of freedom 4 i.e.. df=4 .  We are also using the format() function to reduce the decimal value to three decimal places.

Whenever, we perform a two-tailed test we get two critical values as output. So here in the above code, the T critical values are 2.999 and -2.999. Therefore, if the test score is less than -2.999 or greater than 2.999, the results of the test are statistically significant.

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Critical Value Calculator

This quick calculator allows you to calculate a critical valus for the z , t , chi-square, f and r distributions.

Critical Value for T

Select your significance level (1-tailed), input your degrees of freedom, and then hit "Calculate for T".

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Critical Value for Z

Select your significance level (1-tailed), and then hit "Calculate for Z".

Critical Value for Chi-Square

Select your significance level, input your degrees of freedom, and then hit "Calculate for Chi-Square".

Critical Value for F

Select your significance level (1-tailed), input your degrees of freedom for both numerator and denominator, and then hit "Calculate for F".

Critical Value for R

Select your significance level (1-tailed), input your degrees of freedom ( n - 2), and hit "Calculate for R".

the critical value of t

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AP®︎/College Statistics

Unit 11: lesson 1.

Example finding critical t value

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Video transcript

t-test introduction

Case I represents the null hypothesis (H O : µ 1 = µ 2 ) indicating that the mean of group one equals the mean of group two; both samples come from the same population. This would signify that the drug had no effect on blood pressure. The difference in the means is small, suggesting that they come from the same population. Case II represents the alternate hypothesis (H A :µ 1 ≠ µ 2 ), indicating that the mean of group one does not equal the mean of group two; the two sample means are from different populations. The difference in the means is too large to come from one population in most cases. Hence the means are probably coming from two different populations. A t-test decides which of these hypotheses to accept.

In Figure 2B, the difference in the sample means is larger, therefore, it is likely that the means come from two different populations. However, look at Figure 2C. It is possible that the two means could come from the same population and have the same difference. It is not likely because the probability (area under curve) of getting a small sample mean (x 1 ) or a large sample mean (x 2 ) from population 1 is small. If you accept the alternate hypothesis (H A :µ 1 ≠ µ 2 ), indicating the means come from two different populations (Case II); it is more likely you will be correct. But you could be wrong. There is not a high probability, but the null hypothesis (H O : µ 1 = µ 2 ) could be true (Case III). How many times out of 100 are you willing to be wrong?

Alpha Level (α)

An alpha level represents the number of times out of 100 you are willing to be incorrect if you reject the null hypothesis. If you choose an alpha level of 0.05, 5 times out of 100 you will be incorrect if you reject the null hypothesis. Those five times, both means would come from the same population (Case III). But that's about it. 95 times out of 100, you will be correct because it is more likely that the means come from two different populations (Case II). The difference in the means is large enough that it is most likely that the means come from two different populations.

t-distribution's relation to t-test

Instead of comparing the t-critical and t-statistical values to determine significant difference, you may also compare the alpha level and p-values. In Figure 4, the alpha level would be the area under the curve to the right of the positive t-critical and to the left of the negative t-critical (all gray and light blue). Together, these areas total the alpha-level, 0.05. The p-value is the area under the curve to the right of the purple t-statistic plus the area to the left of the negative, purple t-statistic (the light blue only). For the drug study, this area equals 0.0329. Because the p-value is then less than the alpha level, the alternate hypothesis is accepted. However, if the p-value was greater than the alpha level, p>α, (the blue covered the gray), the null hypothesis would be retained.

Copyright © 2003 Central Virginia Governor's School for Science and Technology Lynchburg, VA

IMAGES

  1. Critical T Value Table 2 Tailed

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  2. T Critical Value: Easy Definition, Calculating

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  3. probability

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  4. Critical value of t for the Student's t Distribution.

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  5. 588 Statistical Tables Appendix A Table A.2 Critical

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  6. Critical values and moments for the individual t-tests.

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VIDEO

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  2. Recording #3517

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