T value or t critical value is used in statistics. T value or t critical value is a type of critical value. Critical values are basically cut-off values that outline areas in which the check statistic is unlikely to lie.

For example, a location in which the critical value is handed with probability alpha if the null hypothesis is true. The hypothesis which tells that there is no statistical significance among the 2 variables in the hypothesis is said to be a null hypothesis. Critical values are also used in mathematics, such as if the derivative of a function is zero or undefined is said to be the critical value.

## What is t critical value?

The value which decides whether we retain or reject the null hypothesis is said to be **t value ort critical value**. If the statistic value is beyond the x-axis, then we reject the null hypothesis otherwise we will accept the null hypothesis. If the null hypothesis is rejected then an alternate hypothesis will be accepted.

The hypothesis which tells that there is no statistical significance among the 2 variables in the hypothesis is said to be a **null hypothesis**.

The **alternative hypothesis** is what you may be agreed with to be true or wish to prove true.

## The formula of t value or t critical value

The formula for t value or t critical value is given below,

**T = (x – µ)/ (s / √n)**

Where **s** is the standard deviation of a sample, **x** is the mean of the sample, **µ** is the population mean, and **n** is the size of the sample.

## Types of t value or t critical value

There are three types of t value or t critical value.

- Right tailed t value
- Left tailed t value
- Two-tailed t value.

## T distribution tables

The table used to find the t values is said to be the **t** distribution table. This table has to forms,

- One tail t value table
- Two tail t value table

### One tail t value table

One tail t value table is given below.

DF |
A = 0.1 |
0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0005 |

∞ |
t_{a}= 1.282 |
1.645 |
1.960 |
2.326 |
2.576 |
3.091 |
3.291 |

1 |
3.078 | 6.314 | 12.706 | 31.821 | 63.656 | 318.289 | 636.578 |

2 |
1.886 | 2.920 | 4.303 | 6.965 | 9.925 | 22.328 | 31.600 |

3 |
1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 10.214 | 12.924 |

4 |
1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 7.173 | 8.610 |

5 |
1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 5.894 | 6.869 |

6 |
1.440 | 1.943 | 2.447 | 3.143 | 3.707 | 5.208 | 5.959 |

7 |
1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.785 | 5.408 |

8 |
1.397 | 1.860 | 2.306 | 2.896 | 3.355 | 4.501 | 5.041 |

9 |
1.383 | 1.833 | 2.262 | 2.821 | 3.250 | 4.297 | 4.781 |

10 |
1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 4.144 | 4.587 |

11 |
1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 4.025 | 4.437 |

12 |
1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 3.930 | 4.318 |

13 |
1.350 | 1.771 | 2.160 | 2.650 | 3.012 | 3.852 | 4.221 |

14 |
1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 3.787 | 4.140 |

15 |
1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 3.733 | 4.073 |

16 |
1.337 | 1.746 | 2.120 | 2.583 | 2.921 | 3.686 | 4.015 |

17 |
1.333 | 1.740 | 2.110 | 2.567 | 2.898 | 3.646 | 3.965 |

18 |
1.330 | 1.734 | 2.101 | 2.552 | 2.878 | 3.610 | 3.922 |

19 |
1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.579 | 3.883 |

20 |
1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.552 | 3.850 |

21 |
1.323 | 1.721 | 2.080 | 2.518 | 2.831 | 3.527 | 3.819 |

22 |
1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.505 | 3.792 |

23 |
1.319 | 1.714 | 2.069 | 2.500 | 2.807 | 3.485 | 3.768 |

24 |
1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.467 | 3.745 |

25 |
1.316 | 1.708 | 2.060 | 2.485 | 2.787 | 3.450 | 3.725 |

26 |
1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.435 | 3.707 |

27 |
1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.421 | 3.689 |

28 |
1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.408 | 3.674 |

29 |
1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.396 | 3.660 |

30 |
1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.385 | 3.646 |

60 |
1.296 | 1.671 | 2.000 | 2.390 | 2.660 | 3.232 | 3.460 |

120 |
1.289 | 1.658 | 1.980 | 2.358 | 2.617 | 3.160 | 3.373 |

1000 |
1.282 | 1.646 | 1.962 | 2.330 | 2.581 | 3.098 | 3.300 |

### Two-tailed t value table

A two-tailed t value table is given below.

DF |
A = 0.2 | 0.10 | 0.05 | 0.02 | 0.01 | 0.002 | 0.001 |

∞ |
t_{a} = 1.282 |
1.645 |
1.960 |
2.326 |
2.576 |
3.091 |
3.291 |

1 |
3.078 | 6.314 | 12.706 | 31.821 | 63.656 | 318.289 | 636.578 |

2 |
1.886 | 2.920 | 4.303 | 6.965 | 9.925 | 22.328 | 31.600 |

3 |
1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 10.214 | 12.924 |

4 |
1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 7.173 | 8.610 |

5 |
1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 5.894 | 6.869 |

6 |
1.440 | 1.943 | 2.447 | 3.143 | 3.707 | 5.208 | 5.959 |

7 |
1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.785 | 5.408 |

8 |
1.397 | 1.860 | 2.306 | 2.896 | 3.355 | 4.501 | 5.041 |

9 |
1.383 | 1.833 | 2.262 | 2.821 | 3.250 | 4.297 | 4.781 |

10 |
1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 4.144 | 4.587 |

11 |
1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 4.025 | 4.437 |

12 |
1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 3.930 | 4.318 |

13 |
1.350 | 1.771 | 2.160 | 2.650 | 3.012 | 3.852 | 4.221 |

14 |
1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 3.787 | 4.140 |

15 |
1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 3.733 | 4.073 |

16 |
1.337 | 1.746 | 2.120 | 2.583 | 2.921 | 3.686 | 4.015 |

17 |
1.333 | 1.740 | 2.110 | 2.567 | 2.898 | 3.646 | 3.965 |

18 |
1.330 | 1.734 | 2.101 | 2.552 | 2.878 | 3.610 | 3.922 |

19 |
1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.579 | 3.883 |

20 |
1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.552 | 3.850 |

21 |
1.323 | 1.721 | 2.080 | 2.518 | 2.831 | 3.527 | 3.819 |

22 |
1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.505 | 3.792 |

23 |
1.319 | 1.714 | 2.069 | 2.500 | 2.807 | 3.485 | 3.768 |

24 |
1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.467 | 3.745 |

25 |
1.316 | 1.708 | 2.060 | 2.485 | 2.787 | 3.450 | 3.725 |

26 |
1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.435 | 3.707 |

27 |
1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.421 | 3.689 |

28 |
1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.408 | 3.674 |

29 |
1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.396 | 3.660 |

30 |
1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.385 | 3.646 |

60 |
1.296 | 1.671 | 2.000 | 2.390 | 2.660 | 3.232 | 3.460 |

120 |
1.289 | 1.658 | 1.980 | 2.358 | 2.617 | 3.160 | 3.373 |

∞ |
1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 3.091 | 3.291 |

## How to calculate t value or t critical value using formula and t table?

Let’s take examples to solve t value or t critical value by using a formula.

**Example 1**

A factory wants to improve its sale of clothes. The previous sales data indicated the mean sale of 36 salesmen was Rs 500 per transaction. After training, the researched data showed an average sale of Rs 800 per transaction. If the standard deviation is Rs 300, find the t value?

**Solution **

**Step 1:**Write t value formula.

t = (x – µ)/ (s / **√n)**

**S****tep 2:**Identify values from given statement.

Mean = x = Rs 800

µ = Rs 500

Standard deviation = s = Rs 300

n = 36

**Step 3:**Put the values in the formula.

t = (x – µ)/ (s / √n)

t = (800 – 500) / (300/ √36)

t = 300 / (300/6)

= 300 / 50

t = 6

this is the degree of freedom (**Degree of freedom** are that amount of information your data provide that you can spend to estimates the value of unknown sample population parameters and denoted by df or Df) for t critical value, we will check the degree of freedom at significance level (**Significance level** also referred as alpha, is the probability that the event could have occurred by chance).

From t table 6 > 1.9432 at alpha = 0.05.

Thus, training boosts the sale.

**Example 2**

A finance company wants to improve the sale of its product. The previous sales data indicated the mean sale of 64 salesmen was Rs 1500 per transaction. After training, the researched data showed an average sale of Rs 800 per transaction. If the standard deviation is Rs 400, find the t value?

**Solution**

**Step 1:**Write t value formula.

t = (x – µ)/ (s / **√n)**

**Step 2:**Identify values from given statement.

Mean = x = Rs 800

µ = Rs 1500

standard deviation = s = Rs 400

n = 64

**Step 3:**Put the values in the formula.

t = (x – µ)/ (s / √n)

t = (800 – 1500) / (400/ √64)

t = -700 / (400/8)

= -700 / 50

t = -14

This is the degree of freedom for t critical value, we will check the degree of freedom at a significance level at 0.05

Since the degree of freedom never be negative, so we reject the null hypothesis.

**Example 3**

Calculate t value for significance level 10% and 25 degrees of freedom?

**Solution **

**Step 1:**Point out the values.

Significance level = 10% = 10/100 = 0.1

Degree of freedom = df = 25

**Step 2:**Look the table for one tailed t value.

T critical value for one-tailed = 1.316

**Step 3:**Look at the table for the two-tailed t value.

T critical value for two-tailed = 1.708

You can verify the result by using the T value Calculator.

**Example 4**

Calculate t value for significance level 5% and 60 degrees of freedom?

**Solution **

**Step 1: **Point out the values.

Significance level = 5% = 5/100 = 0.05

Degree of freedom = df = 60

**Step 2: **Look the table for one tailed t value.

T critical value for one-tailed = 1.671

**Step 3: **Look at the table for the two-tailed t value.

T critical value for two-tailed = 2.0008

## Summary

Now you are witnessed that this topic is not difficult. Once you practice this topic you can easily solve the problems related to the t value. Just remember how to locate the values in the table you can easily do any problem.