What is the t critical value and how to calculate it by using the T table?

What is the t critical value and how to calculate it by using the T table?

Written by Deepak Bhagat, In Education, Published On
March 12, 2022
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Last modified on March 21st, 2022

In statistics, t critical value is used widely to solve the problems of hypothesis testing. The critical value is that value in hypothesis testing which tells whether to accept or reject the null hypothesis.

The null hypothesis is used in hypothesis testing to assure that either statistical relationship or significance exists among two sets of the observed data of the given single observed variable.

In this post, we will learn about the t critical value and how to calculate it by using a formula or by using the t tables.

What is t critical value?

T critical value or simply t value is a value used in statistics to measure the size of the difference relative to the variation in the sample data. It is usually used to determine the change represented in the units of standard error.

The greater magnitude or closer to zero magnitudes of T gives the evidence against the null hypothesis to have the greater significance difference or not. Greater magnitude shows the greater evidence that there is a significant difference. While in closer to zero magnitudes there is no significant difference.

The critical value has two regions one is the rejection region in which the null hypothesis is rejected and the other is the non-rejection region in which the null hypothesis. In simple words, the critical value is that value that cuts the test statistics value on scale into two regions one is the rejection region and the other is a non-rejection region.

While solving the problems of hypothesis testing, if the null hypothesis is rejected then the alternative hypothesis must be accepted. The t critical value can be left-tailed, right-tailed, or two-tailed.

  • Left-tailed test: The critical value from the left to find the area under the density curve is the left-tailed test and is denoted by α.
  • Right-tailed test: The critical value from the right to find the area under the density curve is the right-tailed test and is denoted by α.
  • Two-tailed test: The critical value from the left (α/2) and right (α/2) to find the area under the density curve is a two-tailed test and is denoted by α.

To calculate the t critical value by using the table, the degree of freedom must be known. The value used to calculate p-value and t-value for the t-test problems a term named as the degree of freedom is used.

How to calculate the t value by using the T table?

The t value or t critical value can be calculated easily by using tables if the degree of freedom and significance level is given. Two types of tables are used to calculate the t value. One is for one tail and the other is for two-tail values.

T distribution table for one tail

Degree of

freedom

A = 0.1 0.05 0.025 0.01 0.005 0.001 0.0005
ta= 1.282 1.645 1.960 2.326 2.576 3.091 3.291
1 3.078 6.314 12.706 31.821 63.656 318.289 636.578
2 1.886 2.920 4.303 6.965 9.925 22.328 31.600
3 1.638 2.353 3.182 4.541 5.841 10.214 12.924
4 1.533 2.132 2.776 3.747 4.604 7.173 8.610
5 1.476 2.015 2.571 3.365 4.032 5.894 6.869
6 1.440 1.943 2.447 3.143 3.707 5.208 5.959
7 1.415 1.895 2.365 2.998 3.499 4.785 5.408
8 1.397 1.860 2.306 2.896 3.355 4.501 5.041
9 1.383 1.833 2.262 2.821 3.250 4.297 4.781
10 1.372 1.812 2.228 2.764 3.169 4.144 4.587
11 1.363 1.796 2.201 2.718 3.106 4.025 4.437
12 1.356 1.782 2.179 2.681 3.055 3.930 4.318
13 1.350 1.771 2.160 2.650 3.012 3.852 4.221
14 1.345 1.761 2.145 2.624 2.977 3.787 4.140
15 1.341 1.753 2.131 2.602 2.947 3.733 4.073
16 1.337 1.746 2.120 2.583 2.921 3.686 4.015
17 1.333 1.740 2.110 2.567 2.898 3.646 3.965
18 1.330 1.734 2.101 2.552 2.878 3.610 3.922
19 1.328 1.729 2.093 2.539 2.861 3.579 3.883
20 1.325 1.725 2.086 2.528 2.845 3.552 3.850
21 1.323 1.721 2.080 2.518 2.831 3.527 3.819
22 1.321 1.717 2.074 2.508 2.819 3.505 3.792
23 1.319 1.714 2.069 2.500 2.807 3.485 3.768
24 1.318 1.711 2.064 2.492 2.797 3.467 3.745
25 1.316 1.708 2.060 2.485 2.787 3.450 3.725
1000 1.282 1.646 1.962 2.330 2.581 3.098 3.300

T distribution table for two tails

Degree of

freedom

A = 0.2 0.10 0.05 0.02 0.01 0.002 0.001
ta = 1.282 1.645 1.960 2.326 2.576 3.091 3.291
1 3.078 6.314 12.706 31.821 63.656 318.289 636.578
2 1.886 2.920 4.303 6.965 9.925 22.328 31.600
3 1.638 2.353 3.182 4.541 5.841 10.214 12.924
4 1.533 2.132 2.776 3.747 4.604 7.173 8.610
5 1.476 2.015 2.571 3.365 4.032 5.894 6.869
6 1.440 1.943 2.447 3.143 3.707 5.208 5.959
7 1.415 1.895 2.365 2.998 3.499 4.785 5.408
8 1.397 1.860 2.306 2.896 3.355 4.501 5.041
9 1.383 1.833 2.262 2.821 3.250 4.297 4.781
10 1.372 1.812 2.228 2.764 3.169 4.144 4.587
11 1.363 1.796 2.201 2.718 3.106 4.025 4.437
12 1.356 1.782 2.179 2.681 3.055 3.930 4.318
13 1.350 1.771 2.160 2.650 3.012 3.852 4.221
14 1.345 1.761 2.145 2.624 2.977 3.787 4.140
15 1.341 1.753 2.131 2.602 2.947 3.733 4.073
16 1.337 1.746 2.120 2.583 2.921 3.686 4.015
17 1.333 1.740 2.110 2.567 2.898 3.646 3.965
18 1.330 1.734 2.101 2.552 2.878 3.610 3.922
19 1.328 1.729 2.093 2.539 2.861 3.579 3.883
20 1.325 1.725 2.086 2.528 2.845 3.552 3.850
30 1.310 1.697 2.042 2.457 2.750 3.385 3.646
60 1.296 1.671 2.000 2.390 2.660 3.232 3.460
120 1.289 1.658 1.980 2.358 2.617 3.160 3.373
1.282 1.645 1.960 2.326 2.576 3.091 3.291

Example

Determine the t value for one tail and two tails if the degree of freedom is 15 and the significance level is 1%?

Solution

Step 1: Write the given values of degree of freedom and significance level.

Degree of freedom = Df = 15

Significance level = 1% = 1/100 = 0.01

Step 2: Look at the above table of t value for one tail and see the first row of the table for significance level and the first column for the degree of freedom and match the given values and write the corresponding value where both terms intersect.

T critical value for one tail = 2.602

Step 3: Look at the above table of t value for two tails and see the first row of the table for significance level and the first column for the degree of freedom and match the given values and write the corresponding value where both terms intersect.

T critical value for two tails = 2.947

You can also use the t value calculator to find the t value either by one-tail or two-tailed to avoid using larger tables to find the value. You can use this calculator to find the result of your problem according to the tables.

Step 1: You have to put the degree of freedom and significance level in the required box, select one tail or two tail options, and press the “calculate” button.

Step 2: You can get the result below the calculate button in a couple of seconds. This calculator also shows a table up to the given degree of freedom.

Summary

By using the above-mentioned tables, you can easily solve any problem related to this topic. The T value calculations are not tough, you have to find the values from the table. For a higher degree of freedom, you can simply use the above-mentioned calculator.

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