Video: Grouped Frequency Tables

In this video, we will learn how to construct, read, and interpret frequency tables for a given quantative data set.

10:23

Video Transcript

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 test.

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 frequency table.

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 eight.

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 total.

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 work.

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.

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