OxfordAQA International GCSE Mathematics: Data types, tables and averages (9260) – Revision Notes

These notes condense statements S1, S2, S3 and S5 from section 3.4.1 of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1. Core content is also assessable on the Extension papers. Content marked “Extension only” is extra. For full explanations read the study guide, then try the practice questions. Charts and diagrams (S4) are on a separate page.

Course links: OxfordAQA IGCSE Mathematics hub, printable checklist and the free diagnostics.

Data types (S1)

Type	Meaning	Examples
Qualitative	Described in words, not numbers	colour, nationality, favourite food
Discrete	Numerical, separate values, counted	siblings, shoe size, goals
Continuous	Numerical, any value in a range, measured	time, mass, length, temperature
Ungrouped	Individual values (or values with frequencies)	4, 7, 7, 9
Grouped	Sorted into class intervals	20 ≤ t < 30

Class intervals must not overlap and must leave no gaps. With classes 30 ≤ x < 40 and 40 ≤ x < 50, the value 40 belongs to the second class.

Reading tables and lists (S2)

1. Read the title, headings and units.
2. Find the row, then the column, and read where they meet.
3. Check the wording at boundaries: “up to 3 kg” includes 3 kg; “over 3 kg” does not.
4. For lists, cross out or tick items as you use them so none is counted twice.

Two-way tables (S3)

Two features, one in rows and one in columns, with totals at the ends.

Method to complete a table:

1. Fill in any row or column with only one empty cell.
2. Use totals to find missing totals.
3. Check that the row totals and the column totals give the same grand total.

Designing a table: categories must cover everyone and not overlap (“under 16”, “16 or over”). Label everything. Add a totals row and column.

Reminder: 50 people, 30 female, 18 left-handed, 11 of the left-handed are female. Left-handed males = 18 − 11 = 7. Males = 50 − 30 = 20. Right-handed males = 20 − 7 = 13.

Averages and range (S5, Core)

Measure	Ungrouped list	Frequency table	Grouped table
Mean	total ÷ how many values	Σfx ÷ Σf	Σfx ÷ Σf using midpoints (estimate)
Median	middle of ordered list	value at position (n + 1) ÷ 2	class containing position n ÷ 2
Mode	most common value	value with highest f	modal class: highest frequency (equal class widths)
Range	largest − smallest	largest − smallest	largest possible: top upper end − bottom lower end

Method in steps for a frequency table:

1. Add a column for fx and a column for cumulative frequency.
2. Total f and fx.
3. Divide Σfx by Σf for the mean.
4. Use the cumulative column to find the median position.
5. Pick the mode from the f column, then state the value.

Median position: for an even number of values, the median is the mean of the two middle values. For 10 values, take the 5th and 6th.

Finding the class that holds the median

For grouped data, work out n ÷ 2, then add frequencies down the table until the running total first reaches or passes that number. With frequencies 4, 9, 5, 2 (n = 20), n ÷ 2 = 10. The running totals are 4, 13, 18, 20, so the 10th value is in the second class. You cannot give an exact median, only the class.

Working backwards and combining

• Total = mean × number of values.
• Missing value = new total − total of known values.
• Mean of two groups = (n₁ × mean₁ + n₂ × mean₂) ÷ (n₁ + n₂). Example: 8 items with mean 5 and 2 items with mean 10 give (40 + 20) ÷ 10 = 6.

Worked reminder: frequency table

Value, x	10	11	12	13
Frequency, f	3	7	6	4
Cumulative	3	10	16	20

Σfx = 30 + 77 + 72 + 52 = 231, Σf = 20
Mean = 231 ÷ 20 = 11.55
Median position = 10.5: the 10th value is 11 and the 11th is 12,
so median = (11 + 12) ÷ 2 = 11.5
Mode = 11 (frequency 7)
Range = 13 − 10 = 3

Effect of adding a value

If the mean of 5 values is 12, the total is 60. Adding a sixth value of 18 gives a total of 78, so the new mean is 78 ÷ 6 = 13. Adding a value above the current mean raises the mean. Adding a value below it lowers the mean. A new value equal to the mean leaves the mean unchanged.

Checking your answer

• The mean, median and mode of ungrouped data must lie between the smallest and largest values.
• A mean from a frequency table must lie between the smallest and largest x.
• A grouped mean must lie between the lowest class limit and the highest class limit.
• The range is never negative. If you get a negative range you subtracted the wrong way round.
• In a two-way table, if the two grand totals disagree, there is an arithmetic slip somewhere. Find it before moving on.

Quartiles, IQR and percentiles (S5, Extension only)

Measure	Position in an ordered list of n values
Lower quartile Q1	(n + 1) ÷ 4
Median Q2	(n + 1) ÷ 2
Upper quartile Q3	3(n + 1) ÷ 4
pth percentile	p ÷ 100 × (n + 1)
IQR	Q3 − Q1

If a position is not whole, go that fraction of the way between the neighbouring values. Use the method your teacher shows you. The sets on this site are chosen so common methods agree.

Worked reminder: 4, 6, 9, 12, 15, 18, 25 has n + 1 = 8. Q1 is at position 2, so 6. Q3 is at position 6, so 18. IQR = 18 − 6 = 12.

For a frequency table, make a cumulative column and look up the position, as for the median.

The IQR describes the middle half of the data, so it is not changed by one extreme value. The range is changed by it.

Must-know distinctions

Pair	Difference
Discrete vs continuous	Counted vs measured
Mode vs modal frequency	The value vs how often it occurs
Mean vs median	Mean uses every value; median is the middle value
Exact vs estimate	Grouped mean is an estimate, from midpoints
Range vs IQR (Extension only)	All data vs middle half

Which average? Use the median when there is an extreme value, the mode for qualitative data, and the mean when every value should count.

Quick self-test

1. Classify as qualitative, discrete or continuous: (a) the number of goals in a match, (b) a runner’s reaction time, (c) the colour of a car.
2. Find the mode and median of 7, 3, 12, 9, 3, 15.
3. The mean of 4 numbers is 9.5. Three of them are 6, 11 and 8. Find the fourth.
4. Values 1, 2, 3, 4 have frequencies 5, 8, 4, 3. Find the mean.
5. Classes 0 ≤ x < 3, 3 ≤ x < 6, 6 ≤ x < 9, 9 ≤ x < 12 have frequencies 4, 9, 5, 2. State the modal class and estimate the mean.
6. Of 50 people, 30 are female, 18 are left-handed and 11 left-handed people are female. How many males are right-handed?
7. Which class holds the value 40: 30 ≤ x < 40 or 40 ≤ x < 50?
8. Extension only: find the IQR of 4, 6, 9, 12, 15, 18, 25.
9. Extension only: in a list of 99 values in order, which value is the 90th percentile (using p ÷ 100 × (n + 1))?
10. Eight items have a mean of 5 and two items have a mean of 10. What is the mean of all ten?

Answers

1. (a) discrete, (b) continuous, (c) qualitative.
2. Ordered: 3, 3, 7, 9, 12, 15. Mode 3. Median = (7 + 9) ÷ 2 = 8.
3. Total = 38. Known = 25. Fourth = 13.
4. Σfx = 5 + 16 + 12 + 12 = 45; Σf = 20; mean = 2.25.
5. Modal class 3 ≤ x < 6. Σfx = 4×1.5 + 9×4.5 + 5×7.5 + 2×10.5 = 6 + 40.5 + 37.5 + 21 = 105; mean = 105 ÷ 20 = 5.25.
6. Males = 20; left-handed males = 7; right-handed males = 13.
7. 40 ≤ x < 50.
8. Q1 = 6, Q3 = 18, IQR = 12.
9. Position 0.9 × 100 = 90, so the 90th value.
10. (8 × 5 + 2 × 10) ÷ 10 = 6.

Where marks are usually lost

• Taking a median or quartile from data that has not been put in order.
• Quoting the largest frequency, not its value, as the mode.
• Dividing Σfx by the count of table columns instead of Σf.
• Putting a class limit into Σfx where the midpoint should go.
• Presenting a grouped mean without saying it is an estimate when the question asks you to comment.
• Averaging two means without weighting by group size.
• Reading the wrong row at a boundary such as “up to 3 kg”.
• Leaving out units or totals in a two-way table.
• Giving the position of a quartile instead of the value there (Extension only).
• Rounding part-way through a calculation, rather than at the end.

Official syllabus

Source: OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1, for exams from May/June 2018, published by OxfordAQA. Section 3.4.1, statements S1, S2, S3 and S5.