General Comments on Standard Deviation

Date: 01/15/2004 at 01:23:40
From: Mike Gray
Subject: Understanding Standard Deviation

I have a basic understanding of the Standard Deviation and what it's 
used for, but there are some things about it that I do not quite
grasp.  1. When someone says, "Compute the Standard Deviation of a
data set", is that the same as asking to compute ONE Standard
Deviation?  If they wanted to know the range of data that falls 
within TWO SD's, how would they properly ask that question?  2. When 
we speak of "Standard Deviation", does it only apply to a Normal 
Distribution curve, or can SD also apply to other distribution curves?
 3. When someone asks to compute the SD of a data set, how would a
person know which type of distribution curve to use for the
calculation if the distribution curve is not known?  4. What is the 
history behind the SD?  In other words, who invented it and why? 

I own a business that repairs X-ray equipment.  During testing, I 
record various measured parameters of the repaired equipment.  After 
I record a number of values over time, I calculate the SD to determine
if my measuring methods are precise or "sloppy".  If I had a better
understanding of SD I may decide that I'm wasting my time calculating
it if the numbers have little meaning.  I hope this makes sense!  Thanks!

Date: 01/15/2004 at 13:29:55
From: Doctor Douglas
Subject: Re: Understanding Standard Deviation


Hi Mike.

Thanks for writing to the Math Forum.  Your question is refreshing,
because it seeks understanding, even when "calculating the math is
easy".  Very few of our submitted questions try to do this.

1.  Yes, "computing THE standard deviation" means "computing one
standard deviation".  Implicitly, this means that when you are done,
you also know the value of 2 standard deviations (e.g. for 95.5%
confidence interval problems), or 3 SD's, or 2.5 SD's, and so on.

2.  You can compute a standard deviation for almost any distribution
as the square root of the variance, where the variance is defined in
terms of an integral (for continuous distributions such as the normal
or gaussian distribution, and the uniform distribution), and in terms
of a sum (for discrete distributions such as the binomial and poisson
distributions).  I say "almost" because there are some distributions
for which these integrals or sums do not converge, even though the
distributions do describe probabilities in certain situations.

3.  For a data set {x1,x2,...,xN}, the distribution is (usually)
unknown, but you can still compute its standard deviation:

   SD = sqrt{[(x1-u)^2 + (x2-u)^2 + ... + (xN-u)^2]/N}

where N is the number of sample values and u is their mean.  This
computation does not require a priori knowledge of the distribution. 
In fact, it may be used to INFER the shape of the distribution,
although additional information is needed.  For example, suppose the
values {x1,...,xN} measure the angular position of the sun in degrees
(and for argument's sake let us imagine that the sun moves in a
circular orbit at constant speed around the earth, and that it passes
directly overhead at noon).  You can imagine two situations:

  A.  measurements are taken at "noon" by many different people, 
      each of which defines "noon" according to his or her own
      wristwatch.  The sample values {x} will probably be *normally*
      distributed around a mean of zero degrees (i.e. overhead), 
      and the standard deviation estimate of the typical amount of
      time by which wristwatches are unsynchronized.  If you
      construct a histogram of the measured angles, it should
      approximate the bell-shaped curve of a normal distribution.

  B.  measurements are taken anytime the sun is up.  Now, the sample
      values will probably be *uniformly* distributed between -90
      degrees (sunrise) and +90 degrees (sunset), with no particular
      time being favored over another--even though zero degrees is
      certainly still the mean of this distribution.  In this case
      the standard deviation is related to the (angular) length of
      the day, and has very little to do with wristwatches [in fact
      the standard deviation extracted from the data set will
      likely have the value near [(180 degrees)/sqrt(12)].  The
      histogram of the sample angles in this situation will be flat
      from -90 to +90 degrees.

In both of these cases, one can mechanically compute the standard
deviation of the data.  Interpreting what this value means, though, is
more delicate.  This is relevant to your equipment repair, and your
measurements may indicate, for example, that your measuring methods
are too coarse or just right, or perhaps indicate what particular
components of the equipment are problematic.
  
4.  As for the history of the standard deviation and its usage, the
following webpage

  Earliest Known Usage of Some of the Words of Mathematics (S)
    http://jeff560.tripod.com/s.html 

says that the word was "introduced by Karl Pearson (1857-1936) in 
1893 'although the idea was by then nearly a century old'".  A
bibliography on this topic is at

  History of Mathematics:  History of Probability and Statistics
    http://aleph0.clarku.edu/~djoyce/mathhist/statistics.html 
 
I hope this helps answer your questions.  Feel free to write back
if you need more information.
 
- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/15/2004 at 13:37:48
From: Mike Gray
Subject: Thank you (Understanding Standard Deviation)

Thank you very much for your quick and enlightening answers to my
questions!  I appreciate it very much.  You pick up where the text
books leave off!

My best regards,
Mike