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Let's say I have mean quality of 50 and standard dev of .1. And I the requirements for any product must be of quality 45 or higher. How do I calculate the chances that the quality of a product is greater than 45?

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Is the quality distributed normally? –  copper.hat Nov 29 '12 at 20:49
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up vote 2 down vote accepted

You can use a one-sided version of Chebyshev's inequality, $$\Pr(X - \mu \leq -k \sigma) \leq \frac{ 1 }{ 1 + k^2 }$$ though in this rather extreme case the two sided version will give much the same bound

Here $50-45$ is $50$ times the standard deviation, so the probability that a sample of the product is less than or equal to $45$ is less than or equal to $\frac{1}{2501}$ so the probability it is more than $45$ is greater than or equal to $\frac{2500}{2501} \approx 0.9996$.

That can be a rather loose bound. If you knew the distribution was Gaussian, then that final figure would be about $1-10^{-545}$ which is extremely close to $1$.

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You can't. You need to know the distribution to do that. All you can do if you only know the mean and standard deviation is to bound the probability using Chebyshev's inequality.

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Oh... darn :s I thought it would've been simpler than that. Thanks! –  Shelby. S Nov 29 '12 at 20:52
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