Question

Why are there 2 formulas for standard deviation?

Answer 1

see below

Explanation:

The two formulas, as shown below, are equivalent. They are alternate forms and which one is used depends on which is the most efficient method with the given data.

for a set of numbers

`x_1,x_2,x_3,...x_i,...,x_n`

and mean

`barx`

the standard deviation is

`sd=sqrt((sum(x_i-barx)^2)/n)---(1)`

arrange as follows

`sd^2=1/nsum(x_i-barx)^2`

`sd^2=1/nsum(x_i^2-2x_ibarx+barx^2)`

`sd^2=(sumx_i^2)/n-2barx(sumx_i)/n+(nbarx^2)/n`

`sd^2=(sumx_i)/n-2barxbarx+barx^2`

`sd^2=(sumx_i)/n-2barx^2+barx^2`

`sd^2=(sumx_i)/n-barx^2`

`sd=sqrt((sumx_i)/n-barx^2)--(2)`