aglucks wrote:
>
> I was always taught that at least 3 (three) data points are needed to
> calculate a statistically valid, meaningful standard deviation.
>
> What is the value of a standard deviation that is calculated using
> only 2 data points?
It depends which meaning of "value" you have in mind.
The numerical value is |x1 - x2| / sqrt(2)
The practical value is practically nothing - as is the case for n=3, 4,
... to a diminishing extent. See the following simulations
(in MINITAB, my fave stats-hacking tool)
#anything after a # is a comment!
#anything after a MTB > is what I typed
#MINITAB does the rest automagically
MTB > rand 1000 c1 c2 #1000 samples of size 2 (normal, mean 0, SD 1)
MTB > rstdev c1-c2 c10 #Estimate the SD from the sample
MTB > boxp c10 #boxplot
Boxplot
-------------
-----I + I---------------- ****** ***
-------------
+---------+---------+---------+---------+---------+------C10
0.00 0.70 1.40 2.10 2.80 3.50
MTB > rand 1000 c1-c3 #size 3
MTB > rstdev c1-c3 c10
MTB > boxp c10
Boxplot
--------------
----------I + I-------------------***
--------------
+---------+---------+---------+---------+---------+------C10
0.00 0.50 1.00 1.50 2.00 2.50
MTB > rand 1000 c1-c4 #size 4
MTB > rstdev c1-c4 c10
MTB > boxp c10
Boxplot
---------------
-------------I + I------------------ * * **
---------------
--------+---------+---------+---------+---------+--------C10
0.40 0.80 1.20 1.60 2.00
Note that in every case half the computed standard
deviations lie outside the "box". Thus, for
n=2 half the values 1/3 or greater than 4/3 the true
are less than value
n=3 1/2 5/4
n=4 2/3 6/5
Not good for much, really. It looks better by the time
you get up to (say) n=10:
MTB > rand 1000 c1-c10 #size 10
MTB > rstdev c1-c10 c15
MTB > boxp c15
Boxplot
-------------
*------------------I + I------------------***
-------------
--------+---------+---------+---------+---------+--------C15
0.50 0.75 1.00 1.25 1.50
and by n=30 you're within +- 10% half the time.
MTB > rand 1000 c1-c30
MTB > rstdev c1-c30 c35
MTB > boxp c35
Boxplot
----------
** *------------I + I------------- * *
----------
--------+---------+---------+---------+---------+--------C35
0.60 0.80 1.00 1.20 1.40
So nothing magical happens when n=3; you *can* compute
SD for n=2 and it won't be much good till about n=10.
-Robert Dawson
.
.
=================================================================
Instructions for joining and leaving this list, remarks about the
problem of INAPPROPRIATE MESSAGES, and archives are available at:
. http://jse.stat.ncsu.edu/ .
=================================================================
