# Standard deviation problem

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

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 $1$.