# Lesson Video: The Mean of a Data Set Mathematics

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

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### Video Transcript

In this video, we will learn how to calculate the arithmetic mean of a data set. Data sets can often 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. 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. This arithmetic mean can be calculated using the following formula. It is equal to the sum of all values divided by the number of values. Informally, this is referred to as adding them all up and dividing by how many there are.

Once there are other types of mean which are calculated differently, for example, the geometric mean, when we see the abbreviation “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.

What is the mean of the following numbers six, four, five, three, five, and one?

We begin by recalling that we can calculate the mean of a data set by dividing the sum of all the values by the number of values. In this question, we have six integers, so the number of values is six. We can find the sum of the values by adding six, four, five, three, five, and one. This is equal to 24. The mean of the set of numbers is therefore equal to 24 divided by six. And this is equal to four.

At this stage, it is worth noting that even if the data themselves must be integer values, the mean does not need to be an integer. It is therefore important that we round to a suitable degree of accuracy if necessary. This is one of the key steps when summarizing the process of calculating the mean. Our first step is to determine how many values there are in the data set. Next, we find the sum of all the data values. We then divide the sum of the data values by the number of data values. And finally, we round to a suitable degree of accuracy if necessary.

We will now consider two examples where we need to follow this process and the data set has been presented in different formats. We begin with the data presented in a bar chart.

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.

The bar graph shows the number of points that each one of four houses scored at the end of the year. The blue house scored 30 points. The orange house scored 70 points. The pink house scored 50 points. And the green house scored 90 points. The first part of this question asked us to calculate the mean number of points earned. We recall that the mean of a data set can be calculated by finding the sum of all the values and dividing it by the number of values. In this question, since there are four houses, the denominator will be four. We can find the sum of all the values, that is, the total number of points scored, by adding 30, 70, 50, and 90. This is equal to 240, so this is the numerator of our fraction.

We can then calculate the mean by dividing 240 by four. This is the same as finding a quarter of 240, which is equal to 60. The mean number of points earned by each house is 60.

The second part of this question asked us to describe what this mean represents. We recall that the mean is a measure of central tendency. In particular, it is the average. And as such, in this question, it is the number of points that each house would have earned if they had all earned the same number of points as multiplying the mean number of points of 60 by the number of houses four gives us the total of 240 points that were earned.

In our next example, we are given information in a table and we need to extract the relevant information in order to calculate the mean.

The table shows the marks that four students received in their end-of-year exams. Calculate the mean mark of student (C).

In this question, whilst we’re given information about four students, we are only interested in student (C). This means that we only need to focus on that row of the table. We are asked to calculate the mean mark of this student. And we recall that the mean of a data set is equal to the sum of all the values divided by the number of values. All the students had marks for five exams, mathematics, chemistry, physics, biology, and history. Therefore, the denominator will be equal to five. We can find the sum of all the values by adding eight, 13, seven, five, and 12. This is equal to 45. So the total number of marks that student (C) achieved is 45. We can now calculate the mean mark by dividing 45 by five. This is equal to nine. So the mean mark achieved by student (C) is nine marks.

Whilst it is not required in this question, we could repeat this process for students (A), (B), and (D) in order to compare their average marks. This would be useful information for a teacher when comparing the overall progress of their students.

In our next example, we are given the mean of a data set and all but one of the data values. And we will then work backwards from knowing the mean to calculate the missing value. In order to do this, we will need to form and then solve an equation using the formula for calculating the arithmetic mean.

Given that the mean of the values four, 𝑥, five, eight, and 18 is 10, find the value of 𝑥.

In order to answer this question, we begin by recalling that we can calculate the mean of a data set by dividing the sum of all the values by the number of values. In this question, there are five values in our data set: four, 𝑥, five, eight, and 18. We are also told that the mean of the data set is equal to 10. We can begin by writing an expression for the sum of all the values as shown. It is equal to four plus 𝑥 plus five plus eight plus 18. We know that dividing this by five, the number of values, gives us an answer of 10.

In order to solve this equation for 𝑥, we begin by multiplying both sides by five. This gives us 50 is equal to four plus 𝑥 plus five plus eight plus 18. Collecting like terms on the right-hand side, our equation simplifies to 50 is equal to 𝑥 plus 35. We can then subtract 35 from both sides such that 𝑥 is equal to 15. If the mean of the five values four, 𝑥, five, eight, and 18 is 10, then the value of 𝑥 is 15. By forming and then solving an equation using the formula for calculating the arithmetic mean, we have been able to find the missing value in the data set.

We will now move on to one final example where the data set has already been partially summarized. We are asked to calculate the mean height for a group of school children given the sample means for the height in each grade.

David took samples of students from each grade in his school to investigate the average height of a student. The results are summarized as shown. Use his data to find the mean height of a student in the school. Give your answer to the nearest centimeter.

We begin by recalling that the mean can be calculated by using the following formula. We divide the sum of all the values by the number of values. In this question, in order to find the mean height, we will divide the total height of all the students by the number of students. In order to work out the number of students, we find the sum of the numbers in the second row of our table. There were 14 students in kindergarten, 16 in grade 1, 15 in grade 2, and so on. Adding 14, 16, 15, 19, 17, and 18 gives us a total of 99. There are 99 students represented in the table.

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. After adding an extra row to our table, we can find the combined height of the students in grade K, or kindergarten, by multiplying 14 by 116 centimeters. This is equal to 1,624 centimeters. Repeating this for grade 1, we multiply the number of students, 16, by the mean height of 122 centimeters. This is equal to 1,952 centimeters. In grade 2, the combined height of the 15 students is 1,905 centimeters. In grades 3, 4, and 5, we get values of 2,508, 2,329, and 2,574 centimeters, respectively.

We can now find the total height of all the students by finding the sum of the six values in the bottom row. This is equal to 12,892 centimeters. We are now in a position to calculate the mean height of a student in the school by dividing 12,892 by 99. This is equal to 130.2 recurring. And as we are asked to give our answer to the nearest centimeter, the mean height of a student in the school is 130 centimeters.

We will now finish this video by recapping the key points. We saw in this video that the arithmetic mean of a data set gives a measure of the center of the data. We can calculate this mean using the formula arithmetic mean is equal to the sum of all the values divided by the number of values. We saw that the mean can be found for data presented in a table or a chart. If we know the mean of a data set and all but one of the values, we can work backwards to find the missing value by forming and solving an equation. We do this using the formula for the arithmetic mean.

Finally, in our last example, we saw that we can calculate the overall mean of a data set given individual group means. We do this by calculating 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.