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.

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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