T Distribution Formula
T distribution formula is used when the sample size is small and the population standard deviation is unknown to us and we want to estimate the mean of the normally distributed population. The T distribution formula is also known as Student’s T Distribution. The curve of the t distribution is almost similar to the normal distribution curve, but the only difference arises that it is a bit short and a little wide than that of the normal distribution curve. Let us study the t distribution formula using solved examples.
What is T Distribution Formula?
The t distribution formula tells us that the larger the sample size, the more it will be like the normal distribution. For the t distribution formula, we need to know the degree of freedom = m which is nothing but "n1", where n is the sample size. The t distribution formula for the small sample size is given as:
\( \large \boldsymbol{T Distribution Formula = \frac{\overline{x}\mu}{\frac{s}{\sqrt{N}}}} \)
where,
 \(\mu\) is the expected mean value.
 \(\overline{x}\) is the sample mean.
 s is the standard deviation of the sample.
 n is the sample size.
Let's take a quick look at a couple of examples to understand the tdistribution formula better.
Solved Examples Using T Distribution Formula

Example 1: If the sample mean and expected mean value of the marks obtained by 15 students in a class test is 290 and 300 respectively. What is the tscore if the standard deviation of the marks is 50?
Solution:
To find: tscore
Using the t distribution formula,
\( \large \boldsymbol{t=\frac{\overline{x}\mu}{\frac{s}{\sqrt{N}}}} \)
\( \large \boldsymbol{t=\frac{290300}{\frac{50}{\sqrt{15}}}} \)
t = 10/12.909 = 0.7745Answer: T score of the marks is 0.7745.

Example 2: If the sample mean and expected mean value of the height of 16 friends is 170 and 165 respectively. What is the tscore if the standard deviation of the heights is 21.05?
Solution:
To find: tscore
Using the t distribution formula,
\( \large \boldsymbol{t=\frac{\overline{x}\mu}{\frac{s}{\sqrt{N}}}} \)
\( \large \boldsymbol{t=\frac{170165}{\frac{21.05}{\sqrt{16}}}} \)
t = 0.95Answer: T score of the height is 0.95.
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