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**abc4616****Member**- Registered: 2006-10-01
- Posts: 9

If a set of observations is distributed as N(u, std. dev.^2), what is the percentage of the observations will differ from the mean by:

a) less than 1 standard deviation?

b) less than 2 standard deviation?

C) less than 3 standard deviation?

How can I prove this?

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**fgarb****Member**- Registered: 2006-03-03
- Posts: 89

Pretend that you took an enourmous number of measurements of your variable that was distributed according to N and plotted the results. You would make a graph *of N* itself - exactly the same shape. Call the function f the normalized (unit area) version of this graph.

Then, it should make sense that the probability that a given measurement reveals a result between x and x+dx is f(x)dx. So the total probability to get a result between <i>any<\i> two points is just the integral from the lower point to the upper point of f(x) (the area under your normal distribution between those two points). The trouble is that I don't think you can symbolically integrate your normal distribution. A couple ways you can proceed is to

a) make an approximation (say, Taylor expansion) and work with that

b) Use a computer and find your answer numerically

c) Just look up what the values are in a table and believe them because someone else has already done this work for you!

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