How to Find the Averages and Range From Grouped Data
Finding the mean, median, mode and range from grouped frequency tables
We have seen in a previous article how to find the mean, median, mode and range of a group of data when that data was presented in frequency tables, but what about when we have grouped data?
When collecting information over a large range such as the time taken to complete a task or the age of people visiting a museum, it generally makes sense to record our data in groups. For example, when collecting data on age, instead of tallying all of the one-year-olds in one row, two-year-olds in the next row, etc., we may choose to tally in groups of a set number of years, e.g., 0 ≤ age < 10, 10 ≤ age < 20 and so on. This makes it easier to record and a lot easier to analyse afterward. Imagine drawing a pie chart of this data if we had a separate slice for each age; it's not a good idea.
However, this leads us to a potential problem. By recording our data in groups, we lose a degree of accuracy that would be present if we tallied each item precisely. How do we find the mean if we don't know the exact figure for each item?
In this case, we can find estimates for the averages and range.
When collecting information over a large range such as the time taken to complete a task or the age of people visiting a museum, it generally makes sense to record our data in groups. For example, when collecting data on age, instead of tallying all of the one-year-olds in one row, two-year-olds in the next row, etc., we may choose to tally in groups of a set number of years, e.g., 0 ≤ age < 10, 10 ≤ age < 20 and so on. This makes it easier to record and a lot easier to analyse afterward. Imagine drawing a pie chart of this data if we had a separate slice for each age; it's not a good idea.
However, this leads us to a potential problem. By recording our data in groups, we lose a degree of accuracy that would be present if we tallied each item precisely. How do we find the mean if we don't know the exact figure for each item?
In this case, we can find estimates for the averages and range.
Grouped frequency table showing the ages of people visiting a museum one morning
Age |
Frequency |
0 ≤ age < 20 |
17 |
20 ≤ age < 30 |
8 |
30 ≤ age < 40 |
6 |
40 ≤ age < 50 |
2 |
50 ≤ age < 60 |
14 |
60 ≤ age < 70 |
10 |
70 ≤ age < 80 |
4 |
80 ≤ age < 100 |
zero |
Total |
61 |
The first step: Finding the mid-point of each group
As we need exact values in order to find the mean, our first step is to assume that for each group, the values average out to the mid-point of that group. In our example above, the mid-point of the first group is halfway between 0 and 20, so 10. The mid-point of the group 20 ≤ age < 30 is 25 and so on.
If you are unsure of the mid-point, simply add the two end values together and then divide by 2.
We will now add this to our table.
If you are unsure of the mid-point, simply add the two end values together and then divide by 2.
We will now add this to our table.
Age |
Frequency |
Mid-point |
0 ≤ age < 20 |
17 |
10 |
20 ≤ age < 30 |
8 |
25 |
30 ≤ age < 40 |
6 |
35 |
40 ≤ age < 50 |
2 |
45 |
50 ≤ age < 60 |
14 |
55 |
60 ≤ age < 70 |
10 |
65 |
70 ≤ age < 80 |
4 |
75 |
80 ≤ age < 100 |
zero |
90 |
Total |
61 |
- |
Finding an estimate for the mean from a grouped frequency table
To find the mean, we now use the same method as for an un-grouped frequency table but use out mid-points. So, for each row, we multiply the mid-point by the frequency. We then add these new values together before dividing by the total frequency to give us an estimate for the mean.
Age |
Frequency |
Mid-point |
Mid-point x frequency |
0 ≤ age < 20 |
17 |
10 |
170 |
20 ≤ age < 30 |
8 |
25 |
200 |
30 ≤ age < 40 |
6 |
35 |
210 |
40 ≤ age < 50 |
2 |
45 |
90 |
50 ≤ age < 60 |
14 |
55 |
770 |
60 ≤ age < 70 |
10 |
65 |
650 |
70 ≤ age < 80 |
4 |
75 |
300 |
80 ≤ age < 100 |
zero |
90 |
zero |
Total |
61 |
- |
2390 |
Our estimate for the mean is 2390 ÷ 61 = 39.2 (to 1 d.p.)
Finding the modal group
When dealing with data in a frequency table, we can't find an actual mode as we don't know the exact values. Instead, we look for the group with the highest frequency and call this the modal group.
In our example above, there are more people in the 0 ≤ age < 20 group than in any other group, so this is our modal group.
In our example above, there are more people in the 0 ≤ age < 20 group than in any other group, so this is our modal group.
Finding the median from a grouped frequency table
Finding the median can be approached in two different ways when dealing with grouped data. We are just going to look at the simplest method here, which is to find the group that contains the middle value and state this. This method is perfectly good for comparing data sets and is all that is required at the GCSE level, for example.
To find the middle value, we add one to the total frequency and divide it by 2. In our example above, this gives us (61 + 1) ÷ 2 = 31, and so the median is the 31st value.
By adding a cumulative frequency column to our table, we can see that this is the last value in the 30 ≤ age < 40 group. We now say that the median is contained in this group.
To find the middle value, we add one to the total frequency and divide it by 2. In our example above, this gives us (61 + 1) ÷ 2 = 31, and so the median is the 31st value.
By adding a cumulative frequency column to our table, we can see that this is the last value in the 30 ≤ age < 40 group. We now say that the median is contained in this group.
Age |
Frequency |
Cumulative frequency |
0 ≤ age < 20 |
17 |
17 |
20 ≤ age < 30 |
8 |
25 |
30 ≤ age < 40 |
6 |
31 |
40 ≤ age < 50 |
2 |
33 |
50 ≤ age < 60 |
14 |
47 |
60 ≤ age < 70 |
10 |
57 |
70 ≤ age < 80 |
4 |
61 |
80 ≤ age < 100 |
zero |
61 |
Finding the range from a grouped frequency table
To find the range from a grouped frequency table, we look for the largest and smallest possible values. In our example, the largest possible value is the top of the oldest group which contains a value, 80, while the smallest possible value is the bottom of the youngest group and so is 0.
The range is therefore 80 − 0 = 80.
Note, that just like the mean, this is an estimate as we can't know for certain what the largest and smallest values actually are.
The range is therefore 80 − 0 = 80.
Note, that just like the mean, this is an estimate as we can't know for certain what the largest and smallest values actually are.
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