This month's Challenge Question: Specs & Techs from IEEE Engineering360:
Natural variations in repeated measurements form a normal distribution. Ten recent measurements are made where the average and standard deviation of the measurement is known from many previous measurements. How many times more likely is it that none of the ten recent measurements will occur outside of 2 standard deviations of the average as compared to outside of 1 standard deviation of the average?
And the answer is:
When dealing with normal distributions, the probability of a data point occurring within 1 standard deviation is 68.27%. The probability of a data point occurring within 2 standard deviations is 95.45%. The odds that 10 recent measurements all fall within 1 standard deviation is 0.6827*0.6827*0.6827*0.6827*0.6827*0.6827*0.6827*0.6827*0.6827*0.6827=0.6827^{10}=0.022=2.2%
Similarly, the odds that 10 recent measurements all fall within 2 standard deviations is (0.9545^{10}=0.6277=62.8%). Since 0.6277/0.022=28.5, it is 28.5 times more likely that ten measurements will fall within 2 standard deviations than 1 standard deviation. This makes sense since 2 standard deviations provides more room for measurement variation than 1 standard deviation.

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