Lesson Explainer: The Mean of a Data Set Mathematics • First Year of Preparatory School

In this explainer, we will learn how to calculate the arithmetic mean of a data set.

Often, data sets can be quite large, and it is useful to be able to summarize the data with a single value or a smaller set of values. Such values, which represent the data set, give an indication of the typical or average value. They reveal the approximate position of the center of the data set and also enable us to make comparisons with other sets of data. The arithmetic mean is one of a number of such measures, which are collectively called measures of central tendency because they describe the center of a data set.

Informally, we might think of the process of calculating the arithmetic mean as “adding them all up and dividing by how many there are.” For example, suppose there are three children. One child has 10 pieces of candy, another child has 5 pieces of candy, and the third child has 3 pieces of candy. The total number of pieces of candy is

The mean number of pieces of candy is .

The arithmetic mean represents the number of pieces of candy that each child would get if the total number of pieces of candy was shared equally between them all so that each child had the same amount.

There are other types of mean that are calculated differently, for example, the geometric mean. When we see the term “the mean,” we can assume it is the arithmetic mean we are referring to. In our first example, we will calculate the mean of a set of six values.

Even if the data themselves must be integer values, the mean does not need to be an integer, and sometimes, rounding the calculated value to the nearest integer causes a lot of information to be lost. For example, consider two groups of people. In one group, the mean number of children is calculated to be 1.6. In the other group, the mean number of children is calculated to be 2.4. If these values were rounded to the nearest integer, they would both be rounded to 2, and so it would appear that the mean number of children for the two groups is the same. However, there is actually quite a large difference between the values of 1.6 and 2.4 relative to the size of the numbers themselves, and in truth, the mean number of children is quite a bit lower for the first group of people. Even though the number of children must itself be a whole number, the arithmetic mean does not need to be, unless the question specifies that the answer should be rounded in this way.

We summarize the process of calculating the mean of a data set in the following steps.

Sometimes, the data for which we want to calculate the mean will be presented in other formats, such as a table or some kind of graph. We will now consider two examples in which we calculate the mean of data sets that have been presented in different formats, beginning with data presented in a bar chart.

Example 2: Finding the Mean of Data Presented in a Bar Chart and Interpreting Its Meaning

At a school, every student belongs to one of four houses, and the houses compete throughout the year to earn points.

The graph shows how many points each house had scored at the end of the year.

1. Calculate the mean number of points earned.
2. Describe what the mean represents.

Answer

Part 1

We recall that the mean of a data set is found using the formula

To calculate the mean for this data set, we first need to read the number of points earned by each house from the bar chart. The number of points earned by each house is given by the vertical height of the bar and is as follows:

Blue	Orange	Pink	Green
30	70	50	90

The mean is equal to the sum of these values divided by the number of houses (4):

Part 2

In Part 1, we found that the total number of points scored by the four houses together was 240. To find the mean number of points scored, we then divided this by the number of houses. Because we shared the total number of points equally between all the houses, the mean represents the number of points that each house would have earned if they had all earned the same number of points.

When analyzing data sets, we need to be able to identify the relevant information from a larger data set or table. In our next example, we will be given a table consisting of the examination grades for four students across five different subjects. We will be required to calculate the mean examination grade for one student and will need to extract the relevant information from the table in order to do this.

Sometimes, we may be given the mean of a data set and all but one of the data values. In this instance, we can work backward from knowing the mean to calculate the missing value, as we will see in our next example. This will require us to form and then solve an equation using the formula for calculating the arithmetic mean.

Now, suppose that a data set has already been partially summarized for us. For instance, we may be given the means of some groups within the data set and the number of items within each group. We will now consider how we can find the overall mean for the data set given this information.

Suppose we want to calculate the mean height of a group of people at a party. There are three adults at the party whose mean height is 168 cm, and there are twenty-four children at the party whose mean height is 115 cm.

Can we then conclude that the mean height of everybody at the party is ?

We know that the mean height of everybody at the party should be calculated by dividing the total height of everybody at the party by the total number of people at the party. The total number of people is . The total height can be found by considering the total, or combined, height for each group.

For the adults, there are three of them and their mean height is 168 cm. It follows that

The combined height of the adults can be found by multiplying both sides of this equation by 3:

The mean height of the twenty-four children at the party is 115 cm, and so it follows that

The combined height of the children can be found by multiplying both sides of this equation by 24:

The total height of all the people at the party is the sum of the combined height of the adults and the combined height of the children:

The mean height is found by dividing this total height by the total number of people at the party (27):

This value is significantly different from the value of 141.5 cm found by simply calculating the mean of the means for the two groups. This is because there are different numbers of people in the two groups. The mean height for the adults is much larger than the mean height for the children, but there are only three adults at the party. If we think of the mean as sharing the total height equally over the group, then the adults’ “extra” height needs to be shared between all twenty-four children. This leads to quite a small increase in the overall mean compared with the mean height for just the children.

Let us now consider an example in which we calculate the mean height for a group of school children, given the sample means for the height in each grade.

Let us finish by recapping some key points.