The shape of a distribution

The mean and standard deviation describe important aspects of a distribution of values (its centre and spread) but many distributions have the same mean and standard deviation.

The mean and standard deviation do not fully define the shape of a distribution.

In particular, the mean and standard deviation give not indication of whether a distribution is symmetric or skew with a long tail to one side.

Sketching a symmetric distribution

However from the 70-95-100 rule-of-thumb, you should be able to sketch a symmetric distribution to match and given mean and standard deviation. Such a distribution would have about 70% of values within 1 standard deviation of the mean and 95% within 2 standard deviations of the mean.

If you properly understand what the mean and standard deviation tell you about a distribution of values, you should be able to sketch such a histogram or box plot.

Sketching a distribution from its mean and standard deviation

In the exercises below, try to sketch a symmetric distribution with the target mean and standard deviation. (The actual distribution may be skew, but a symmetric distribution is the best we can do from only the mean and standard deviation.)

For the data set decribed above, most of the data will be within 2 standard deviations of the mean, so

  1. Use the mean and standard deviation to work out the likely minimum and maximum for the data.
  2. Use the pop-up menu to select a suitable axis to cover this range of values.
  3. Drag the histogram bars to give a suitable distribution.
  4. Click Check.

Click Another Question and repeat until you can sketch a reasonable distribution consistently.


The following exercise is similar, but you should sketch a box plot of the data.

Again, select a suitable axis to cover a distribution with the mean and standard deviation in the question. Then drag the five vertical lines in the box plot to give an approximate distribution and click Check.

Repeat until you master the technique.