In this video, we’ll learn how to construct, read, and interpret grouped frequency tables for a given data set. We should already be familiar with a standard frequency table. So let’s consider why we might want to have a grouped frequency table.
Let’s consider that we have collected the information about the ages of children attending a birthday party. We could order this data and then put it into a frequency table. The frequency table has the benefit of allowing us to represent a data set much more simply. Recall that the frequency of a value is simply the number of times that value occurs. For example, we know that one child at the birthday party had an age of one and there were three children at the birthday party with an age of two. We can of course see this from the original data, whether ordered or unordered, but if we have a very large data set, then a frequency table is very helpful.
So let’s consider a different scenario. This time, let’s say that we need to record the ages of adults completing a survey. If we want to record this information in a frequency table, then we can already see that this table is going to be huge. The solution to this then is to group the data. That means instead of recording every individual age, for example, 18, 19, 20, and so on, we group the ages instead, for example, an age group from 18 to 27, one from 28 to 37, and so on. This allows us to still make statistical conclusions about the ages and any variable under study, but the data is much more manageable. This is what we would call a grouped frequency table, and that’s simply a frequency table with the data organized into smaller groups or classes.
However, there are some very important points to consider when creating a grouped frequency table. We must ensure that a grouped frequency table meets the following criteria. Firstly, classes should be exhaustive. That means that there should be a class for every data value in the set. We saw this in the previous example of the age-grouped table because we made sure that every single age had a class. As this was a survey for adults, then we didn’t need to include a class for under eighteens. But a very common mistake is to not include classes for very low or very high values. The second criteria is that classes should be mutually exclusive. That means there should be no overlapping data values between classes. Thirdly, classes should be continuous. There should be no gaps between classes.
In the example that we saw with the ages, this is an example of discrete data. We wouldn’t say that a person is 27 and three months. We might say that they are 27 or 28. But if the data is continuous, for example, if we’re measuring something like heights or weights, then we need to make sure that we include all the values in the data set. And before we look closer at how we can present the data, there’s one final criterion for a grouped frequency table, and that is that classes are usually of an equal width. Now, let’s see some of the different ways in which we might see the groups presented in a grouped frequency table.
Here are three different ways in which we may see the groups presented in a grouped frequency table. In a colloquial way, we might say that these groups deal with values between 10 and 20 and 20 and 30. As noted previously, this first example table would only work with discrete data where the values are integers, for example. This would not work however for continuous data because, for example, the value 19.5 would not be able to be allocated to a group. If we look at the second table, the class which is labeled with 10 dash really indicates values which are 10 or greater up to, but not including, 20, which is the lower boundary of the next class. This would have the same meaning in the third table. This would have values of 10 or greater but not including the value of 20.
We’ll now see some examples of grouped frequency tables. And in the first example, we’ll recap how to read a tally chart and how to calculate the frequencies in a grouped frequency table.
The tally chart shows the marks that students in a class received in a mathematics test. Using the information from the tally chart, complete the frequency table.
In this question, we need to complete the five missing values in the lower table. But first, let’s begin by looking at the information which is presented in the first table. We can see from the groupings here that this is a grouped frequency table. The value of sets represents the marks that the students in a class received in the test. The first group in the table, five hyphen, represents values or marks which are five or greater up to, but not including, 10 marks. What we need to do is to convert the tally in the first table into a frequency.
Recall that a group of four lines crossed through with a fifth represents five. In the group five hyphen, we have a tally mark of five and one of four, so together that gives us a frequency of nine. The second group, 10 dash, has a five, a five, and a four, which gives us a total frequency for that group of 14. The third group has a frequency of 12, the fourth group has a frequency of 16, and the fifth group has a frequency of six. We can therefore list the five values as nine, 14, 12, 16, and six.
In this example, we were given the tally chart data and calculated the frequency. However, in the next example, we will need to carry out the tallying.
Using the data given for number of absences, complete the frequency table.
In this question, we need to take the data values indicated in the top table and fill these in to the grouped frequency table below. If we look at the first group in this frequency table, one to two, that means the number of values which are one or two. In other words, it will be the number of people who had either one day absent or two days absent. The most efficient way in which we can determine these frequencies is by first creating a tally. We can extend the table to create a tally row. However, if we were creating this question in our books, an ideal place for this row is in between the number of absences row and the frequency row. It’s always good practice to take the data values in order and assign a tally mark to the correct group rather than trying to count the number of ones and twos, for example, within the data set.
Beginning with the data set then, the first value is seven. So that goes in the group seven to eight. The next value is 10, and that goes in the group nine to 10. Crossing through the first row of data, we get the following tallies. We can then complete the tally as shown, remembering that a group of four tally marks with a diagonal line through them indicates five. We then simply need to convert the tallies into frequencies. The first frequency would be six, since that’s made up of a tally of five plus one. The second group has a frequency of three; the third group, a frequency of five; the fourth group, a frequency of 10; and the fifth group, a frequency of eight.
It’s worth noting that although we don’t need to calculate the total frequency, it can be a good check of our working. The total frequency can be obtained by adding the given frequencies. In this case, this would give us an answer of 32. Looking at the original data values, we know that there were eight columns by four rows, which would give us a total number of 32 data values. It’s always worthwhile checking this just in case we have accidentally missed any data values. We can therefore give the answer that the five missing frequencies are six, three, five, 10, and eight.
In the next example, we’ll see how we can interpret the information given in a grouped frequency table.
The table shows the marks of 100 students in a recent science exam. How many students received 40 marks or more?
If we look at the given table, we can see that the frequencies indicate the number of students who achieved particular marks in a science exam. The first frequency of six indicates that six students got a mark between zero and nine. 15 students got a mark which was 10 up to and including 19 marks. We need to use this table to determine how many students received 40 marks or more. We can observe that in the class 40 to 49 marks, there were 23 students. However, it would be a mistake to think that the answer is 23 because we also need to include the next group in the table. Everyone in the group 40 to 49 and any higher group would have achieved 40 marks or more. Therefore, to find the total students who received 40 marks or more, we add 23 and 18, which gives us an answer of 41 students.
So far in this video, we have seen examples of how we can create a frequency table and an example of how we can interpret a frequency table. In the next example, we’re going to do both. We’ll construct a grouped frequency table, taking into account the information that we need to obtain from it.
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.
The data values in this table represent the number of days of absence. For example, somebody took 16 days off work, another person took 18 days off work, another had five, and so on. If we consider the range of data in the table, the smallest value is five and the highest value is 30. So if we were to create a frequency table to represent every individual data value from five to 30, the table would be very large. So we are going to create a grouped frequency table with classes that group the data values together.
One way in which we can do this is to create classes which have 10 days of absences and then create a tally chart. We need to be careful however and consider what we’re asked to calculate. We need to work out the absences of 20 days or more. Therefore, we don’t want the value of 20 to appear inside of a group boundary. So one of the groups must begin with 20 days’ absence. There are also some important considerations when creating a grouped frequency table. Firstly, the classes should be exhaustive. That means there should be a class for every data value in the set. Secondly, they should be mutually exclusive, which means there are no overlapping data values between classes. And thirdly, classes should be continuous. There should be no gaps between classes.
Therefore, using the groups zero to nine, 10 to 19, 20 to 29, and 30 to 39 should fulfill these criteria. Notice however that if we were using this frequency table for a different data set of absences, then we may need to make the upper class have an open interval. Doing so would take account of any values which are higher than 39. But we can now turn our attention towards working out the tally for each group. The first value in the table is 16, and 16 would fall in the class 10 to 19. The next value of 18 also falls within this class. We can then continue by creating tallies for all the data values. Once we’ve done that, we can then determine the frequencies from each of the tallies. The first group has a tally mark of five plus one, which is six. We then have frequencies of 15, 17, and two.
Notice that we can confirm that the total frequency is also 40 by adding the frequencies. And we were given that this is the data for 40 workers. This is always a good check to make sure that we haven’t missed out any data values. We can now use the frequency table to calculate the number of workers who took 20 or more days off work. This will be the total frequency of classes with 20 or more days absent. Adding 17 and two gives us 19. Therefore, by using this frequency table, we found that there were 19 workers who took 20 or more days off work.
However, before we finish with this question, there are a few points to note. This type of approach to a grouped frequency table is really useful if we know that there are several problems we need to answer involving the data. But if it’s just this question we need to answer, then we could create a slightly different grouped frequency table. This grouped frequency table could just have two groups, a group for zero to 19 days absent and one of 20 or more days absent. And therefore, once we have created the tally, then the frequency of the group 20 or more would give us the required answer. Hence, either method would demonstrate that the number of workers who took 20 or more days off work is 19.
We can now summarize the key points of this video. A grouped frequency table is a frequency table with data organized into smaller groups or classes. Grouped frequency tables allow for easier analysis of large data sets and those with a wide range of values. We must ensure that the classes in a grouped frequency table are exhaustive, mutually exclusive, and continuous. Finally, we saw that there are different ways in which class groups may be defined. Discrete data can be written as 10 to 19, 20 to 29, and so forth. Discrete and continuous data can be represented by groups such as 10 dash, 20 dash, and so on or by groups such as 10 is less than or equal to 𝑥 is less than 20 for a variable 𝑥.