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 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.

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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