# T-Test Essentials: Definition, Formula and Calculation

## T-Test Formula

This article describes the t-test formula for one-sample, two-independent samples and paired samples t-test. The t-test formula is also referred as:

• t-test equation,
• t-score equation,
• t-test statistic formula
• formula for t-test

Contents:

#### Related Book

Practical Statistics in R II - Comparing Groups: Numerical Variables

## One sample t-test formula

The one sample t-test formula is used to compare the mean of one sample to a known standard mean. The the one-sample t-test formula can be written as follow:

$t = \frac{m-\mu}{s/\sqrt{n}}$

where,

• $m$ is the sample mean
• $n$ is the sample size
• $s$ is the sample standard deviation with $n-1$ degrees of freedom
• $\mu$ is the theoretical mean

The p-value, corresponding to the absolute value of the t-test statistics (|t|), is computed for the degrees of freedom (df): df = n - 1.

How to interpret the one-sample t-test results?

If the p-value is inferior or equal to the significance level 0.05, we can reject the null hypothesis and accept the alternative hypothesis. In other words, we conclude that the sample mean is significantly different from the theoretical mean.

## Independent t-test formula

The independent t-test formula is used to compare the means of two independent groups.The independent samples t-test comes in two different forms:

• the standard Student’s t-test, which assumes that the variance of the two groups are equal.
• the Welch’s t-test, which is less restrictive compared to the original Student’s test. This is the test where you do not assume that the variance is the same in the two groups, which results in the fractional degrees of freedom.

In this article, you will learn the Student t-test formula and the Weltch t-test formula.

### Student t-test formula

If the variance of the two groups are equivalent (homoscedasticity), the t-test value, comparing the two samples ($A$ and $B$), can be calculated as follow.

$t = \frac{m_A - m_B}{\sqrt{ \frac{S^2}{n_A} + \frac{S^2}{n_B} }}$

where,

• $m_A$ and $m_B$ represent the mean value of the group A and B, respectively.
• $n_A$ and $n_B$ represent the sizes of the group A and B, respectively.
• $S^2$ is an estimator of the pooled variance of the two groups. It can be calculated as follow :

$S^2 = \frac{\sum{(x-m_A)^2}+\sum{(x-m_B)^2}}{n_A+n_B-2}$

with degrees of freedom (df): $df = n_A + n_B - 2$.

### Welch’s t test formula

If the variances of the two groups being compared are different (heteroscedasticity), it’s possible to use the Welch t-test, which is an adaptation of the Student t-test. The Welch t-statistic is calculated as follow :

$t = \frac{m_A - m_B}{\sqrt{ \frac{S_A^2}{n_A} + \frac{S_B^2}{n_B} }}$

where, $S_A$ and $S_B$ are the standard deviation of the the two groups A and B, respectively.

Unlike the classic Student’s t-test, the Welch t-test formula involves the variance of each of the two groups ($S_A^2$ and $S_B^2$) being compared. In other words, it does not use the pooled variance $S$.

The degrees of freedom of Welch t-test is estimated as follow :

$df = (\frac{S_A^2}{n_A}+ \frac{S_B^2}{n_B})^2 / (\frac{S_A^4}{n_A^2(n_A-1)} + \frac{S_B^4}{n_B^2(n_B-1)} )$

A p-value can be computed for the corresponding absolute value of t-statistic (|t|).

If the p-value is inferior or equal to the significance level 0.05, we can reject the null hypothesis and accept the alternative hypothesis. In other words, we can conclude that the mean values of group A and B are significantly different.

Note that, the Welch t-test is considered as the safer one. Usually, the results of the classical student’s t-test and the Welch t-test are very similar unless both the group sizes and the standard deviations are very different.

## Paired t-test formula

The paired t-test is used to compare the means of two related groups of samples.

The procedure of the paired t-test analysis is as follow:

1. Calculate the difference ($d$) between each pair of value
2. Compute the mean ($m$) and the standard deviation ($s$) of $d$
3. Compare the average difference to 0. If there is any significant difference between the two pairs of samples, then the mean of d ($m$) is expected to be far from 0.

The paired t-test statistics value can be calculated using the following formula:

$t = \frac{m}{s/\sqrt{n}}$

where,

• m is the mean differences
• n is the sample size (i.e., size of d).
• s is the standard deviation of d

We can compute the p-value corresponding to the absolute value of the t-test statistics (|t|) for the degrees of freedom (df): $df = n - 1$.

If the p-value is inferior or equal to 0.05, we can conclude that the difference between the two paired samples are significantly different.

## Related article

T-test in R 