Understanding Standard Deviation

Date: 09/14/98 at 17:33:25
From: Nick Basehore
Subject: Pre-Calculus - Variance and standard deviation

The question, verbatim from my book:

   Suppose two samples have the same mean, but different standard 
   deviations s_1 and s_2, with s_1 < s_2. Which sample will show more 
   variability?

How do I determine the variability? What is the book asking for? 
Can you give me another example of this problem with numbers instead 
of variables?

Date: 09/14/98 at 18:37:56
From: Doctor Pat
Subject: Re: Pre-Calculus - Variance and standard deviation

Nick,

"Variability" is used in two different but related ways in talking 
about data. The technical meaning is a mathematical measure of the 
average of the squares of the distances from the mean of the data.  
This is also the square of the standard deviation.  

I think your question is more about the non-technical meaning, which is 
more like the common English use of variability - how much are the 
measures spread out from the center? This is essentially what the 
technical variability and standard deviation try to measure, but they 
are just two of the many ways that spread could be quantified.  

In either case, it is safe to assume that, as a rule of thumb, more 
spread out data will mean a greater variability and a larger standard 
deviation. These sort of go together.  

I hope this helps you understand. Because the language of Statistics 
is mostly 20th century, it is still changing and at times seems 
inconsistent. Many of the terms you hear and use in statistics were 
created by people who are still alive and still practicing statistics. 
This "newness" leads to some differences in interpretation that make it 
a little more difficult sometimes, but being able to see the multiple 
meanings (and uses) helps to understand the power of what statistics 
can do.  

Good luck,
  
- Doctor Pat, The Math Forum
Check out our web site! http://mathforum.org/dr.math/