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T-Distribution

This page provides indispensable tables containing critical values for the Student's t-distribution. The t-distribution is a probability distribution that is symmetrical and bell-shaped, much like the standard normal distribution (Z-distribution). However, it is characterized by having heavier tails than the normal distribution. This difference in shape reflects the greater uncertainty that arises in statistical inference when the population standard deviation ($\sigma$) is unknown and must be estimated from a potentially small sample of data.

A key characteristic of the t-distribution is that its precise shape depends on a single parameter known as the degrees of freedom (df). The degrees of freedom are directly related to the sample size used to estimate the population standard deviation. In many common statistical applications, such as estimating the mean of a single population or comparing the means of two populations using a t-test, the degrees of freedom are calculated as one less than the sample size, denoted as $n-1$, where 'n' is the sample size. As the degrees of freedom increase (typically meaning larger sample sizes), the t-distribution becomes progressively more similar in shape to the standard normal distribution.

These tables are specifically designed to provide the critical t-values. A critical t-value is a threshold value from the t-distribution that corresponds to a specific area in the tails of the distribution. This area represents the significance level, commonly denoted by $\alpha$ (alpha). The tables list critical t-values for various standard significance levels (e.g., $\alpha = 0.10, 0.05, 0.025, 0.01, 0.005$) and a range of degrees of freedom. The significance level $\alpha$ typically represents the probability of a Type I error in hypothesis testing (incorrectly rejecting a true null hypothesis) and is often split between the two tails for two-sided tests ($\alpha/2$).

In hypothesis testing using t-tests, a calculated t-statistic from sample data is compared to the appropriate critical t-value found in these tables. Based on this comparison and the chosen significance level $\alpha$, a decision is made whether to reject the null hypothesis or fail to reject it. If the absolute value of the calculated t-statistic exceeds the critical t-value from the table for the given degrees of freedom and $\alpha$, the result is considered statistically significant at that level.

Beyond hypothesis testing, these critical t-values are also indispensable for constructing confidence intervals for the population mean ($\mu$) when the population standard deviation ($\sigma$) is unknown. The critical t-value helps define the margin of error for the confidence interval, which provides a range of plausible values for the population mean based on the sample data and a chosen confidence level (e.g., 95% confidence corresponds to $\alpha=0.05$).

These tables are fundamental tools in statistical inference, particularly vital in research fields like biology, medicine, psychology, and social sciences, where sample sizes are often relatively small and population standard deviations are rarely known. They allow researchers to perform valid statistical analyses and make informed conclusions even when faced with the uncertainty inherent in small samples.

$\alpha$ (1 tail)
$\alpha$ (2 tail)
df		 0.05
0.10		 0.025
0.05		 0.01
0.02		 0.005
0.01		 0.0025
0.005		 0.001
0.002		 0.0005
0.001
1 6.3138 12.7065 31.8193 63.6551 127.3447 318.4930 636.0450
2 2.9200 4.3026 6.9646 9.9247 14.0887 22.3276 31.5989
3 2.3534 3.1824 4.5407 5.8408 7.4534 10.2145 12.9242
4 2.1319 2.7764 3.7470 4.6041 5.5976 7.1732 8.6103
5 2.0150 2.5706 3.3650 4.0322 4.7734 5.8934 6.8688
6 1.9432 2.4469 3.1426 3.7074 4.3168 5.2076 5.9589
7 1.8946 2.3646 2.9980 3.4995 4.0294 4.7852 5.4079
8 1.8595 2.3060 2.8965 3.3554 3.8325 4.5008 5.0414
9 1.8331 2.2621 2.8214 3.2498 3.6896 4.2969 4.7809
10 1.8124 2.2282 2.7638 3.1693 3.5814 4.1437 4.5869
11 1.7959 2.2010 2.7181 3.1058 3.4966 4.0247 4.4369
12 1.7823 2.1788 2.6810 3.0545 3.4284 3.9296 4.3178
13 1.7709 2.1604 2.6503 3.0123 3.3725 3.8520 4.2208
14 1.7613 2.1448 2.6245 2.9768 3.3257 3.7874 4.1404
15 1.7530 2.1314 2.6025 2.9467 3.2860 3.7328 4.0728
16 1.7459 2.1199 2.5835 2.9208 3.2520 3.6861 4.0150
17 1.7396 2.1098 2.5669 2.8983 3.2224 3.6458 3.9651
18 1.7341 2.1009 2.5524 2.8784 3.1966 3.6105 3.9216
19 1.7291 2.0930 2.5395 2.8609 3.1737 3.5794 3.8834
20 1.7247 2.0860 2.5280 2.8454 3.1534 3.5518 3.8495
21 1.7207 2.0796 2.5176 2.8314 3.1352 3.5272 3.8193
22 1.7172 2.0739 2.5083 2.8188 3.1188 3.5050 3.7921
23 1.7139 2.0686 2.4998 2.8073 3.1040 3.4850 3.7676
24 1.7109 2.0639 2.4922 2.7970 3.0905 3.4668 3.7454
25 1.7081 2.0596 2.4851 2.7874 3.0782 3.4502 3.7251
26 1.7056 2.0555 2.4786 2.7787 3.0669 3.4350 3.7067
27 1.7033 2.0518 2.4727 2.7707 3.0565 3.4211 3.6896
28 1.7011 2.0484 2.4671 2.7633 3.0469 3.4082 3.6739
29 1.6991 2.0452 2.4620 2.7564 3.0380 3.3962 3.6594
30 1.6973 2.0423 2.4572 2.7500 3.0298 3.3852 3.6459
31 1.6955 2.0395 2.4528 2.7440 3.0221 3.3749 3.6334
32 1.6939 2.0369 2.4487 2.7385 3.0150 3.3653 3.6218
33 1.6924 2.0345 2.4448 2.7333 3.0082 3.3563 3.6109
34 1.6909 2.0322 2.4411 2.7284 3.0019 3.3479 3.6008
35 1.6896 2.0301 2.4377 2.7238 2.9961 3.3400 3.5912
36 1.6883 2.0281 2.4345 2.7195 2.9905 3.3326 3.5822
37 1.6871 2.0262 2.4315 2.7154 2.9853 3.3256 3.5737
38 1.6859 2.0244 2.4286 2.7115 2.9803 3.3190 3.5657
39 1.6849 2.0227 2.4258 2.7079 2.9756 3.3128 3.5581
40 1.6839 2.0211 2.4233 2.7045 2.9712 3.3069 3.5510