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### Standard Deviation and Conditional Probability

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Date: 07/14/98 at 22:45:22
From: Angela C. Williams
Subject: Quantitive stats

I have a couple of questions:

Find the standard deviation of the following sample: 5.

My answer is the square root of five. Am I suppose to use some formula?

Also, here are data on 246 stocks:

Price Increase     No Increase     Total

Dividends paid          34              78           112
No dividends paid       85              49           134

Total                   119            127           246

Given that a stock increased in price, what is the probability that it
also paid dividends?

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Date: 10/22/98 at 21:52:24
From: Doctor Andrewg
Subject: Re: Quantitive stats

Hi Angela!

Okay, first of all let's look at the standard deviation. Standard
deviation is a measure of variability in the observations - a higher
standard deviation means more variability, and a lower standard
deviation means less variability.

Here is a sample with high variability and one with low variability:

High variability       Low variability
2                     15
8                     16
14                     14
13                     12
26                     15
13                     14
3                     11

Can you see what I mean? The first column's values are spread out much
more than the second's.

We need a way of quantifying (giving a number that measures) the
variability of a sample of observations. This way we can talk about a
sample with a standard deviation of 100, say, not just a large standard
deviation, since in some cases 100 may be a small standard deviation as
well, depending on the units we use to measure the observations.
Imagine heights measured in meters and then measured in centimeters.
Which one looks more variable?

For a sample we define the standard deviation as the sum of the squares
around the mean, divided by the number of observations. If n is the
number of observations, x-bar (the x with a bar over the top) is the
mean of the observations, and xi is the ith observation, then the
standard deviation is:

n
---\          2
\     (x  - x)
/       i
---/
i=1
------------------------
n

If you put the single number 5 into that formula, n = 1, x-bar = 5, and
x1 = 5. The answer should be zero. This isn't very useful, but it is
the only answer I can see here.

So, if 5 is the only number in the sample then the standard deviation
for that sample would be zero - there is no variability at all.

If you were using the sample standard deviation to estimate the
population standard deviation, then you wouldn't be able to do this
with only one observation. The division by n in the formula above would
be replaced with a division by (n-1) which is 1-1 = 0 in this case. The
standard deviation would then be equal to 0/0 (which is what we call an
indeterminate form).

The second question is a question of conditional probability. What you
are asking is the following: What is the probability that a stock
paid dividends if it increased in price? We can write this as P(A|B)
where A is "paid dividend," the "|" symbol can be read as "given
that," and B is "stock increased in price." So in total that reads
"Probability(paid dividends given that stock increased in price)."

So what you are saying is: how many (as a proportion) of the stocks
that increased in price also paid dividends? Can you see what I mean?
We are not interested in companies whose stocks did not increase in
price, only those whose prices did increase. So we can ignore the "no
increase" and "total" columns and just look at the "price increase"
column.

There are 119 companies that had a stock price increase, and 34 of them
paid dividends, so the probability that a company whose price increased
also paid dividends is 34/119. Does this make sense to you?

I'll finish this message now and let you read through my suggestions.
If you have any more questions or if I'm not clear enough in my

- Doctor AndrewG, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Probability
High School Statistics

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