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Statistical measures

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Pearson Edexcel International GCSE Mathematics A (4MA1)

Statistical measures

The big idea: An average tells you where the data sits; a measure of spread tells you how much it varies. A comparison of two data sets needs one of each, written in context.

Summary — What This Chapter Covers

Mean = total ÷ number of values; median = middle value in order; mode = most common; range = largest − smallest.
From a frequency table: mean = Σfx ÷ Σf.
Grouped data: estimate the mean using midpoints; the modal class has the highest frequency.
Higher: estimate the median and quartiles from a cumulative frequency diagram.
Higher: interquartile range = upper quartile − lower quartile, a measure of spread unaffected by extreme values.

1. Averages and Range from a List

Foundation & Higher

Mean, median, mode, range

An average is a single value that represents the data. Put the data in order before finding the median. With n values, the median is the ((n + 1)/2)th value; for an even number of values, it is halfway between the two middle ones.

Measures mean = sum ÷ number of values median = middle value mode = most common range = highest − lowest

Worked Example

Find the mean, median, mode and range of: 6, 3, 9, 7, 3, 10, 5, 13.

Practice Question 1

The mean of five numbers is 8. Four of the numbers are 6, 7, 9 and 11. Find the fifth number.

2. Averages from a Frequency Table

Foundation & Higher

Discrete data in a table

Multiply each value by its frequency to get fx. The mean is the total of fx divided by the total frequency. For the median, find the position (n + 1)/2 and use running totals.

Goals x	0	1	2	3	4
Frequency f	4	7	5	3	1

Worked Example

Using the table, find the mean, median and mode number of goals.

3. Grouped Data: Estimated Mean and Modal Class

Foundation & Higher

Using midpoints

The exact values are unknown, so use each class midpoint as x. The answer is an estimate. The modal class is the class with the highest frequency.

Mass m (g)	f	Midpoint x	fx
0 < m ≤ 20	5	10	50
20 < m ≤ 40	12	30	360
40 < m ≤ 60	18	50	900
60 < m ≤ 80	10	70	700
80 < m ≤ 100	5	90	450

Worked Example

Work out an estimate for the mean mass and write down the modal class.

Sense check

An estimated mean must lie inside the range of the data — here between 0 and 100. If it does not, you probably divided by the number of classes instead of Σf.

4. Median, Quartiles and Spread from Cumulative Frequency

Higher tier

Reading across the curve

A measure of spread shows how varied the data is. The range uses only the extremes; the interquartile range (IQR) uses the middle half, so outliers do not affect it. From a cumulative frequency graph with total n, read across at n/2 for the median, n/4 for the lower quartile and 3n/4 for the upper quartile.

From a cumulative frequency graph median at n/2 LQ at n/4 UQ at 3n/4 IQR = UQ − LQ

Worked Example

5. Interquartile Range from a Discrete Data Set

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Quartiles from a list

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Statistical measures

Statistical measures

Mean, median, mode, range

Discrete data in a table

Using midpoints

Reading across the curve

Past papers that test Statistical measures

More Mathematics A topics