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In probability and statistics, the t-distribution or Student's distribution arises in the problem of estimating the mean of a normally distributed population when the sample size is small, as well as when (as in nearly all practical statistical work) the population standard deviation is unknown and has to be estimated from the data.
Where $\mu $ is the population mean, $\overline{x}$ is the sample mean and s is the estimator for population standard deviation (i.e., the sample variance) defined by
If $\sigma =s$ , $t=z$ , the distribution becomes the normal distribution. As N increases, Student’s t distribution approaches the normal distribution . It can be derived by transforming student’s z -distribution using $$z\equiv \frac{\overline{x}-\mu}{s}$$ and then defining $$t=z\sqrt{n-1}.$$
The resulting probability and cumulative distribution functions are:
where,
The effect of degree of freedom on the t distribution is illustrated in the four t distributions on the Figure 1 .
In general, it is difficult to evaluate the distribution function of T . Some values are usually given in the tables. Also observe that the graph of the p.d.f. of T is symmetrical with respect to the vertical axis t =0 and is very similar to the graph of the p.d.f. of the standard normal distribution $N\left(\mathrm{0,1}\right)$ . However the tails of the t distribution are heavier that those of a normal one; that is, there is more extreme probability in the t distribution than in the standardized normal one. Because of the symmetry of the t distribution about t =0, the mean (if it exists) must be equal to zero. That is, it can be shown that $E\left(T\right)=0$ when $r\ge 2$ . When r =1 the t distribution is the Cauchy distribution , and thus both the variance and mean do not exist.
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