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Statistics: Standard Deviation for a Normal Distribution

  1. Jun 13, 2017 #1
    1. The problem statement, all variables and given/known data
    A company allows a maximum failure rate of 1 out of 250,000 parts. To insure this quality goal, failed parts must be how many standard deviations from the mean? Use Excel to solve.

    2. Relevant equations
    z= (X-μ)/σ

    3. The attempt at a solution
    Hi! So I'm assuming that this is a normal distribution. I'm a little confused, I kind of feel like there wasn't enough information provided to find how many standard deviations need to be away from the mean.

    So far I've tried finding the z-value using excel and assuming that 1/250000 is my alpha value:
    =NORM.S.INV(1/250000) = -4.46518

    I was thinking about trying to plug it into this formula to find σ:
    z= (X-μ)/σ

    Am I on the right track with this? I wasn't given an X or a μ, so I don't know how I would go about solving this.
  2. jcsd
  3. Jun 13, 2017 #2


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    You are likely correct that you should be assuming a normal distribution. However you are not determining your z-value correctly. You need to account for the two sided nature of the error rate. Your alpha is the area under the bell curve on both sides of the mean z standard deviations out. So the area on each tail is alpha/2.
    The inverse norm function is the inverse CDF and so gives the upper bound on the area under the bell curve of the input value.

    Visualize the bell curve with the two tails, alpha/2 is the area of the upper tail (above the critical z value) and so 1 - alpha/2 is the area to the left of the critical z value.
    You can then take inverse norm of (1-alpha/2) or equivalently the negative of the inverse norm of alpha/2.

    Remember your Z score is standardized in terms of the mean and SD so it is the number of standard deviations above the mean so once you find the critical z-value that is your answer.
  4. Jun 13, 2017 #3
    Thank you!! Super helpful!!
  5. Jun 13, 2017 #4


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    Just for a bit of context, remember the (1-2-3) -68-95-99.7 rule, which will give you 27 failures per 10,000. Times 25 that is 675 per 250,000. So you have some way to go beyond that.
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