Given below are two T-tables (also known as T-Distribution Tables or Student’s T-Table). There are two T Tables provided below for you to use depending on whether you’re dealing with an one-tailed T-distribution or a two-tailed T-distribution
T Table (One Tail)
T Table ( Two Tail)
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How to Use the T Table
Step 1: To calculate the score for a T Distribution, find out the ‘df’ that is the ‘degrees of freedom’. Finding out df is easy as all you have to do is subtract one from your sample size and what you get will be your df or degrees of freedom.
Step 2: For using the table given above look up the df in the left hand side of the respective, one tail or two tailed T Table. Then locate the column under your alpha level which is usually provided to you.
When is T Distribution used?
T Distribution is used when you have a small sample size because otherwise the T Distribution is almost identical to normal distribution with the only difference being that the T distribution curve is shorter and fatter than normal distribution curve
T statistic formula or T Score formula
T statistic = (Sample mean – hypothesised mean)/sample standard error
Hence we can see that how large or how small the T statistic is depends on how close or far away the sample mean is from the hypothesised mean. If the sample mean is close to hypothesised mean, we will get a T statistic close to zero. Whereas if the sample mean if far away from the hypothesised mean, we will get a larger T statistic.
Why is T Table called as Student’s T Table or Student’s T Distribution?
The term ‘Student’ has nothing to do with the literal term student as used in the English language per se. But rather from William Sealy Gosset to whom the T-distrubtion is attributed to. William Sealy Gosset’s pen name ‘Student’ was used in 1908 to publish the distrubtion for the first time in 1908 in the paper Biometrika. The Student’s t-distribution was also initially referred to as ‘Student’s Z’ and ‘Student’s test of statistical significance’ before being commonly called Student’s t-distribution as it is known today.
What is one tail vs two tail?
Let us understand first what a ‘tail’ is when it comes to t distribution and then let us figure out when to use a one-tailed t test vs two-tailed t test.
The ‘tail’ in terms of any distribution refers to end of the distribution of the test statistic. As you can see in the image alongside, the black shaded areas of the distributions are the tails. In the image where both the ends of the distribution is shaded it is said to be two-tailed and where only one end of the distribution is shaded, it is one-tailed. Usually distribution patterns like t distributions and z distributions are two tailed. Whereas asymmetrical distributions like Chi-square distributions and F distribution will have only one tail. One-tailed tests are also known as directional tests whereas two-tailed tests are also known as non-directional tests.
So how do you choose whether you want to use a one-tailed t test or a two-tailed t test? A simple way to determine that is by checking if you want to use both the negative and the positive end of the distribution (use two-tail) or if you only want to use a one directional comparison (use one-tail)
For example if you want to want to check whether Group A is both taller and shorter than Group B, then you must use a two-tailed test. Whereas if you only want to see if Group A is taller than Group B but without any interest in checking if Group A is shorter than Group B, then use a one-tailed test.
But if you are in doubt and are unsure if whether you should use a one-tailed test or a two-tailed test, then it is better to go with a two-tailed test generally.
T Table vs Z Table vs Chi Square Table
The T distribution, Z distribution and Chi Squared distribution are few of the most commonly used probability distribution patterns and it is important to know the differences between them and when to use which distribution pattern
Usually a Z Table is used when the population standard deviation and mean are known. Whereas a T Table is used when the T score is calculated without the knowledge of the mean and the population standard deviation. Generally T Table is also preferred over the Z Table to be used when the sample size is small (N<30)
A chi square distribution on the other hand, with k degrees of freedom is the distribution of a sum of squares of k independent standard normal variables. And is used in test for the independence of two variables in a contingency table and for tests fir goodness of fit of an observed data to see if it matches to a theoretical one.
History of T Table
Both the t-statistic and the t-distribution were discovered around the 19th century. T-distribution was first penned by Helmert and Lüroth in the year 1876. Friedrich Helmert born in the year 1843 in Kingdom of Saxony penned ‘Die mathematischen und physikalischen Theorieen der höheren Geodäsie’ which formed the foundation of modern Geodesy.
Jacob Lüroth born in 1844 in Germany was a mathematician known for proving Lüroth’s theorem, for introducing Lüroth quartics and his thesis on Pascal’s theorem. Although the discovery of T-distribution is credited to William Sealy Gosset, it is believed Helmert and Lüroth played a key role and derived it first. The t-distribution is also found in Karl Pearson’s 1895 paper but in a very general form known then as the Pearson Type IV.
The t-statistic however is named after and attributed to William Sealy Gosset. Gosset was born in 1876 was the Head Brewer at Guinness and is considered the father of modern British statistics. The t-statistic was introduced by William Gosset in the year 1908 under his pen name ‘Student’.
The distribution was first published in 1908 paper in Biometrika under his pseudonym ‘Student’. Hence, Student’s t-distribution gets it’s name from his pseudonym ‘Student’ and has nothing to do with the literal term student as used in the English language. The Student’s t-distribution was also initially referred to as ‘Student’s Z’ and ‘Student’s test of statistical significance’ before being commonly called Student’s t-distribution as it is known today.
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