The standard deviation is “interpreted" with statements about the proportions of the data that fall within 1, 2, or 3 standard deviations of the mean.

**Chebychev’s rule** applies to
*any* set of data. It is summarized in the following table:

intervalz-scoreguaranteed proportion of the datamean – s to mean + s –1 to 1 0% -- 100% (could be anything) mean – 2s to mean + 2s –2 to 2 75% -- 100% ( at least75%)mean – 3s to mean + 3s –3 to 3 89% -- 100% ( at least89%)mean – 4s to mean + 4s –4 to 4 94% -- 100% ( at least94%)

For instance, consider the data set {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2}. The mean is 1, and the standard deviation turns out to be s = 0.97. So the only observations that are within 1 standard deviation of the mean are the two 1's in the middle, and they comprise only 10% of the data. However, all of the data are within 2s of the mean.

For a more extreme example, consider the data set {0, 0, ... , 0, 0, 1, 1, 2, 2, ... , 2, 2}, where there are 99 ones and 99 twos. The mean is 1, and the standard deviation turns out to be s = 0.99975. So the only observations that are within 1 standard deviation of the mean are again the two 1's in the middle, and now they comprise only 1% of the data. However, again, all of the data are within 2s of the mean.

So we see that it is *possible* to have a very small
proportion of a data set within 1 standard deviation of the mean.
However, Chebychev's rule says that it is *impossible* for
*less* than 75% of the data to be within 2s of the mean, and it
is *impossible* for *less* that 89% of the data to be
within 3s of the mean.

**The empirical rule** applies to sets
of data that have approximately “*bell-shaped*"
histograms.

intervalz-scoreguaranteed proportion of the datamean – s to mean + s –1 to 1 approx 68% mean – 2s to mean + 2s –2 to 2 approx 95% mean – 3s to mean + 3s –3 to 3 approx 99.7% (nearly all)

*So the empirical rule says that this picture is * typical *for data
with an approximately bell-shaped histogram:*

Below is a histogram that represents a set of 100 measurements. The mean of the data is approximately 23, and the standard deviation is approximately 7. Notice that the histogram is approximately bell-shaped.

In this picture, we estimate that about 70% of the data are within
1 standard deviation of the mean (between 16 and 30), about 95% are
within 2 standard deviations of the mean (between 9 and 37), and
*all, or almost all,* of the data are within 3 standard
deviations of the mean (between 2 and 44). These estimates are
consistent with the empirical rule.

Here’s a histogram for another set of data, in which the mean is approximately 71, and the standard deviation is approximately 10. Notice that the histogram is again approximately bell-shaped (though not as nicely as the one above).

About 68% of the data are within 1 standard deviation of the mean
(between 61 and 81), about 93% are within 2 standard deviations of
the mean (between 51 and 91), and (this time) *all* of the data
are within 3 standard deviations of the mean (between 41 and 101).
Again, these estimates are consistent with the empirical rule.

**Remember** that the Empirical rule
applies only to data sets with symmetric, bell-shaped histograms.
For any other set of data, we have to rely on Chebychev's rule. __This
includes data for which the histogram shape is unknown.__