Edexcel IGCSE Maths 复习笔记 6.1.3 Averages from Grouped Data

What is grouped data and why use it?

• Some data for a particular scenario can vary a lot
• For example, the heights of people, particularly if you include a mixture of children and adults
• Because data like height is also continuous (essentially data that can be measured) it would be difficult, even using a table, to list every height that gets recorded – also, there is little difference between someone who is 176 cm tall and someone who is 177cm tall
• So we often group data into classes but that leads to one important point.…

What do I need to know?

• When data is grouped we lose the raw data
• With height data this means we might know, how many people have a height of between 150 cm and 160 cm but not the specific heights of those 10 people
• This means we cannot find the actual mean, median and mode from their original definitions – but we can estimate the mean and we can also talk about the class the median lies in and the modal class

1. Estimating the Mean

• There is one extra stage to this method compared to finding the mean from tables with discrete data – use the class midpoints as our data valueseg. the heights of 25 members of a youth club were recorded and the results are summarised in the table below – estimate the mean height

• Note that the class widths (group sizes) are not all equal (this is not a problem so do not let it put you off) and be careful with the inequality signs: a height of exactly 130 cm would be recorded in the second row not the first
• As we don’t know the original data we use the midpoint of each group – this is the height that is half way between the start and the end of the group
• Usually these are easy to ‘see’ but you can always work it out if in doubt (eg halfway between 140 and 145 is (140 + 145) ÷ 2 = 285 ÷ 2 = 142.5)
• We then use theses midpoints as the heights for all the people in the 120 ≤ h < 130 class – so we assume that all 4 people in the class will have a height of 125 cm
• This is why the mean will be an estimate – we assume the heights in all the classes ‘average out’ at the midpoint height

• Now we can proceed as if it were discrete data and multiply, including a total row as well

• And finally we can find the mean:Mean = 3612.5 ÷ 25 = 144.5Mean height is 144.5 cm

2. Median

• Rather than find an actual value for the medium you could be asked to find the class in which the median lies
• The process for finding its position is the same as before so for the above example:

Position of median = (25 + 1) ÷ 2 = 13

The median is the 13^th value

• From looking at the frequency column we can see the 13^th value would fall in the 145 ≤ h < 150 class (it is the last value in this class in fact)So we would say the median lies in the 145 ≤ h < 150 class

3. Modal Class (Mode)

• Similar to finding the median we are only interested in the class the modal value lies within.
• Again using the example above we can see from the table the highest frequency is 6
• So the modal class is 145 ≤ h <150