# T Distribution

The **T Distribution** also called the student’s t-distribution and is used while making assumptions about a mean when we don’t know the standard deviation. In probability and statistics, the normal distribution is a bell-shaped distribution whose **mean is μ and the standard deviation is σ**. The t-distribution is similar to normal distribution but **flatter and shorter** than a normal distribution. Here, we are going to discuss what is t-distribution, formula, table, properties, and applications.

## T- Distribution Definition

The t-distribution is a hypothetical probability distribution. It is also known as the student’s t-distribution and used to make presumptions about a mean when the standard deviation is not known to us. It is symmetrical, bell-shaped distribution, similar to the standard normal curve. As high as the degrees of freedom (df), the closer this distribution will approximate a standard normal distribution with a mean of 0 and a standard deviation of 1.

## T Distribution Formula

A t-distribution is the whole set of t values measured for every possible random sample for specific sample size or a particular degree of freedom. It approximates the shape of normal distribution.

Let x have a normal distribution with mean ‘μ’ for the sample of size ‘n’ with sample mean \(\bar{x}\) and the sample standard deviation ‘s’, then the t variable has student’s t-distribution with a degree of freedom, d.f = n – 1. The formula for t-distribution is given by;

## T-Distribution Table

The t-distribution table is used to determine proportions connected with z-scores. We use this table to find the ratio for t-statistics. The t-distribution table shows the probability of t taking values from a given value. The obtained probability is the area of the t-curve between the ordinates of t-distribution, the given value and infinity.

In the t-distribution table, the critical values are defined for degrees of freedom(df) to the probabilities of t-distribution, α.

df/α |
0.9 |
0.5 |
0.3 |
0.2 |
0.1 |
0.05 |
0.02 |
0.01 |
0.001 |

1 |
0.158 | 1 | 2 | 3.078 | 6.314 | 12.706 | 31.821 | 64 | 637 |

2 |
0.142 | 0.816 | 1.386 | 1.886 | 2.92 | 4.303 | 6.965 | 10 | 31.598 |

3 |
0.137 | 0.765 | 1.25 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 12.929 |

4 |
0.134 | 0.741 | 1.19 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 8.61 |

5 |
0.132 | 0.727 | 1.156 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 6.869 |

6 |
0.131 | 0.718 | 1.134 | 1.44 | 1.943 | 2.447 | 3.143 | 3.707 | 5.959 |

7 |
0.13 | 0.711 | 1.119 | 1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 5.408 |

8 |
0.13 | 0.706 | 1.108 | 1.397 | 1.86 | 2.306 | 2.896 | 3.355 | 5.041 |

9 |
0.129 | 0.703 | 1.1 | 1.383 | 1.833 | 2.263 | 2.821 | 3.25 | 4.781 |

10 |
0.129 | 0.7 | 1.093 | 1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 4.587 |

11 |
0.129 | 0.697 | 1.088 | 1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 4.437 |

12 |
0.128 | 0.695 | 1.083 | 1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 4.318 |

13 |
0.128 | 0.694 | 1.079 | 1.35 | 1.771 | 2.16 | 2.65 | 3.012 | 4.221 |

14 |
0.128 | 0.692 | 1.076 | 1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 4.14 |

15 |
0.128 | 0.691 | 1.074 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 4.073 |

16 |
0.128 | 0.69 | 1.071 | 1.337 | 1.746 | 2.12 | 2.583 | 2.921 | 4.015 |

17 |
0.128 | 0.689 | 1.069 | 1.333 | 1.74 | 2.11 | 2.567 | 2.898 | 3.965 |

18 |
0.127 | 0.688 | 1.067 | 1.33 | 1.734 | 2.101 | 2.552 | 2.878 | 3.922 |

19 |
0.127 | 688 | 1.066 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.883 |

20 |
0.127 | 0.687 | 1.064 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.85 |

21 |
0.127 | 0.686 | 1.063 | 1.323 | 1.721 | 2.08 | 2.518 | 2.831 | 3.819 |

22 |
0.127 | 0.686 | 1.061 | 1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.792 |

23 |
0.127 | 0.685 | 1.06 | 1.319 | 1.714 | 2.069 | 2.5 | 2.807 | 3.767 |

24 |
0.127 | 0.685 | 1.059 | 1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.745 |

25 |
0.127 | 0.684 | 1.058 | 1.316 | 1.708 | 2.06 | 2.485 | 2.787 | 3.725 |

26 |
0.127 | 0.684 | 1.058 | 1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.707 |

27 |
0.137 | 0.684 | 1.057 | 1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.69 |

28 |
0.127 | 0.683 | 1.056 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.674 |

29 |
0.127 | 0.683 | 1.055 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.649 |

30 |
0.127 | 0.683 | 1.055 | 1.31 | 1.697 | 2.042 | 2.457 | 2.75 | 3.656 |

40 |
0.126 | 0.681 | 1.05 | 1.303 | 1.684 | 2.021 | 2.423 | 2.704 | 3.551 |

80 |
0.126 | 0.679 | 1.046 | 1.296 | 1.671 | 2 | 2.39 | 2.66 | 3.46 |

120 |
0.126 | 0.677 | 1.041 | 1.289 | 1.658 | 1.98 | 2.358 | 2.617 | 3.373 |

Infinity |
0.126 | 0.674 | 1.036 | 1.282 | 1.645 | 1.96 | 2.326 | 2.576 | 3.291 |

### Properties of T Distribution

- It ranges from −∞ to +∞.
- It has a bell-shaped curve and symmetry similar to normal distribution.
- The shape of the t-distribution varies with the change in degrees of freedom.
- The variance of the t-distribution is always greater than ‘1’ and is limited only to 3 or more degrees of freedom. It means this distribution has a higher dispersion than the standard normal distribution.

### T- Distribution Applications

The important applications of t-distributions are as follows:

- Testing for the hypothesis of the population mean
- Testing for the hypothesis of the difference between two means. In this case, the t-test can be calculated in two different ways, such as
- Variances are equal
- Variances are unequal

- Testing for the hypothesis of the difference between two means having the dependent sample
- Testing for the hypothesis about the Coefficient of Correlation. It is involved in three cases. They are:
- When the population coefficient of correlation is zero, i.e. ρ = 0.
- When the population coefficient of correlation is zero, i.e. ρ≠ 0.
- When the hypothesis is examined for the difference between two independent correlation coefficients

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