In this video, we will learn how to
construct, read, and interpret grouped frequency tables. We’ll begin by looking at the word
frequency and looking at a standard frequency table.
Let’s imagine that a student is
carrying out a survey in their class to find out how many books had read in the last
month. They would ask their classmates the
question, and they could record the results in the table below. The word frequency means the number
of times a value occurred.
If the student obtained the
following results, the frequency five under zero books would mean that five students
had each read zero books or no books in the last month. Eight students would have read one
book each in the last month. Four students would have read two
books. Two students would have read three
books. And two students would have read
four or more books. We don’t know exactly how many
books. So it could be four and four, for
example, or four and six or even 10 and 15 books. We’ll now look at how this method
of recording frequency works when the data is grouped.
Let’s imagine that a teacher is
looking at the marks of a class test. The teacher has very lightly
recorded the class scores. But they want to do something more
useful with the information of the scores. For example, they might want to see
how many students got 50 marks or more. In which case, a grouped frequency
table would be an ideal way to show this information.
Looking at the table, we could see
that a frequency in the score 20 to 29 means that two people scored between 20 and
29 in the test. We don’t know exactly what these
values are from looking at the table. For example, it could be a 21 score
and a score of 23 or even two scores of 26. But, here, it doesn’t really matter
what the individual scores are. After all, if the teacher wanted to
know individual scores, they’d simply look at the list.
We can see that grouped frequency
tables are really good for showing an overall pattern in any data. For example, we could see that most
students scored between 60 and 69 in this test. Grouped frequency tables are also
very good for continuous data. That’s data that can be
measured. For example, measuring the height
and weight of people. We’ll now look at a few different
questions on grouped frequency tables and we’ll begin by interpreting a table.
The frequency table below shows
the weights of 40 students in a class. How many students weigh less
than 50 kilograms?
We can see in this table that
we have the weights of the students along with the frequency. Here, the word frequency will
mean how many students there are in each weight category. If we look at the categories,
we can see that 30 dash will indicate a weight of 30 kilograms up to, but not
including, 35. This means that a student
weighing 35 kilograms would be put into the second category. In this category, we would have
any student weighing 35 kilograms up to, but not including, 40 kilograms, since
that would be in the third category.
So if we’re looking to find how
many students that weigh less than 50 kilograms, that would be all the people in
the first four categories. Since we know that everyone in
this final category would weigh 50 kilograms or more. And therefore, we add our four
values, five, eight, 12, and nine, which gives us an answer of 34 students.
We’ll now look at a question where
we find the missing value in a grouped frequency table.
Complete the frequency table
which shows the marks a group of students received in a test.
Let’s begin by looking at this
table. We can see, for example, that
the frequency of 15 in the marks category 30 to 34 means that 15 students
received between 30 and 34 in the test. Equally, the frequency of 10
here means that 10 students received between 40 and 44 in the test. We can also see that there is a
total of 50, which means that there must have been 50 students that sat the
And we can use this fact to
help us work out the missing value in the category 45 to 49. If we add together all the
remaining values, we would have four plus five plus 15 plus nine plus 10 plus
four, which is equal to 47. So 47 students received the
other marks. And then 50 subtract 47 will
give us the number of students who got between 45 and 49, which is equal to
three. And so, we’ve completed the
Using the data given for the
number of absences, complete the frequency table.
We can see that this table is
made up of, in fact, two sections. The top half represents all the
absences per student. For example, one student had
seven absences and another had 10 absences. We need to fill this data into
the grouped frequency table at the bottom of this table. In the first column of this
lower part of the table, we’ll have students that have either one or two
absences, in the second, three or four absences, and so on. Perhaps the quickest way to do
this is to create a tally column and go through each piece of data in turn
filling it in to the table.
Beginning with seven, we can
fill that into our table. Next, we have 10 absences for
one student and then another seven absences. And we can continue crossing
off and filling in the table as we go. Once we have completed our
tallies, we can then fill in the frequency column. So we’d have six for the first
value, three for the second value, and so on. And therefore, we have
completed our frequency table with the values six, three, five, 10, and
A good check of our answer at
this point is to check our frequency row and add up the values. Here we can see that these
would add to give us a total frequency of 32. And as we had eight values in
each row and four in each column, that means we must’ve had 32 values in
In the next question, we’ll look at
constructing our own grouped frequency table. And we’ll pay close attention to
the groups that we use.
The table shows the number of
days taken off work by 40 workers in a year. By constructing a frequency
table or otherwise, calculate the number of workers who took 20 or more days off
We could look at the table and
see, for example, that the first value means that a worker took 16 days off. The second value means a worker
took 18 days off, and so on. We’re asked to calculate the
number of workers who took 20 or more days off work. We’re asked to do this by using
a frequency table or otherwise. But let’s start by looking at
our frequency table and then think about how we could’ve done it in an
alternative way. So let’s set up a table with a
row for the days absent and a row for the frequency. The word frequency here will
refer to the number of workers who took those number of days absent.
Now, let’s think about the
headings or groups that we might have in this frequency table. As we’re asked to work out the
number who had 20 or more days off work, we could, in theory, just have two
categories, less than 20 days absent or 20 or more days absent. But let’s assume we want to
create a useful frequency table other than just for answering this question
about the 20 or more days off work. We can think about grouping the
days absent into groups of 10.
If we started with our
groupings of zero to 10 and 10 to 20, we might then have a potential
problem. If we had a worker who had 10
days absent, we wouldn’t know which group that that should go into. Therefore, we need to have
groups that don’t have overlapping values. We can continue to create
groups. And as it looks like our
highest value is in the thirties, then 30 to 39 can be our highest group. We can now go through the table
and put each individual value into the appropriate grouping. It can be helpful to add in a
tally row to help us.
Beginning with our first value
of 16, that would fall in the grouping 10 to 19. Next, we’d have a value of 18
which also falls into the same group. And we can continue adding our
values to the groups. When we have completed the
tallies, we can then fill in the frequency values. When we have done this, a good
check at this point is to work out the total frequency. Six plus 15 plus 17 plus two
would give us the value 40. We can see that there are 40
values in the table above. And we were also told that
there were 40 workers.
So now, to calculate the number
of workers who took 20 or more days off work, we can see that this would be in
the group 20 to 29 and in the group 30 to 39. If we add together 17 and two,
we get that there would be 19 workers who took 20 or more days off work.
Returning to the question of
constructing a frequency table or otherwise, how else could we have answered
this question? If we look at our original
data, we could simply count the number of workers who took 20 or more days off
work. In which case, we would find
that there are 19 values, which confirms our original answer that 19 workers
took 20 or more days off work.
Now, let’s summarize what we’ve
learnt in this video. We recalled that the word frequency
means the number of times a value occurs. We saw that a grouped frequency
table is used for displaying data in groups. The values are not displayed
individually. But, instead, the range of values
is split into groups. And we record how many data points
are in each group. And finally, we saw how we need to
be careful when creating a grouped frequency table to make sure that the end points
of the groups don’t overlap.