How to Find the Average From a Frequency Table
A frequency table showing pet ownership
Finding the mean, median, mode and range from frequency tables
When people initially learn how to find the mean, median, mode and range of a group of data, we always start with data presented as a list of numbers e.g. 5, 7, 2, 4, 9, 5, 6
When dealing with large groups of data, however, it is generally much more useful to present that data in a frequency table as it can then be read and analysed much more quickly than a long list of digits can.
To find the three averages and the range from a frequency table uses exactly the same ideas as from before; we just need to adapt the method to fit the new form of presentation.
When dealing with large groups of data, however, it is generally much more useful to present that data in a frequency table as it can then be read and analysed much more quickly than a long list of digits can.
To find the three averages and the range from a frequency table uses exactly the same ideas as from before; we just need to adapt the method to fit the new form of presentation.
Finding the mean from a frequency table
For this example, 31 members of a class were all asked how many siblings they have, and the results were recorded in the frequency table shown below.
Number of siblings |
Frequency |
Zero |
7 |
1 |
11 |
2 |
9 |
3 |
3 |
4 |
1 |
If we think about the usual method of finding the mean, we add all of the values together and then divide by the total number of values. We will be using the same method here. The trick is how we get that information from the table.
When the data is in a table, we can quickly sum the values by working out how much each row is worth.
For example, we have 7 people, each with no siblings. This gives a total of 7 × 0 = 0 siblings. There are 11 people who each have one sibling, so 11 × 1 = 11 siblings.
By multiplying the number of siblings in the left-hand row by its frequency and then adding these together, we can quickly sum up the total number of siblings in the whole table.
To find the total number of values, we can see that there are 7 people in the first row, 11 people in the second and so on. Therefore, we add the numbers in the frequency column to give us our total.
Finally, dividing the first total by the second gives us our mean. Keeping it in the table, the calculation will look like this:
When the data is in a table, we can quickly sum the values by working out how much each row is worth.
For example, we have 7 people, each with no siblings. This gives a total of 7 × 0 = 0 siblings. There are 11 people who each have one sibling, so 11 × 1 = 11 siblings.
By multiplying the number of siblings in the left-hand row by its frequency and then adding these together, we can quickly sum up the total number of siblings in the whole table.
To find the total number of values, we can see that there are 7 people in the first row, 11 people in the second and so on. Therefore, we add the numbers in the frequency column to give us our total.
Finally, dividing the first total by the second gives us our mean. Keeping it in the table, the calculation will look like this:
Number of siblings |
Frequency |
Number of siblings x frequency |
Zero |
7 |
Zero |
1 |
11 |
11 |
2 |
9 |
18 |
3 |
3 |
9 |
4 |
1 |
4 |
Total |
31 |
42 |
This then gives us a mean of 42 ÷ 31 = 1.4 (to 1 d.p.)
It is a good idea at this point to check that the answer is a sensible one. 1.4 looks like a sensible answer as we can see it's pretty much in the middle of our data. If we got an answer of 5, for example, we could quickly see that that would not be possible; hence, we would go back and check our calculations.
It is a good idea at this point to check that the answer is a sensible one. 1.4 looks like a sensible answer as we can see it's pretty much in the middle of our data. If we got an answer of 5, for example, we could quickly see that that would not be possible; hence, we would go back and check our calculations.
Finding the median from a frequency table
Just like with the mean, finding the median from a frequency table requires only a small adaptation from the usual method. When finding the median from a list of numbers, we find the middle number (or middle two if there are an even number of values) and this is our median.
When doing this from a table, we need to know where the middle is. To do this, we take our total frequency, add 1 and then divide by 2.
For our example, there are 31 values in the table; hence the middle value is (31 + 1) ÷ 2 = 16th.
Referring back to our table of values, we can see that the first 7 values are in the group of 0 siblings. As we want the 16th value, it is clear that the median is not in this group.
The second group has a further 11 values in it, hence must go up to value number 7 + 11 = 18. As the 16th value comes before the 18th, our median must lie in this group; hence, the median is 1.
When doing this from a table, we need to know where the middle is. To do this, we take our total frequency, add 1 and then divide by 2.
For our example, there are 31 values in the table; hence the middle value is (31 + 1) ÷ 2 = 16th.
Referring back to our table of values, we can see that the first 7 values are in the group of 0 siblings. As we want the 16th value, it is clear that the median is not in this group.
The second group has a further 11 values in it, hence must go up to value number 7 + 11 = 18. As the 16th value comes before the 18th, our median must lie in this group; hence, the median is 1.
Finding the mode from a frequency table
The mode is the easiest average to find from a frequency table as long as you are careful about its meaning. Remember that the mode is the most common value. In a frequency table, this means the value with the highest frequency, not the largest number in the table.
In our example, the highest frequency is the 11 people with 1 sibling each, so the mode is 1.
In our example, the highest frequency is the 11 people with 1 sibling each, so the mode is 1.
Example 2: Find the mean, median, mode and range from the frequency table
Score on an arithmetic test |
Frequency |
Score x frequency |
Zero |
1 |
Zero |
1 |
2 |
2 |
2 |
5 |
10 |
3 |
Zero |
Zero |
4 |
7 |
28 |
5 |
3 |
15 |
6 |
8 |
48 |
7 |
1 |
7 |
8 |
2 |
16 |
9 |
Zero |
Zero |
10 |
1 |
10 |
Total |
30 |
136 |
Mean
Sum of score × frequency = 136
Sum of frequencies = 30
Mean = 136 ÷ 30 = 4.5
Sum of frequencies = 30
Mean = 136 ÷ 30 = 4.5
Median
Middle value = (30 + 1) ÷ 2 = 15.5
So we need the 15th and 16th values.
Counting up the frequencies, we get to 1 + 2 + 5 + 0 + 7 = 15 on the row for score 4, hence the 15th value is 4. The 16th value must be the first number in the next row, hence 5.
The median then equals the average of these: (4 + 5) ÷ 2 = 4.5
So we need the 15th and 16th values.
Counting up the frequencies, we get to 1 + 2 + 5 + 0 + 7 = 15 on the row for score 4, hence the 15th value is 4. The 16th value must be the first number in the next row, hence 5.
The median then equals the average of these: (4 + 5) ÷ 2 = 4.5
Mode
The mode is the value with the highest frequency, which in this case is 6 (this has a frequency of 8, which is higher than any of the other frequencies).
Range
The highest test score is 10; the lowest is 0; therefore, the range = 10 − 0 = 10.
Summary
So there you have it, finding the three averages and the range from a frequency table just involves adapting the usual methods slightly to take into account the presentation of the data. We can see from the two examples that this is much quicker than rewriting all of the data into a list in order to use our old methods.
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