Statistical Analysis - z and t Statistics

Date: 07/06/2004 at 13:32:52
From: Carl
Subject: Research Applications/Statistics

Dr. Math,

Could you tell me the differences between z-statistics and t-
statistics?

Date: 07/06/2004 at 14:53:25
From: Doctor Mitteldorf
Subject: Re: Research Applications/Statistics

Hi Carl -

Suppose you have a large population, and you want to determine the
mean of that large population.  It is impractical to measure all of
them and take the average longhand, because there are so many.  The
standard approach is sampling, which means you select n of them at
random, and take the average of those.

The first question you ask is: what's my best estimate of the average
of the entire population, and the answer is, as you expect, that the
average of your sample of n is your best estimate of the average of
the entire population.

The second question you want to ask is: after measuring a sample of n,
how well do I know the average of the entire population?  The answer
as a tolerance, or "plus or minus" is the "standard deviation of the
mean".  It is standard deviation of your sample of n, divided by the
square root of (n-1).

  tolerance in estimate of population average = std dev of mean
  = sqrt((<x^2> - <x>^2)/(n-1))

The third question you may want to ask--a deeper level of detail--is
a full probability distribution for estimating the population average.
In other words, given your sample with its mean m and standard
deviation s, can you assign a probability that the population mean is
equal to any arbitrary value mu?

It is here that the Student-t distribution comes into play.  In many
situations you may have reason to believe that the large population is
Normally distributed (what you have called z statistics).  If so, then
mu is t-distributed, with a mean of m and a standard deviation equal
to the standard deviation of mean calculated above.

The curve for the t distribution has the bell shape reminiscent of a
Normal distribution.  But the t distribution is not just one shape,
but a family of shapes dependent on n.  For small n, the tails of the
distribution are much longer than the tails of a Normal distribution,
reflecting a broad uncertainty of the value of the true population
mean.  For large n, the t distribution approaches the Normal
distribution as a limit.  In practice, a rule of thumb says that if
your n is greater than 50, you needn't bother with the t-distribution,
but can use the Normal distribution as an approximation.

You can read more about the t distribution in stat text books, and at
many web sites.  Here is a sample:

  The Student t Distribution     
http://www.fmi.uni-sofia.bg/vesta/Virtual_Labs/special0/special4.htm 

  Student's t Distribution
    http://mathworld.wolfram.com/Studentst-Distribution.html 

  Sampling
    http://astro.temple.edu/~mjoshua/Assgn-6.doc 

  Statistical Distributions - Student t Distribution - Example
    http://www.xycoon.com/studentt.htm 

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/