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.
Formula: The Arithmetic Mean
The arithmetic mean of a data set is calculated using the formula
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.
Example 1: Finding the Mean of a Simple Data Set
What is the mean of the following numbers: ?
The mean of a set of data is calculated using the following formula:
The given list contains six values. The mean of this data set is therefore equal to
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.
How To: Calculating the Mean of a Data Set
- Determine how many values there are in the data set.
- Find the sum of all the data values.
- Divide the sum of the data values by the number of data values.
- Round to a suitable degree of accuracy if necessary.
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.
- Calculate the mean number of points earned.
- Describe what the mean represents.
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:
The mean is equal to the sum of these values divided by the number of houses (4):
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.
Example 3: Interpreting a Table and Calculating a Mean
The table shows the grades that four students received in their end-of-year exams. Calculate the mean grade of student (C).
We need to calculate the mean grade for student C across their five examination subjects. We can ignore the other rows of the table and just consider the row for student C:
To calculate their mean grade, we need to find the total of their grades across all subjects and divide this by the number of examinations they took (5):
This tells us that if the student had performed equally well in each subject, they would have scored 9 points in each examination.
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.
Example 4: Finding a Missing Value in a Data Set given the Arithmetic Mean
Given that the mean of the values , and 18 is 10, find the value of .
We know that the mean of a set of data is calculated by finding the sum of the values and dividing by how many values there are:
In this data set, there are five values, four of which are known and one that we wish to calculate. We also know that the mean of all five values is 10, so we can form an equation. We substitute 10 for the mean and 5 for the number of values to give
By multiplying both sides of this equation by 5, we find that the sum of all the values is 50. We can find an expression involving the unknown for the sum of the values:
By setting this expression equal to 50, we have an equation in :
To find the value of , we subtract 35 from each side of the equation, giving .
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.
Example 5: Calculating an Overall Mean given Individual Means of Different Groups
Maged took samples of students from each grade in his school to investigate the average height of a student. The results are summarized as shown:
|Number of Students||14||16||15||19||17||18|
|Mean Height||116 cm||122 cm||127 cm||132 cm||137 cm||143 cm|
Use his data to find the mean height of a student in the school. Give your answer to the nearest centimetre.
We recall first that the mean is calculated using the formula
To find the mean height for the entire group of students, we need to divide the total height of all the students by the total number of students.
The total number of students can be found by summing the values in the second row of the table, which are often called the frequencies:
The combined height of the students in each grade can be found using the number of students in each grade and the mean height for that grade. For example, in grade K, there are fourteen students and their mean height is 116 cm. Hence, we know that
The combined height of students in grade K can be found by multiplying both sides of this equation by 14:
In the same way, the combined height of the students in each of the other grades can be found by multiplying the number of students in that grade by the mean height of the students in that grade:
= 1 624 cm
= 1 952 cm
= 1 905 cm
= 2 508 cm
= 2 329 cm
= 2 574 cm
The total height of all the students is the sum of the combined heights for the children in each grade:
The overall mean height is found by dividing this by the total number of students:
Let us finish by recapping some key points.
- The arithmetic mean of a data set gives a measure of the center of the data.
- The mean is calculated using the formula
- The mean of a data set can be found for data presented in a table or chart by first identifying the individual data values and how many values there are.
- If we know the mean of a data set and all but one of the values, we can work backward to find the missing value by forming and solving an equation using the formula for the arithmetic mean.
- To calculate the overall mean of a data set given individual group means, we need to calculate the total of all the data values by first finding the total for each group. We then divide by the total number of items, which is the sum of the number of items in each group.