Further examples for Intermediate and Higher tier
Example 1
7 people were asked to record how many cans of fizzy drink they drank in one month. The range of the data is 152; the mode is 3; the median is 31 and the mean is 41.
Find a possible set of values that satisfies these conditions.
Which average do you think best represents this list of numbers?
Solution
The median is 31 and there are 7 numbers – so the 4th number is 31.
The mode is 3 so we need at least 2 of these, so let’s have 2 of them. We’ll put these 3s at the start of the list.
The range is 152 so the highest must be 155, because 155 – 3 = 152.
So far we have : 3, 3, ?, 31, ? , ? , 155
The mean is 41 so the total is 41 × 7 = 287
Our total so far = 3 + 3 + 31 + 155 = 192
We need the other three numbers to add up to 287 – 192 = 95 and the 3rd number must be less than 31.
We could choose 28, 33 and 34 which gives us a list of numbers as 3, 3, 28, 31, 33, 34, 155.
There are many solutions for the last three numbers we chose.
The average that best reflects the numbers is the median. The mean is distorted by 155 and the mode is much lower than all of the rest.
Example 2
Aled carried out a survey to see how many text messages pupils in his year sent per week in school. This is his data:
Calculate an estimate for the mean number of text messages (to the nearest whole number) and find the median and modal group.
Solution
Estimate of mean = 3255 ÷ 130 = 25.03846154 = 25 (nearest whole number)
The total frequency = 130.
(130 + 1) ÷ 2 = 65.5
So the median value is the 65.5th or in other words, the (65th + 66th) ÷ 2.
13 people sent 0 – 9 messages (total people so far = 13)
44 people sent 10 – 19 messages (total people so far = 44 + 13 = 57)
35 people sent 20 – 29 messages (total people so far = 44 + 13 + 35 = 92)
Therefore the 65th and 66th values are in the group sending 20 – 29 messages.
The median group is the 20 – 29 text messages group.
The modal group is the group with the highest frequency.
The modal group is the 10 – 19 text messages group.
