Re: Standard Deviation & the 68-95-99.7 rule

On Dec 21, 3:25 pm, Maya <maya_s...@xxxxxxxxxxx> wrote:
On Dec 21, 3:12 pm, Virgil <Vir...@xxxxxxx> wrote:



In article
<befa0c10-aa71-454d-8b95-60c00767b...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,

Maya <maya_s...@xxxxxxxxxxx> wrote:
On Dec 21, 1:58 pm, Maya <maya_s...@xxxxxxxxxxx> wrote:
On Dec 21, 1:02 pm, "FredJeffr...@xxxxxxxxx" <FredJeffr...@xxxxxxxxx>
wrote:

On Dec 21, 11:39 am, Maya <maya_s...@xxxxxxxxxxx> wrote:

At the bottom of the intro to the Wikipedia entry on the 68-95-99.7
rule, it states:

"This rule is often used to quickly get a rough estimate of
something's probability, given its standard deviation."

What an awful sentence.

What " thing's " probability could I estimate, given the thing's
standard deviation? Let's say I have this data set: {6, 6, 8, 8} .
It's standard deviation is 1. So, given its "1", I can estiate the
probability of ..... what?

http://en.wikipedia.org/wiki/68-95-99.7_rule

You also need the mean, in the case of your data set 7. So the
68-95-99.7 rule says that about 68% of observations will be within 1
of 7 (between 6 and 8), 95% within 2 of 7 (between 5 and 9) and 99.7%
within 3 of 7 (between 4 and 10) IF your data set were distributed
normally.

Say you have a normally distributed data set with mean 7 and standard
deviation 1. Pick an element at random from your set. The probability
of that element's being between 6 and 8 is 68%, the probability of its
being between 5 and 9 is 95%, etc.

There is a better example at the bottom of this
page:http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.h
tml

Thanks Fred.

This stuff seems to be going in a circle. The Empirical Rule applies
only to Normal Distributions. So I can ascertain some things about the
data points in a normal distribution by applying the empirical rule,

No. If you know the distribution is normal, or nearly so, you can get
exact probabilities, so there is no need at all for an empirical rule
(although it may still be useful if you lack access to a scientific
calculator or statistical tables, etc.). The so-called "empirical
rule" is /derived/ from exact calculations on normal distributions.


but I should only apply the empirical rule if I'm first sure that the
data set is a normal distribution!

If the distribution is a bit different from normal and not too badly
skewed, the empirical rules may at least give you figures in the right
ballpark. (Of course, it is hard to say exactly what is meant by "a
bit different" and "not too badly ... ".) Typically, these things are
tested experimentally, perhaps using simulation



I'm trying to find a real-world use for Standard Deviation

Well, for one thing, it can give you bounds that are often useful and
are certainly better than nothing. Chebychev's Inequality says that /
for any kind of distribution at all/ (discrete, continuous,...,
anything) that has finite mean m and finite standard deviation s, the
probability that the random variable X deviates from m by more than k
standard deviations is <= 1/k^2. So, the probability that you are more
than 1 standard deviation from m is <= 1 (not very useful!), that you
are more than 2s away from m is <= 1/4, that you are more than 3s away
from m is <= 1/9, etc. In other words, for any distribution at all,
the chance of being within 2s of m is at least 3/4, of being within 3s
of m is at least 8/9, etc. In the special case of the normal
distribution we can give better answers, but the bounds are sometimes
good enough in applications. Also, in some types of applied models,
quantities like system costs, etc., may be quadratic functions of the
random variable, in which case evaluating the expected cost needs
only the mean and the variance. Also, the standard deviation is, in
general, a type of "spread" measure that is often used in real-world
applications. For example, Markowitz won the Nobel Prize in Economics
for his work on mean-variance investment-portfolio analysis. These
concepts are widely used in designing mutual funds and other types of
investment instruments.

and the
Empirical Rule, but so far it seems the only uses are to tell me
things about a data set if and only if I already know those very
things about the data set are already true.- Hide quoted text -

- Show quoted text -

I thought the result of calculating the Standard Deviation of a data
set would tell me whether the data set's distribution is Normal,
Continuous, or Discrete?

Absolutely not. If you give me the mean and the standard deviation I
can construct several different discrete and several different
continuous distributions that have that mean and variance.


If it can't tell me that, then what good is it to know that data
points are either: 1)close to the mean, or 2)not so close to the
mean ?

It depends on the application: who wants to know the answer to that
question, and why is it important to them?


Among other things, this sort of information is used in statistical
hypothesis testing, q.v.- Hide quoted text -

- Show quoted text -

IN what way? What would be a simple example of the good of knowing
only the SD without knowing which type of distribution the data set is?

See some of the reasons I listed above.

R.G. Vickson

.

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