EXAMPLE 2.10.6

Euclid is soliciting donations to fund his spring break trip to Cancun. More than half of his donations have been in the amount of 50¢. The others have been equally divided among values of 25¢ and values of 75¢. Select the statement that correctly gives a relationship between measures of central tendency for this distribution.

A. The median is greater than the mean.

B. The mean is less than the mode.

C. The mean is equal to the median.

D. The mode is less than then median.

SOLUTION

Although we don't know exactly how many donations have been collected, we have enough information to draw a useful bar graph summarizing the distribution. The bar graph will show three values: 25, 50 and 75. Since half of the donations are in the amount of 50¢, the column above the value 50 will be as tall as the other two columns combined. Moreover, since there were exactly as many 25¢ donations as there were 75¢ donations, the column above the value 25 will be the same height as the column above the value 75. The bar graph looks like this:

Recall that a distribution such as this one, which is symmetric with respect to the mode, is called approximately normal.

In such a distribution, things "balance out" so that the mean, median and mode are all equal.

We see that choice C is correct.