Mean when data set is presented in a frequency table

The following table gives the number of children in each of 20 families. Calculate the mean number of children per family.

Example 2

 Number of children Frequency 0 1 2 3 4 5 7 4

Solution

There are 4 families with 0 children. This gives a total of 0 children.
There are 5 families with 1 child. This give a total of 1 × 5 = 5 children.
Continuing in this way, we have:

\begin{align}\text{Mean}\ &=\ \dfrac{0 \times 4 + 1 \times 5 + 2 \times 7 + 3 \times 4}{4 + 5 + 7 + 4}\\\\ &=\ \dfrac{31}{20}\\\\ &=\ 1.55\end{align}

It is obviously impossible for a family to have 1.55 children, though this figure can be used by governments and other institutions for planning purposes. In general, the mean or average of a data set is not one of the original values.

Mean when data is presented in a stem-and-leaf plot

Example 3

The stem-and-leaf plot below gives the marks of 12 students in a test.

 1 8 9 2 2 4 5 6 3 1 4 9 4 2 3 6 4$$\mid$$3 means 43

Calculate the mean of the marks.

Solution

\begin{align}\text{Mean}\ &=\ \dfrac{18 + 19 + 22 + 24 + 25 + 26 + 31 + 34 + 39 + 42 + 43 + 46}{12}\\\\ &=\ \dfrac{369}{12}\\\\ &=\ 30.75\end{align}

Mean when data is presented in a column graph

Example 4

The marks obtained for a quiz by a group of students are displayed in the column graph.
Find the mean of the marks.

Solution

\begin{align}\text{Total number of students} &=\ 2 + 6 + 6 + 4 + 4 + 2\\\\ &=\ 24\\\\ \text{Mean} &=\ \dfrac{2 × 4 + 6 × 5 + 6 × 6 + 4 × 7 + 4 × 8 + 2 × 10}{24}\\\\ &=\ \dfrac{154}{24}\\\\ &=\ 6.417\ \text{correct to 3 decimal places}\end{align}