EXAMPLE 2.10.5

The graph below shows the distribution of a group of Gator fans according the number junk cars in their back yards. Select the statement that correctly gives a relationship between measures of central tendency for this distribution.

A. The mean is the same as the median.

B. The mean is less than the mode.

C. The mode is less than the mean.

D. The median is less than the mean.

SOLUTION

Recall that this is an example of data skewed to the left.

It is easy to identify the mode: the mode is 4, since 4 is the value under the tallest column.

Notice also that 4 is greatest number in this distribution. This indicates that the mean and the median should both be less than 4 (because in a collection of numbers that are not all identical the average number must be larger than the smallest number and smaller than the largest number).

The following statements are all reasonable:

The mean is less than the mode.

The mode is greater than the mean.

The median is less than the mode.

The mode is greater than the median.

We see that choice B is correct.

Also, recalling the fact that in a skewed distribution the median tends to fall between the mean and the mode, the following statements would be reasonable:

The median is greater than the mean.

The mean is less than the median.

Finally, in the special case where the height of the column above the mode was greater than the combined heights of all of the other columns, the median would be equal to the mode.