As my question says, when I look at the empirical rule for a normal distribution, it says that 95% percent of values lie between 2 standard deviatons but if I look at the z-score for the 95% confidence interval, the z-score is1.96 which is close to 2 standard deviations from the mean but not exactly 2 standard deviations. What does the 95% mean in the both cases?
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asked Oct 24, 2020 at 18:41
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2 Answers
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For the empirical rule, we say “two” because it’s more convenient to say that than to say 1.96. The correct value is 1.96, not two. (Even 1.96 has some amount of rounding.)
The 95% for a confidence interval is a separate issue. A 95% confidence interval means that, if you took new samples from your population over and over, if you follow the procedure to calculate a 95% confidence interval, 95% of calculated confidence intervals will contain the true population value.
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$\begingroup$ What I am really confused is that the probability between 2 standard deviation in the satndard normal distribution is 95% but the empirical rule says that 95% of values lie between 2 standard deviation. It sounds like a totally different thing to me. Can you stretch your explanation a bit further? $\endgroup$ Commented Oct 25, 2020 at 5:26
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$\begingroup$ It has to do with looking at data (empirical rule) versus looking at an abstract “standard normal” distribution that only exists in math. $\endgroup$– DaveCommented Oct 25, 2020 at 5:43
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The area under the $2$ standard deviation (std) according to the z table is $0.9772$.
So, the area between $-2$ std and $+2$ std is $$0.9772 - (1-0.9772) = 0.9772-0.0228 = 0.9544,$$ which might be more accurate than the $0.95$.
But commonly, the $\pm 2$ std area is considered as the $95\%$ area according to the empirical rule.
