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Averages and measures of spread

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Cambridge O Level Mathematics (Syllabus D) 4024

Averages and measures of spread

The big idea: An average gives one typical value; a measure of spread says how far the data wanders from it. Choosing the right average matters as much as working it out.

Summary — What This Chapter Covers

Mean = total of the values ÷ number of values. Median = middle value in order. Mode = most common value. Range = largest − smallest.
Choose the average that suits the data: the median resists outliers, the mode works for categories, the mean uses every value.
From a frequency table, the mean is Σfx ÷ Σf.
For grouped data, use the midpoint of each class to calculate an estimate of the mean.
The modal class is the class with the highest frequency.

1. Mean, Median, Mode and Range

Syllabus 9.3 · Paper 1 & 2

Four numbers from one list

Put the data in order first; it makes the median, mode and range easy and helps you spot mistakes.

Definitions (none of these are on the List of formulas) mean = sum of all the values ÷ number of values
median = the middle value when the data is in order; for n values it is the (n + 1)/2 th value
mode = the value that occurs most often range = largest value − smallest value

Worked Example

Find the mean, median, mode and range of: 12, 7, 3, 9, 15, 7, 4, 10, 8, 5

Practice Question 1

(a) The mean of five numbers is 12. Four of the numbers are 8, 15, 10 and 13. Find the fifth number.
(b) In a class, the 12 boys have a mean mark of 64 and the 18 girls have a mean mark of 71. Find the mean mark of the whole class.

2. Which Average to Use

Syllabus 9.3 · Paper 1 & 2

Matching the average to the data

Average	Good for	Weakness
Mean	uses every value; good when there are no extreme values	pulled a long way by outliers
Median	data with outliers or a skewed spread, such as salaries or house prices	ignores most of the actual values
Mode	categorical data (favourite colour) and the most popular size	may not exist, or there may be several

The range is a quick measure of spread, but one outlier changes it completely. The interquartile range (see Cumulative frequency) uses the middle half of the data and is not affected by outliers.

Worked Example

The weekly wages, in dollars, of six workers at a small firm are: 410, 380, 450, 395, 420, 2500
(a) Find the mean and the median. (b) Which average better describes a typical wage? Give a reason.

3. The Mean from a Frequency Table

Syllabus 9.3 · Paper 1 & 2

Σfx ÷ Σf

In a frequency table each value x appears f times. Multiply each value by its frequency to get the total, then divide by the total frequency. The median is found by counting through the cumulative totals.

Mean from a frequency table mean = Σfx ÷ Σf (Σ means “the sum of”)

Worked Example

The table shows the number of children in 30 families.

Number of children, x	0	1	2	3	4
Frequency, f	4	9	11	5	1

Find the mean, the median and the mode.

4. An Estimate of the Mean for Grouped Data

Syllabus 9.3 · Paper 1 & 2

Use the midpoints

midpoint estimate modal class

Estimated mean estimate of mean = Σ(f × midpoint) ÷ Σf

Worked Example

Time (t minutes)	Frequency
0 < t ≤ 10	5
10 < t ≤ 20	12
20 < t ≤ 30

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Averages and measures of spread

Averages and measures of spread

Four numbers from one list

Matching the average to the data

Σfx ÷ Σf

Use the midpoints

Past papers that test Averages and measures of spread

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