GCSE Maths: Averages From Frequency Tables
Introduction
In order to find averages from frequency tables you are generally shown a table with two columns. The first column will show you say goals scored and the second column will show you the frequency which is the same thing as the number of times something has happened.
Just as if you were given a list of numbers, you are still required to determine the mode, median and mean from a frequency table as well as a grouped frequency table.
These types of questions can appear on both the foundation and higher gcse maths papers.
Sometimes when there is a lot of information it is easier to put the data into a frequency table. Once this has been done it is then possible to find averages such as the median, mode, mean and also the range.
Finding Averages From Frequency Tables
Example:
A questionnaire was done to see how many people arrived in each car to the local town. The results are in the following table:
Number of people in each car | Frequency |
1 | 45 |
2 | 198 |
3 | 121 |
4 | 76 |
5 | 52 |
6 | 13 |
From the above table find:
- The mode
- The median
- The mean
Solution:
a) The mode is the most common so from the table you can see that 198 is the highest frequency so that means the modal number of people per car is 2.
b) The median is the value in the middle and we can use the rule \frac{n+1}{2} where n is the total number of, in this case, cars.
To find the total number of cars you must find the total of the frequency of column which is 45 + 198 +121 +76 +52 +13 = 505
So this means that the median is \frac{505+1}{2}=253 i.e. the 253rd term.
If you add the frequencies to see where the 253rd term is you will see that this is in the group where 3 people arrived in a car. So the median is 3 people.
c) In order to find the mean number of people in a car you need to create a new column which shows the result of the number of people in a car multiplied by the frequency.
Number of people in each car | Frequency | Total in cars = Number of people x Frequency |
1 | 45 | 1 x 45 = 45 |
2 | 198 | 2 x 198 = 396 |
3 | 121 | 3 x 121 = 363 |
4 | 76 | 4 x 76 = 304 |
5 | 52 | 5 x 52 = 260 |
6 | 13 | 6 x 13 = 78 |
Total | 505 | 1446 |
So the mean = \frac{\text { Total number of people incars }}{\text { Frequency }}=\frac{1446}{505}=\mathbf{2 . 8 6}
Finding Averages From Grouped Frequency Tables
There will be some instances when the data that you are given is grouped. If you need to find the mean then the answer will only be an estimate because you do not have the exact information.
In order to find the mean from grouped data it is important to take the midpoint of the intervals as the following example will show.
Example:
x | Frequency |
0 < p 1 | 2 |
1 < p 2 | 5 |
2 < p 3 | 5 |
3 < p 4 | 9 |
4 < p 5 | 15 |
From the above find the mean.
Solution:
In order to find the mean we will take the midpoint of each interval as follows:
x | Frequency (f) | Midpoint (m) | f x m |
0 < p 1 | 2 | 0.5 | 2 x 0.5 = 1 |
1 < p 2 | 5 | 1.5 | 5 x 1.5 = 7.5 |
2 < p 3 | 5 | 2.5 | 5 x 2.5 = 12.50 |
3 < p 4 | 9 | 3.5 | 9 x 3.5= 31.50 |
4 < p 5 | 15 | 4.5 | 15 x 4.5 = 67.50 |
Totals | 36 | 120 |
So the estimated mean here is \frac{120}{36}=3.33
It is important to remember that with a grouped frequency table you are estimating the mean because you do not know the actual values. You are just assuming that the value is the middle one. The actual values could be more or less than this value.
When you are finding the mean, it is important to also remember not to round up or round down your answer, even if it does not make sense to the actual question, just remember to leave it.
When you are doing practice averages from frequency tables, gcse maths questions remember to show your working and all the required calculations.
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