I have a set of students with mean height of 100cm with standard deviation of 10cm. I want to be able to qualify a student based on his/her height relative to the mean height for the group. Would it be possible to look at the deviation of a student's height from the mean height for the group in terms of number of standard deviations and say a certain students height is different, very different from the height of the group - like 1 standard deviation or 2 standard deviations or so - are there typical measures for this ? The distribution is not necessarily normal.
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Short answer: Typically if it's over 2 standard deviations, it's considered "different."
Longer answer: If you approximate the distribution of heights as normal, you can look up the proportion of individuals that fall outside a given number of standard deviations from the mean. For instance, roughly only 5% of students will be over 2 standard deviations away. This lets you quantify just how "extreme" such heights are, and is the basis of the "2 standard deviations" rule of thumb. (Section 3.1.5 of OpenIntro Statistics, Page 136).
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$\begingroup$ could you please add references $\endgroup$ Commented Jul 22, 2017 at 2:06
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$\begingroup$ Sure, I added a text for the first two answers. I deleted my "longer answer" since it doesn't seem as relevant upon re-reading your question. $\endgroup$– KevinCommented Jul 22, 2017 at 2:41
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$\begingroup$ I looked at section 3.1.5 which has title
68-95-99.7 rule- not able to find the reference where it says 2 sds is the right range to consider the sample to be different. $\endgroup$ Commented Jul 22, 2017 at 9:17 -
$\begingroup$ This is because what is "different" is ultimately subjective. Do you think a 95th percentile is "different?" Or would you require a 99th percentile. $\endgroup$– KevinCommented Jul 22, 2017 at 15:32