OxfordAQA International AS and A-level Economics: Ratios, percentages, mean, median and quantiles (9640)

This guide teaches quantitative skills QS1, QS2 and QS3 from section 3.5 of the OxfordAQA International AS and A-level Economics (9640) specification, version 1.2. The specification says these skills could be assessed in any of the assessments. Sections 3.1 and 3.2 are International AS content and sections 3.3 and 3.4 are the extra International A2 content. The one A2 addition here is in QS3: for A2 assessments you also need to understand and use relevant quantiles. Every country, firm and number below is invented for practice. Use the OxfordAQA Economics hub and the printable checklist to track coverage. For recall, use the revision notes. To test yourself, use the practice questions or the free diagnostics.

What the specification asks for

Skill	Wording in 3.5	Where it is used
QS1	Calculate, use and understand ratios and fractions	Sharing budgets, trade ratios, shares of a workforce
QS2	Calculate, use and understand percentages and percentage changes	Growth, inflation, profit margins, changes in price and quantity
QS3	Understand and use the terms mean and median	Average income, wages, prices
QS3 (A2)	Also understand and use relevant quantiles	Quartiles, deciles, quintile shares, income spread

These skills sit underneath later topics such as elasticity and growth. See the operation of markets and measurement of macroeconomic performance guides for where they are applied.

QS1: ratios and fractions

A ratio compares parts. A fraction compares one part with the whole. In a ratio of 5:3:1 there are 5 + 3 + 1 = 9 parts, and the first part is 5/9 of the whole.

Method for sharing in a ratio

1. Add the parts to get the total number of parts.
2. Divide the total amount by that number to find one part.
3. Multiply by each ratio figure.
4. Check that the shares add back to the total.

Worked example 1. Country A shares a 360 cu billion budget between health, education and transport in the ratio 5:3:1.

Parts = 5 + 3 + 1 = 9
One part = 360 / 9 = 40
Health = 5 x 40 = 200; education = 3 x 40 = 120; transport = 1 x 40 = 40
Check: 200 + 120 + 40 = 360

Worked example 2. Country A exports 48 cu billion of goods and imports 60 cu billion.

Ratio exports : imports = 48 : 60 = 4 : 5   (divide both by 12)
Exports as a fraction of imports = 4/5 = 0.8
Gap = 60 - 48 = 12, which is 12/60 = 1/5 of imports

The ratio 4:5 says exports cover four-fifths of imports. Always state which quantity comes first. 4:5 and 5:4 mean different things.

Worked example 3. Country A has a workforce of 2.4 million. Three-eighths work in services and one-sixth in manufacturing. The rest work in farming.

Services = 3/8 x 2,400,000 = 900,000
Manufacturing = 1/6 x 2,400,000 = 400,000
Farming = 2,400,000 - 900,000 - 400,000 = 1,100,000
As a fraction: 1 - 3/8 - 1/6 = 9/24 and 4/24 removed, leaving 11/24
Check: 11/24 x 2,400,000 = 1,100,000

Use a common denominator (here 24) when adding or subtracting fractions. To compare fractions, convert both to decimals or percentages.

QS2: percentages and percentage changes

A percentage is a fraction out of 100. Percentage of a total = part/total × 100. Percentage change = (new value − old value) / old value × 100. The starting value is always the base.

Worked example 4: percentage change and share. A small bakery’s revenue rises from 18,000 to 20,700 cu. Its costs are 17,595 cu.

Change = 20,700 - 18,000 = 2,700
% change = 2,700 / 18,000 x 100 = 15%
Profit = 20,700 - 17,595 = 3,105
Profit margin = 3,105 / 20,700 x 100 = 15%

The two 15% figures are a coincidence of the chosen numbers. One uses the old revenue as the base. The other uses the new revenue.

Percentage points versus percentage change. If the unemployment rate in Country B falls from 8% to 6%:

Change in percentage points = 6 - 8 = -2 percentage points
Percentage change = (6 - 8) / 8 x 100 = -25%

Use “percentage points” for the gap between two rates. Use “per cent” for the change relative to the starting rate.

Reverse percentages. After a 12% price rise, a good costs 140 cu. The new price is 112% of the original, so:

Original = 140 / 1.12 = 125
Check: 125 x 1.12 = 140

Do not take 12% off 140. That would divide by the wrong base.

Successive changes are multiplied, not added. Use multipliers: +20% is 1.20 and −20% is 0.80.

Up 20% then down 20%: 1.20 x 0.80 = 0.96, a fall of 4%
Growth of 2%, 3% then 1%: 1.02 x 1.03 x 1.01 = 1.0611, a rise of 6.11%
(Adding the rates would give 6%, which is slightly wrong.)

Composition. If Country A’s GDP is 800 cu billion and consumption is 520, consumption is 520/800 = 0.65, or 65% of GDP. Shares of a whole must add to 100%.

Using the base in economics. When a price rises, work out the percentage change in price and the percentage change in quantity separately, each from its own old value. Later skills, such as elasticity, divide one of these by the other. Getting the base wrong here spoils every later step.

QS3: mean and median

The mean adds all values and divides by the number of values. The median is the middle value once the data are in order. If there is an even count of values, take the point midway between the two central ones.

Worked example 5. Monthly incomes of nine households in a village, in thousands of cu: 12, 14, 15, 16, 18, 19, 21, 24, 95.

Sum = 234; mean = 234 / 9 = 26
Ordered list has 9 values, so the median is the 5th = 18
Without the 95: mean = 139 / 8 = 17.375; median = (16 + 18) / 2 = 17

One very high income lifts the mean by 8.6 (from 17.375 to 26). It moves the median by only 1. In a skewed distribution the mean is pulled towards the long tail, so the median often describes a typical household better.

Worked example 6: data in a frequency table. A firm has 20 workers on weekly pay of 40, 50, 60 and 150 cu.

Weekly pay (cu)	Workers	Pay x workers
40	8	320
50	6	300
60	4	240
150	2	300
Total	20	1,160

Mean = 1,160 / 20 = 58
Median: the 10th and 11th workers are both on 50, so the median = 50

The mean is above the median because of the two high earners. Say which measure you use and why.

QS3 for A2: quantiles

Quantiles split ordered data into equal-sized groups. Quartiles give four groups: Q1 (lower quartile, 25th percentile), the median (50th) and Q3 (upper quartile, 75th). Deciles give ten groups and quintiles five. A percentile is a position out of 100. The interquartile range is Q3 − Q1. It shows how widely the central half of the data is spread, and extreme values do not affect it.

Textbooks place quantiles slightly differently. This guide puts each quantile at position (n + 1) × fraction and interpolates between neighbouring values. State the method you use.

Worked example 7. Eleven households’ incomes (thousands of cu), in order: 8, 10, 11, 13, 15, 18, 20, 24, 30, 45, 120.

n + 1 = 12
Q1 position = 12 x 0.25 = 3, so Q1 = 11
Median position = 12 x 0.5 = 6, so median = 18
Q3 position = 12 x 0.75 = 9, so Q3 = 30
IQR = 30 - 11 = 19
Mean = 314 / 11 = 28.55 (to 2 d.p.), well above the median

Worked example 8: interpolation. Eight values: 22, 25, 27, 31, 36, 40, 52, 90. Here n + 1 = 9.

Q1 position = 9 x 0.25 = 2.25: 25 + 0.25 x (27 - 25) = 25.5
Median position = 4.5: (31 + 36) / 2 = 33.5
Q3 position = 6.75: 40 + 0.75 x (52 - 40) = 49
IQR = 49 - 25.5 = 23.5

Quintile shares of income. Suppose the poorest to richest fifths of Country C receive 5%, 9%, 14%, 22% and 50% of national income.

Check: 5 + 9 + 14 + 22 + 50 = 100
Top fifth : bottom fifth = 50 : 5 = 10 : 1
Bottom two fifths together = 5 + 9 = 14%; top two fifths = 22 + 50 = 72%

If the 90th percentile income is 64 and the 10th is 16 (thousand cu), the 90:10 ratio is 64/16 = 4. A higher ratio means a wider spread. Shares and ratios describe inequality but say nothing about the level of income. Two countries can have the same ratio and very different mean incomes.

Evaluating a measure. Mean income can hide inequality. The median is more robust but ignores how large the top values are. The IQR ignores the extremes entirely. The strongest answers say what each measure leaves out and then say which fits the question.

Common errors

• Reading the ratio 4:5 as a fraction of 4/5 of the total, when the total is 9 parts.
• Dividing by the final figure rather than the starting figure when finding a percentage change.
• Writing a percentage change for a variable that is already a rate when the question asks for the change in percentage points.
• Adding successive percentage changes instead of multiplying the multipliers.
• Forgetting to order the data before picking a median or quartile.
• Dividing an even-sized set at the wrong place. Average the two middle values.
• Treating the mean as typical when one value is extreme.
• Quoting a quantile without saying the position rule.

Next steps

Work through the revision notes, then the practice questions. The free diagnostics show where to begin.

Official syllabus

OxfordAQA International AS and A-level Economics (9640), specification version 1.2, section 3.5 Quantitative skills (QS1, QS2 and QS3). First teaching September 2020; first International AS exams May/June 2021; first International A-level exams May/June 2022. Published by OxfordAQA.