# T Distribution/Students T distribution

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The T distribution is a distribution that is used to model the probability distribution of the sample mean drawn from a normal population when the sample size is small (less than 30). It looks very similar to the normal distribution (“bell-shaped”) except for the fact that it has comparatively fatter tails. As the sample size n increases, it approaches closer and closer to the normal distribution.

Suppose you have a population which is normally distributed and we define,

t = (X̄ – µ)/(S/√n)     where,

X̄ is the sample mean, µ is the population mean,

S is the sample standard deviation and n is the sample size.

Then t follows the Student’s T distribution with (n-1) degrees of freedom. The above quantity is called the t score which is used when we conduct T test for equality of means.

Why is the T distribution called Students T distribution?

This is because the T distribution was first defined by William Gosset in 1908 in his paper “The Probable Error of Mean” which he wrote under the pen-name Student. He obtained this distribution by empirically studying the sampling distribution of the mean for small samples.

Properties of the T distribution:

1. The T distribution has mean 0. In fact all odd central moments of the distribution are zero.
2. The T distribution is symmetric with zero skewness.
3. The T distribution has higher kurtosis then the normal distribution therefore it has fatter tails.

Uses of Students T distribution:

1. The single sample T test for equality of mean.
2. The two sample T test to test for difference of two population means.
3. To test for the significance of correlation coefficient and regression coefficients.

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