Maths
Measures of average
A measure of average is a number that is typical for a set of figures. Finding the average helps you to draw conclusions from data. The main types are mean, median and mode.
This Revision Bite covers:
The mean is the most common measure of average.
To calculate the mean add the numbers together and divide the total by the amount of numbers:
Mean = sum of numbers ÷ amount of numbers
Five friends compare their marks in a French test. These are shown below:
Name | Mark |
---|---|
Shabana | 41 |
Bea | 54 |
Liam | 79 |
Dave | 26 |
Ed | 65 |
What is the mean mark?
Mean = (41 + 54 + 79 + 26 + 65) ÷ 5
= 265 ÷ 5
= 53
Check your answer by asking yourself if it looks about right. If you had forgotten to divide by 5 then the answer would have been 265, which would not have made sense. Always check that your answer seems sensible.
A die is thrown 10 times. The following results are obtained:
3, 5, 1, 2, 6, 4, 2, 5, 6, 1
What is the mean score?
To find the answer, add the numbers together and divide by the amount of numbers:
Mean = (3 + 5 + 1 + 2 + 6 + 4 + 2 + 5 + 6 + 1) ÷ 10
35 ÷ 10 = 3.5
If you place a set of numbers in order, the median number is the middle one.
If there are two middle numbers, the median is the mean of those two numbers.
Find the median of each of the following sets of numbers:
a) 2, 4, 7, 1, 9, 3, 11
b) 4, 1, 3, 10, 6, 9,
a) Place the numbers in order:
1, 2, 3, 4, 7, 9, 11
The middle number is 4. Therefore the median is 4.
b) Place the numbers in order:
1, 3, 4, 6, 9, 10
there are two middle numbers (4 and 6), so we find the mean of these two numbers. The median is therefore:
(4 + 6) ÷ 2 = 5
In general:
If there are n numbers, the median will be the (n + 1) ÷ 2 = th number
If there are 3 numbers, the median will be the (3 + 1) ÷ 2 = 2nd number.
If there are 4 numbers, the median will be the (4 + 1) ÷ 2 = 2^{1}/_{2}^{th} number
Rachel records the number of goals scored by her five-a-side team in their first 20 matches.
The results are shown in the frequency table below:
Numbers of goals | Frequency |
---|---|
0 | 7 |
1 | 5 |
2 | 6 |
3 | 0 |
4 | 2 |
4 | 0 |
What is the median number of goals scored?
20 matches were played, so the median will be the 21 ÷ 2 ^{th} value.
7 matches scored 0 goals, and 5 matches scored 1 goal.
The 21 ÷ 2 ^{th} value lies in the '1 goal' category.
21 ÷ 2 = 10 ^{1}/_{2}
Therefore, the median number of goals is 1.
The mode is the value that occurs most often.
The mode is the only average that can have more than one value.
When finding the mode, it helps to order the numbers first.
Find the mode of each of the following sets of numbers:
a) 3, 7, 1, 3, 4, 8, 3
b) 2, 7, 2, 1, 4, 7, 3
a) Start by placing the numbers in order:
1, 3, 3, 3, 4, 7, 8
The number 3 occurs most often so the mode is 3.
b) Start by placing the numbers in order:
1, 2, 2, 3, 4, 7, 7
The numbers 2 and 7 occur more often than all other numbers so the modes are 2 and 7.
In this frequency table, the mode is the value with the highest frequency:
Shoe size | 5 | 6 | 7 | 8 | 9 |
Frequency | 2 | 5 | 11 | 4 | 1 |
In this example the modal shoe size is 7, because more people take size 7 than any other size.
The range is the difference between the highest and lowest values in a set of numbers.
To find it, subtract the lowest number in the distribution from the highest.
Find the range of the following set of numbers:
a) 23, 27, 40, 18, 25
b) 25, 26, 57, 15, 47
a) The largest value is 40 and the smallest value is 18. Therefore, the range is 40 - 18 = 22.
b) The largest value is 57 and the smallest value is 15. Therefore the range is 57 - 15 = 42.
Complete the following table. The data shows the scores in two tests, English and Maths, across a set of ten students.
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If you were to compare the scores in the two subjects, English and Maths, which measure of average would you use and why?
Range
The range of scores in English (37) is far greater than that in the Maths (4).
Mean
The mean score in each subject is 78 which implies that the scores of the students are more-or-less identical in English and Maths. But looking at the actual scores, you can see that this is not the case.
Median
If you compare the medians (71 and 78), you might assume that the students generally scored less in English, (which is partly true, but there are also some much higher scores there too).
Mode
If you just state the modal score for each subject (101 and 78), you have no information about the scores of the other students.
So, which is best? It seems that to give maximum information, a combination of the median and the range would be best.
So, in summary, English has a median score of 71 and a range of 37, and Maths has a median score of 78 and a range of 4.
Sara wanted to know the ages (in whole years) of children on her school bus. She conducted a survey and her results are shown below:
13 | 14 | 11 | 12 | 12 | 15 |
13 | 14 | 12 | 16 | 15 | 11 |
11 | 12 | 11 | 12 | 14 | 16 |
14 | 15 | 14 | 14 | 13 | 12 |
13 | 11 | 11 | 14 | 12 | 13 |
To find the mean add all the ages together and divide by the total number of children.
If you type all those ages into a calculator it is highly likely that you would make a mistake or forget where you were up to.
It would be better if you could see these results displayed in a frequency table:
Age | Frequency |
---|---|
11 | 6 |
12 | 7 |
13 | 5 |
14 | 7 |
15 | 3 |
16 | 2 |
The frequency table shows us that there are six children aged 11, seven children aged 12, five children aged 13...etc.
To find the sum of their ages, calculate:
(6 × 11) + (7 × 12) + (5 × 13) + (7 × 14) + (3 × 15) + (2 × 16) = 390
The total number of children is 6 + 7 + 5 + 7 + 3 + 2 = 30
So the mean age is 390 ÷ 30 = 13
You could also write this information into the table as shown below:
Age | Frequency | Age × Frequency |
---|---|---|
11 | 6 | 66 |
12 | 7 | 84 |
13 | 5 | 65 |
14 | 7 | 98 |
15 | 3 | 45 |
16 | 2 | 32 |
Totals | 30 | 390 |
Mean = 390 ÷ 30 = 13
Calculating the mean and modal class for grouped data is very similar to finding the mean from an ungrouped frequency table, except that you do not have all the information about the data within the groups so can only estimate the mean.
This table shows the weights of children in a class.
Using this information:
a) Estimate the mean weight
b) Find the modal class
Mass (m) kg | Frequency |
---|---|
30 ≤ m < 40 | 7 |
40 ≤ m < 50 | 6 |
50 ≤ m < 60 | 8 |
60 ≤ m < 70 | 4 |
To estimate the mean weight, you know that 7 children are between 30kg and 40kg, but you don't know exactly how much they weigh, so assume that they all weigh 35kg (the midpoint of the group).
Do the same for all the other groups:
Mass (m) kg | Midpoint | Frequency | Midpoint × Frequency |
---|---|---|---|
30 ≤ m < 40 | 35 | 7 | 245 |
40 ≤ m < 50 | 45 | 6 | 270 |
50 ≤ m < 60 | 55 | 8 | 440 |
60 ≤ m < 70 | 65 | 4 | 260 |
Totals | 25 | 1215 |
a) Estimate of mean = 1215 ÷ 25 = 48.6 kg
b) The modal class is the class that has the highest frequency. In this case the modal class is 50 ≤ m < 60
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