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.
