Associated Topics || Dr. Math Home || Search Dr. Math

### Standard Deviation of Uniform Distribution

```
Date: 02/16/98 at 23:51:16
From: Darin Parker
Subject: Std. deviation of uniform distribution

My problem is not in the calculation of the standard deviation of a
uniform distribution, but in where they got the formula.  I find it
easier for me to learn if I understand the formulas as well as knowing
where to plug in the numbers.

In the equation Std. deviation = (b-a)/ square root of 12, where did
the square root of 12 come from?

I have looked through every statistics text I have, but they all give
the formula with no explanation for where the square root of 12 came
from.  My professor doesn't even know.  I would greatly appreciate any
insight you can give me on the subject.

Thank you,
Darin Parker
```

```
Date: 02/17/98 at 15:52:58
From: Doctor Anthony
Subject: Re: Std. deviation of uniform distribution

The uniform distibution on the interval from a to b is given by

f(x) = 1/(b-a)   for a < x < b

= 0  elsewhere.

E(X) =  INT(from a to b)[x.dx/(b-a)]

=  (1/(b-a)) x^2/2 from a to b

b^2 - a^2      (b+a)
----------  =  ------
2(b-a)          2

E(X^2) =  INT(from a to b)[x^2.dx/(b-a)]

=  (1/(b-a)) x^3/3  from a to b

b^3 - a^3      b^2+ab+a^2
=  ----------  =  ------------
3(b-a)            3

Var(X) =  E(X^2) - [E(X)}^2

b^2+ab+a^2       b^2 + 2ab + a^2
=  -----------  -   ---------------
3                  4

4b^2 + 4ab + 4a^2 - 3b^2 - 6ab - 3a^2
=  ---------------------------------------
12

b^2 -2ab + a^2
=  ----------------
12

(b-a)^2
=  --------
12

(b-a)
and so the s.d. is   --------
sqrt(12)

You can do this without all the algebra if you work on the interval
0 to 1.

E(X) = INT[x.dx]    = x^2/2  from 0 to 1

= 1/2

E(X^2) = INT[x^2.dx]  = x^3/3  from 0 to 1

= 1/3

Var(X) = 1/3 - 1/4

= 1/12

s.d. = 1/sqrt(12)   = length of interval/sqrt(12)

-Doctor Anthony,  The Math Forum
Check out our web site http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search