# T distribution vs Normal Z Distribution

-

In this article, we explain the main differences between the T distribution and the Z distribution.

Suppose that a given normal population has mean \mu and variance \sigma^2. Suppose we draw a sample of size n such that the sample mean is \bar{x}. Then the quantity Z= \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} follows the standard normal distribution also known as the Z distribution.

When we do not assume knowledge of the standard deviation of a studied
random variable, we substitute the sample standard deviation s for the population standard deviation in the earlier equation. That is we obtain, t= \frac{\bar{x} - \mu}{s/\sqrt{n}}. In particular, due to using the estimate s of the standard deviation in the above equation, the resulting test statistic t in it does not follow a standard normal distribution but instead follows a different distribution. The distribution of the right-hand side of the above equation is referred to as the t-distribution.

Each one of these distributions differs from any other of the class of t-distributions by an associated quantity that is called degrees of freedom, commonly denoted as df. This quantity, df, is always an integer number, i.e., a whole positive number. The degrees of freedom (df) associated with the t-ratio in the above equation is df=n-1. 

### 1. The Shape of the Distribution:

The Z distribution had a shape similar to that of a bell-shaped curve. Note that the normal distribution is always symmetric.

The shape of the T distribution depends on the number of subjects studied, i.e., on the number of subjects in the sample. Another property of the t-distribution is that its shape approaches the standard normal distribution with increasing degrees of freedom. The probability density function (pdf) of the t-distribution is somewhat flatter and ‘‘shorter’’ than that of the standard normal. A characteristic property of the t-distribution is that its tails are thicker than those of the standard normal distribution, and thus it rises to a lower level.

### 2. Use of the Distribution:

Both of the distributions are used when testing the value of the population mean for any given population. The difference is that the Z test is used when the population standard deviation is known whereas the T distribution is used when the population parameter is unknown.

Also, the T distribution is generally used when the sample size is small (less than 30) whereas the Z distribution is used if the sample size is large (more than 30).

#### Reference:

Basic Statistics, An Introduction with R – TENKO RAYKOV, GEORGE A. MARCOULIDES

Hey 👋

I'm currently pursuing a Ph.D. in Maths. Prior to this, I completed my master's in Maths & bachelors in Statistics.

I created this website for explaining maths and statistics concepts in the simplest possible manner.

If you've found value from reading my content, feel free to support me in even the smallest way you can.