Interpreting the standard deviation

The mean and standard deviation of a variable summarise important aspects of its distribution — its centre and spread. It is relatively easy to interpret the value of a variable's mean (its point of balance), but the formula for the standard deviation is more complex so it is harder to 'understand' what its numerical value tells you about the distribution. Luckily, understanding its definition is much less important than knowing its properties and having a feel for what its numerical value means.

Guessing the standard deviation from a histogram or box plot

If you have understood the 70-95-100 rule, you should be able to make a fairly accurate guess at the standard deviation of a batch of values from a histogram, dot plot or box plot (without doing any calculations). About 95% of the values should be within 2 standard deviations of the mean, so after dropping the top 2.5% and bottom 2.5% of the crosses (or area of the histogram), the remainder should span approximately 4 standard deviations. So dividing this range by 4 should approximate the standard deviation.

In a similar way, about 70% of the values should be within 1 standard deviation of the mean. The central box of a box plot includes 50% of the values, so the length of the central box should be a bit under 2 standard deviations.

If a quarter of the range is much different from a bit more than half the width of the central box in a box plot (or middle 50% of the histogram area), you could average these two estimates of s.

Estimating the standard deviation 'by eye'

Although you cannot accurately evaluate the standard deviation of a data set without access to the raw data, it should be possible to obtain a rough estimate by looking at a graphical summary of the data. In the exercises below, you should estimate the standard deviation from a histogram or box plot of a data set.

Use the 70-95-100 rule-of-thumb to estimate the standard deviation from the histogram, then type your guess into the Answer box and click Check.

Most of a data set are usually within 2 standard deviations of the mean unless the distribution is very skew, so a good guess is usually a quarter of the range (max - min).

Click Another Question and repeat until you can obtain a reasonable estimate consistently.


The following exercise is similar, but shows a box plot of the data.

Again repeat until you obtain a reasonable estimate consistently.