 # Uniform Distribution Formula

## Theoretical Mean Formula

 $\mu$ = $\frac{x+y}{2}$

## Standard Deviation Formula

 $\sigma = \sqrt{ \frac{(y-x)^{2}}{12}}$

## Example of Uniform Distribution

Example: The data in the table below are 55 times a baby yawns, in seconds, of a 9-week-old baby girl.

 10.4 19.6 18.8 13.9 17.8 16.8 21.6 17.9 12.5 11.1 4.9 12.8 14 22.8 20.8 15.9 16.3 13.4 17.1 14.5 19 22.8 1.3 0.7 8.9 11.9 10.9 7.3 5.9 3.7 17.9 19.2 9.8 5.8 6.9 2.6 5.8 21.7 11.8 3.4 2.1 4.5 6.3 10.7 8.9 9.7 9.1 7.7 10.1 3.5 6.9 7.8 11.6 13.8 18.6
• The sample mean = 11.49
• The sample standard deviation = 6.23.

As assumed, the yawn times in secs, it follows a uniform distribution between 0 to 23 seconds(Inclusive).

So, it is equally likely that any yawning time is from 0 to 23.

In this example,

The theoretical mean = $\mu$ = $\frac{x+y}{2}$ $\mu$ = $\frac{0+23}{2}$ = 11.50

standard deviation = $\sigma = \sqrt{ \frac{(y-x)^{2}}{12}}$

standard deviation = $\sqrt{\frac{(23-0)2}{12}}$ = 6.64 seconds.