Re: Standard Deviation Question....


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Re: Standard Deviation Question....



Dear Mike:

In a sense, the TI-92 is correct. You see, there are two different
definitions of standard deviation:

1. Suppose you have 9 cards in a bag, with the numbers 4,7,10,11,12,14,15,21,32
written on them. Then you start pulling out cards and random, and
replacing them back.
This experiment is modeling a random variable with mean 14 and variance
7.83. To compute the variance, you subtract the mean from each value,
square the results, add them and divide by n=9 (the number of
equiprobable outcomes), finally you take a square root.
Thus, I understand why you expect the answer to your question to be 7.83.

2. Now look at the situation you have. The numbers 4,7,10,11,12,14,15,21,32
are (most probably) obtained as observations of an unknown random
variable, whose
mean and variance you do not know. (It need NOT be the random variable
described in the paragraph above). You want to estimate the mean and the
variance. The best estimate for the mean is the average of the empirical data.
(This is called the "empirical mean", and it is different from
the real mean which you don't know).
The best estimate for the variance is the so-called "empirical standard
deviation", which you obtain in the following way. You subtract the
empirical mean from each value, square the results, add them and divide
by (n-1)=8 (the number of equiprobable outcomes
minus one), and finally you take a square root.Thus you get the answer
8.31, which the TI gives you. You can check that 8.31=7.83*sqrt(9/8).

Why divide by (n-1) instead of n? The answer to this question is rather
tricky. It has
something to do with the so-called "unbiased estimates" in statistics.
The rationale is that, since you dom't know the real mean, and are
replacing it by the empirical mean, you
"lose one degree of freedom". This can all be found in books on
statistics, and has to do with some deep results like Cochran's theorem.


I realize that this might not be very helpful, but at least it lets you
know that there is a reason for the answer you are getting, other than a
bug in the calculator.

Yours,

Bogdan Doytchinov
------------------------------------------


Mike Herald <oasis9@HOTMAIL.COM> wrote:

>Can someone explain to me why TI-92 gives me the wrong answer for
>standard deviation.
>
>I put in:
>stdDev({4,7,10,11,12,14,15,21,32})
>
>and I get 8.30662
>
>but the answer is 7.83
>Try to keep in mind I kinda new at using a TI-92.  So, why do I get 8.31
>instead of 7.83
>
>Mike
>
>
>______________________________________________________
>Get Your Private, Free Email at http://www.hotmail.com
>


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