Averages & Range (Cambridge IGCSE Maths): Revision notes

Section 1

What are the three types of average?

Average	Definition	How to find it
Mean	the total of all values divided by how many values there are	$\text{mean} = \frac{\text{sum of values}}{\text{number of values}}$
Median	the middle value when data is ordered	order the data, find the middle value (average the two middle values if there is an even number)
Mode	the most frequently occurring value	identify the value(s) that appear most often

Distinguish between when each is most useful: the mean uses every value but is affected by extreme outliers; the median is unaffected by outliers; the mode works even for non-numerical (categorical) data and shows the most popular choice.

Key termsmeanmedianmode

Common mistake

Forgetting to order the data before finding the median — an unordered 'middle' value is meaningless.

Section 2

What is the range?

The range measures spread, not average: $\text{range} = \text{highest value} - \text{lowest value}$

A small range means the data is clustered closely together; a large range means the data is spread out.

Key termsrange

Section 3

How do I find averages from a frequency table?

For data in a frequency table:

Mode: the value with the highest frequency
Median: find the position $\frac{n+1}{2}$ using cumulative frequencies to locate which value that position falls in
Mean: $\text{mean} = \frac{\sum (\text{value} \times \text{frequency})}{\sum \text{frequency}}$ — multiply each value by its frequency, sum these, then divide by the total frequency

At Core level, data will not be grouped; individual discrete values are given with their frequencies.

Key termsfrequency table

Example

Values 2,3,4,5 with frequencies 3,5,4,2 (14 items total): mean $= \frac{2(3)+3(5)+4(4)+5(2)}{14} = \frac{6+15+16+10}{14} = \frac{47}{14} \approx 3.36$ .

Section 4

How do I find quartiles and interquartile range? (Extended)

For individual data, ordered:

Lower quartile (Q1): the value a quarter of the way through the ordered data
Upper quartile (Q3): the value three-quarters of the way through
Interquartile range (IQR) $= Q3 - Q1$ , a measure of spread of the middle 50% of the data, less affected by outliers than the full range

For grouped data, an estimate of the mean is found using the midpoint of each class multiplied by its frequency: $\text{estimated mean} = \frac{\sum(\text{midpoint} \times \text{frequency})}{\sum \text{frequency}}$ . The modal class is the class with the highest frequency.

Key termsquartileinterquartile rangemodal classclass midpoint

Exam tip

For grouped data, the word 'estimate' must appear in your answer for the mean — it cannot be exact because individual values within each class are unknown.

Must Know

Mean = sum of values / number of values; affected by outliers
Median = middle value of ordered data; unaffected by outliers
Mode = most frequent value; works for non-numerical data too
Range = highest - lowest value, a measure of spread
From a frequency table: mean $= \frac{\sum(\text{value} \times \text{frequency})}{\sum \text{frequency}}$
Extended: IQR $= Q3 - Q1$ ; estimated mean for grouped data uses class midpoints; modal class is the class with highest frequency

That's the notes covered.

Carry on to the next subtopic.