T Distribution

The T Distribution also called the student’s t-distribution, is utilised 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.

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 a 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 formula

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.