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I know that for a normal distribution, only 5 percent of the sample is outside mean plus or minus 2 times the standard deviation.

I wonder for a non-normal distribution, is there still a specific proportion of data that is filtered out? I know it won't be 5 percent, but is the percentage still fixed?

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The stuff about Chebyshev's inequality is both important and relevant but I wanted to respond more directly to the question being asked here:

  • if you mean population proportion relative to the interval based on population mean and standard deviation then there will be a fixed proportion for each distribution (as long as it's fully specified up to location and scale). You figure out the proportion by working out the quantiles of $\mu-2\sigma$ and $\mu+2\sigma$.

    For example if we look at an exponential distribution, $\mu-\sigma$ is the lower limit of the distribution (I mean that $F(\mu-\sigma)=0$) so certainly nothing lies below $\mu-2\sigma$, and about 5% lies above $\mu+2\sigma$ ($\exp(-3)\approx 0.04979$).

    With the uniform, 100% of the distribution is within $\sqrt{3}\approx 1.732$ standard deviations of the mean. (The other answer gives the lower bound on the proportion across all distributions.)

    We can do the same with any number of other distributions; it will always be between 0 and 25% outside two standard deviations from the mean, and for continuous unimodal densities I think the bounds are 0 to 11.1% (1/9) outside two standard deviations from the mean.

    exponential and uniform distributions, with mean plus or minus two standard deviations

    By comparison, if you asked about the Weibull distribution or the gamma distribution, it would depend in which shape parameter you had (but not on which scale parameter).

  • if you're looking at sample proportions outside limits based on sample mean and sample standard deviation then it's not true even at the normal, the proportion outside those varies from sample to sample. Indeed if you mean to use sample values for any of mean, standard deviation or the proportion outside then it will vary from sample to sample (though in very large samples generally not by much).

    [There are bounds that relate to sample-mean, sample-sd and sample proportion, but they're basically like the Chebyshev bounds adjusted for the fact that the sample standard deviation has the $n-1$ denominator.]

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For a general distribution, at most 25% of the data can be outside 2 standard deviations. Note this is the absolute worst case scenario, so a lot of distributions (pretty much every one you encounter in practice) will do better.

In general, for $k$ standard deviations, it's $1 / k^2$ percent of the data that can be outside.

Here's a link to the general rule: https://en.wikipedia.org/wiki/Chebyshev%27s_inequality

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  • $\begingroup$ @ Kevin Thanks Kevin! What if k is larger than one? By the way, do you mean that for a specific distribution model, there is always a fixed proportion of data lying outside mean+2*stddev? $\endgroup$
    – zero_yu
    Commented Jul 31, 2017 at 17:21
  • $\begingroup$ I'll clarify my answer to "at most 25%" since that is more accurate. If $k$ is smaller (I think that is what you meant) than one, then there's no statement. Take a uniform distribution on the set $\{-1, 1\}$ - then the standard deviation is one, so all of the data lies outside $k$ standard deviations for any $k < 1$. $\endgroup$
    – Kevin
    Commented Jul 31, 2017 at 17:25
  • $\begingroup$ To answer your questions about specific distributions - every distribution has a fixed proportion, it just depends on the distribution. But this rule states that no matter which distribution, that proportion can't be above 25% - not a very strong rule (most distributions have less than 25% of values within 2 standard deviations), but it covers all distributions. $\endgroup$
    – Kevin
    Commented Jul 31, 2017 at 17:27
  • $\begingroup$ @ Kevin Sorry Kevin, one more question. I have a bunch of data which is all between 0 and 1 for each day. if the type of data distribution does not change, is the data proportion outside 2 standard deviation is fixed on every day? $\endgroup$
    – zero_yu
    Commented Jul 31, 2017 at 18:19
  • $\begingroup$ If it's the proportion of observed data outside 2 s.d's of the population distribution, then the answer is no. Suppose you have a distribution with mean 1/2 and s.d. 1/8. Then it is theoretically possible (though unlikely) to draw 1's every single day. However, if you take the sample standard deviation, then the proportion outside $k$ sample standard deviations from the sample mean will be no more than a given proportion. However, the actual proportion will differ day to day. $\endgroup$
    – Kevin
    Commented Jul 31, 2017 at 19:39

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