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

### Mean and Standard Deviation

```
Date: 03/01/99 at 19:08:48
From: Maria Alejandra Martino
Subject: Mean and Standard Deviation

I have to prove that the formula for the mean and standard deviation
for a binomial distribution are: u = np and s = the square root of
(npq), where n is the total number of trials, p is the probability of
success, and q is the probability of failure, u is the mean and s is
the standard deviation.

I know that u = summation of x*P(X), where P(x) is the probability of
x. I also have that P(x) = (combination of n and x)*[(p)^(x)]*[(q)^(n-
x)] and that p + q = 1 (or, q = 1 - p) so that u = Summation of
{x*(combination of n and x)*[(p)^(x)]*[(q)^(n-x)]} from this part on I
do not know what to do.

```

```
Date: 03/02/99 at 10:14:34
From: Doctor Anthony
Subject: Re: Mean and Standard Deviation

One of the easiest proofs is as follows:

In any one trial the number of successes = 1 with probability p
"                      "          "     = 0        "         q

So if x1 = number of successes in trial 1
x2 =    "        "            "   2
........................................
xn =    "        "            "   n

If X = number of successes in n trials then

X = x1 + x2 + ....... + xn

E(X) = E(x1) + E(x2) + ..... + E(xn)

but E(x1) = 1.p + 0.q  = p  similarly E(x2) = E(x3) = .... = p

Therefore  E(X) = p + p + p + ...... + p

= np

Var(x1) = E(x1^2) - p^2

and  E(x1^2) = 1^2.p + 0^2.q  =  p   and so

Var(x1) = p - p^2  = p(1-p) = pq

Var(X) = Var(x1 + x2 + x3 + ..... + xn)  and since the x's are
independent

= Var(x1) + var(x2) + .... + Var(xn)

= pq + pq + pq + ..... + pq

= npq

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Probability
College Statistics
High School Probability
High School Statistics

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