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Vannek
09-07-2004, 04:58 PM
If I have an normal distribution how can I calculate the % of a 1.5SD for example?Whats the formula to calculate the probability of an XSD swing?

topspin
09-07-2004, 10:10 PM
Take the Q-function of the swing in standard deviations. This question was asked before in this thread (http://forumserver.twoplustwo.com/showflat.php?Cat=&Board=probability&Number=953830) .

pzhon
09-07-2004, 10:24 PM
[ QUOTE ]
If I have an normal distribution how can I calculate the % of a 1.5SD for example?Whats the formula to calculate the probability of an XSD swing?

[/ QUOTE ]

The probability that a normally distributed random variable is between A and B standard deviations above from the mean is

Integral from A to B (1/sqrt(2pi)) exp((-x^2)/2) dx.

There isn't an elementary formula for this, but this is common so people have made tables of the values, and now web calculators (http://www-stat.stanford.edu/~naras/jsm/FindProbability.html) . Numerical integration tools work very well. Another technique when A and B are relatively small is to use a power series. The power series for exp(-x^2/2) converges, and so does its termwise integral. <font color="white"> (... and these converge to the right values.)</font>

exp(z) = 1 + z + z^2/2 + z^3/6 + z^4/24 + z^5/5! + ...

exp(-x^2/2) = 1 - x^2/2 + x^4/(4*2) - x^6/(8*6) + x^8/(2^4*4!) - ...

Integral exp(-x^2/2) = C + x - x^3/6 + x^5/40 - x^7/(7*2^3*3!) + x^9/(9*2^4*4!) -...

If you want the probability that you are between 0 and 1 standard deviations above the mean, you could integrate termwise from 0 to 1: (1 - 1/6 + 1/40 - 1/336 + 1/3456 - ...)/sqrt(2pi). By hand, I find the sqrt(2pi) portion harder to estimate.

BeerMoney
09-08-2004, 09:35 AM
Find a normal table.

Vannek
09-08-2004, 10:09 AM
I coundlt properly use those calculators. On the first one I put 1 and clicked left, it gives prob of 0.84, isnt a 1SD change withing 2/3(0.66) of the results?On the second one I inserted 0.66 on the prob, it gives 0.41 percentile, not sure what this means

pzhon
09-09-2004, 05:16 AM
[ QUOTE ]
I coundlt properly use those calculators. On the first one I put 1 and clicked left, it gives prob of 0.84, isnt a 1SD change withing 2/3(0.66) of the results?On the second one I inserted 0.66 on the prob, it gives 0.41 percentile, not sure what this means

[/ QUOTE ]
Normal calculator link (http://www-stat.stanford.edu/~naras/jsm/FindProbability.html).

If you enter 1 and press &lt;Left&gt;, you get the probability the random variable is less than 1 standard deviation above the mean. If you want the probability the variable is between -1 and 1, type in -1 and 1 and press &lt;Between&gt;. See the shaded region.

You saw that 84% of the time, the random variable is at most 1 standard deviations above the mean. What corresponds to 90%? Type 0.9 in the second applet, keeping Mean: 0.0 and Std. Dev.: 1.0. You get 1.28... That means 1 time in 10, a normally distributed random variable is 1.28 or more standard deviations above the mean. One time in 5, a normally distributed random variable is either 1.28 or more standard deviations below or 1.28 or more standard deviations above the mean. 1.28 standard deviations above the mean is the 90th percentile.