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Advanced Math/Standard Deviation & Confidence Interval


Hi there,

I would really appreciate some help with some concepts in stats that I am having trouble grasping. I cannot understand what is meant by 3 standard deviations from the mean. According to my Introductory Stats unit lecture notes, to get a 99.7% confidence interval, you add 3 standard deviations of the distribution to the mean of the distribution but the z-score corresponding to 99.7% is 2.78.

Thanks in advance

I think the confusion is arising from finding the z-score for 2 different confidence intervals:

1. probability that the z-score is strictly less than 3 standard deviations above the mean, and

2. the probability that the z-score is within 3 standard deviations of the mean.

For the first case, I get a z-score (from tables) for the probability of 0 ≤ z ≤ 2.78 of 0.4973. Adding a probabiity of 0.5 for all the z-scores less than 0 gives 99.73%, which is what you get.

For the second case, a z-score of 0 ≤ z ≤ 3 gives a probability of 0.4987. To get the corresponding probability for -3 ≤ z ≤ 3, multiply by 2 to get 99.74%.  

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randy patton


college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography


26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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