# Finding standard deviation or error from normalized data.

1. Sep 26, 2011

### doublemint

Hello All,

I am trying to figure out how to find the standard deviation or error in sets of data. So lets say I have sets x1, x2, x3 and x4 with various values and I found the average and standard deviations for it. Now I have to take the averages, lets say a1, a2, a3, a4, and normalize a2,a3,a4 to a1. Now how do I find the standard deviation or error in the normalized sets? Forgive my ignorance, but I am suppose to do this for a project and I have never taken any stats course before..

Thanks
DoubleMint

2. Sep 28, 2011

### Stephen Tashi

What do you mean by "normalize"? For example, do you mean that multiply each datum in the data set x2 by the factor (a2/a1) ?

Let the data be the $d_i$. Let the sample mean be $m$ . Let the scaling factor be $k$

The mean of the scaled data $k d_i$ is $m k$

The variance of the scaled data is:

$\sum \frac { ( k d_i - m k )^2}{n} = \sum \frac{k^2 (d_i - k)^2 }{n} = k^2 \sum \frac{(d_i - m)^2}{n}$

This is $k^2$ times the variance of the original sample.

So the sample standard deviation of the scaled data is $|k|$ times the standard deviation of the original data.