# How to Calculate Critical t-Value in R ?

A critical-T value is a “cut-off point” on the t distribution. A t-distribution is a probability distribution that is used to calculate population parameters when the sample size is small and when the population variance is unknown. T values are used to analyze whether to support or reject a null hypothesis.

After conducting a t-test, we get its statistics as result. In order to determine the significance of the result, we compare the t- score obtained by a critical t value. If the absolute value of the t-score is greater than the t critical value, then the results of the test are statistically significant.

**Formula:**

t = [ x̄ - μ ] / [ s / sqrt( n ) ]

where,

- t = t score
- x̄ = sample mean,
- μ = population mean,
- s = standard deviation of the sample,
- n = sample size

**Function used:**

In order to find the T Critical value we make use of qt() function provided in R Programming Language.

Syntax:qt(p=conf_value, df= df_value, lower.tail=True/False)

Parameters:

p:-Confidence leveldf:degrees of freedomlower.tail:If TRUE, the probability to the left of p in the t distribution is returned. If FALSE, the probability to the right is returned. By default it’s value is TRUE.

There are three methods for calculating critical t value, all of them are discussed below:

**Method 1: **Right tailed test

A right-tailed test is a test in which the hypothesis statement contains a greater than (>) symbol i.e. the inequality points to the right. Sometimes it is also referred to as the upper test.

Here we are assuming a **confidence value** of **96%** **i.e.. p= .04** and **degree of freedom 4 i.e.. df=4**. We are also using the **format()** function to reduce the decimal value to three decimal places. For the Right tail test, we are setting the value of **lower.tail** as **FALSE**.

**Example:**

## R

`rm` `(list = ` `ls` `())` ` ` `conf<-.04` `daf<-4` ` ` `value<-` `formatC` `(` `qt` `(p=conf, df=daf, lower.tail=` `FALSE` `)) ` ` ` `print` `(` `paste` `(` `"Critical T value is : "` `,value))` |

**Output:**

Critical T value is : 2.333

The t critical value is 2.333. Thus, if the test score is greater than this value, the results of the test are statistically significant.

**Method 2: **Left tailed test

A left-tailed test is a test in which the hypothesis statement contains a less than (<) symbol i.e… the inequality points to the left. Sometimes it is also referred to as the lower test.

Here we are assuming a **confidence value of 95% i.e.. p= .05** and **degree of freedom 4 i.e.. df=4**. We are also using the** format()** function to** **reduce the decimal value to three decimal places. For the Left tail test, we are setting the value of **lower.tail **as **TRUE**.

**Example:**

## R

`rm` `(list = ` `ls` `())` ` ` `conf<-.05` `daf<-4` ` ` `value<-` `formatC` `(` `qt` `(p=conf, df=daf, lower.tail=` `TRUE` `)) ` ` ` `print` `(` `paste` `(` `"Critical T value is : "` `,value))` |

**Output:**

Critical T value is : -2.132

The t critical value is -2.132. Thus, if the test score is less than this value, the results of the test are statistically significant.

**Method 3: **Two-tailed test

A two-tailed test is a test in which the hypothesis statement contains both a greater than (>) symbol and a less-than symbol(<) i.e. the inequality points between a certain range.

In a two-tailed test, we simply need to pass in half of our confidence level in “p” parameter. Here we are assuming **confidence value of 96% i.e.. p= .04** and **degree of freedom 4 i.e.. df=4**. We are also using the** format()** function to** **reduce the decimal value to three decimal places.

**Example**

## R

`rm` `(list = ` `ls` `())` ` ` `conf=0.04 / 2` ` ` `daf<-4` ` ` `value<-` `formatC` `(` `qt` `(p = conf , df = daf)) ` ` ` `print` `(` `paste` `(` `"Critical T value is : "` `,value))` |

**Output:**

Critical T value is : -2.999

Whenever, we perform a two-tailed test we get two critical values as output. So here in the above code, the T critical values are 2.999 and -2.999. Therefore, if the test score is less than -2.999 or greater than 2.999, the results of the test are statistically significant.