# How many values are in one standard deviation?

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?

• 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. Dec 14 '19 at 4:01
• 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.
– Noah
Dec 14 '19 at 4:05
• 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]. Dec 14 '19 at 5:27

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?