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I have 20 Score values:

1, 3, 4, 6, 10, 14, 16, 19, 23, 32, 34, 38, 43, 48, 53, 59, 63, 69, 74, 85.

So, I calculate the Standard Deviation using:

$$ \sigma = \sqrt{\frac{\sum(x-\bar x)^2}n} $$

.. which is 25.4 and mean is 34.7.

Now, from 68-95-99.7% rule:

  • How many values and what are the values in one standard deviation?
  • How many values and what are the values in the second standard deviation?

How do I calculate all that?

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    $\begingroup$ Well, what do you mean by "the values in one standard deviation" and "the values in the second standard deviation"? I haven't heard that kind of phrasing before. Did you get that phrasing from somewhere? The standard deviation is just a number which can be used as a unit of measurement; it's not a set of values. $\endgroup$
    – Sophie Swett
    Commented Dec 14, 2019 at 4:01
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    $\begingroup$ I'm certain OP means "within one standard deviation of the mean" since that is the context in which the 68-95-99.7% rule is meant to apply. $\endgroup$
    – Noah
    Commented Dec 14, 2019 at 4:05
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    $\begingroup$ The rule assumes a normal distribution. .Add the self study tag. Two standard deviations from the mean for a normal distribution is actual 95.4%.So this must be the intervals that contain 1 & 2 standard deviations from the mean. So although it is still ambiguous I think the first answer is [34.7-25.4, 34.7+25.4} =[9.3, 60.1] and for the second [34.7-2(25.4), 34.7+2(25.4)]= [-16.1,85.5]. $\endgroup$
    – Michael R. Chernick
    Commented Dec 14, 2019 at 5:27

1 Answer 1

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The 68-95-99.7% rule can only be validly applied to a normal distribution. Your data are from a finite sample, so the rule does not apply.

You don't need the rule though. You can just count. "Within one standard deviation of the mean" means within the interval $[\bar{x} - \sigma, \bar{x} + \sigma] = [34.7 - 25.4, 34.7 + 25.4] = [9.3, 60.1]$. How many and which values are between 9.3 and 60.1?

You can then apply the same principle to find the values within two standard deviations of the mean. I'll let you figure those out since this is clearly a homework problem and we're not here to give you homework answers.

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  • $\begingroup$ Shouldn't he calculate standard deviation with n-1 since his "data are from a finite sample?" $\endgroup$
    – Ron Jensen
    Commented Dec 14, 2019 at 4:20
  • $\begingroup$ My formula assumes that it is based on population. Okay, thanks. As I understand there are 12 values that lie within the range. @Noah: Can you explain a bit more why I don't need that rule? Should I have like 100 values or 500 values or 1000 values to qualify for that? $\endgroup$
    – user963241
    Commented Dec 14, 2019 at 4:27
  • $\begingroup$ You don't need that rule because you can count. That rule is only useful when you can't count the number of data points because you don't have the data in front of you. But again, it only works for theoretically normal distributions. You can't, shouldn't, and don't need to use it when you have the data and can simply count how many data points are within the interval. There is no number of data points at which this becomes useful if you have the data in front of you. $\endgroup$
    – Noah
    Commented Dec 14, 2019 at 4:47

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