OxfordAQA International AS and A-level Economics: Ratios, percentages, averages (9640) – Revision Notes

Quick-recall notes for quantitative skills QS1 to QS3 in section 3.5 of the OxfordAQA International AS and A-level Economics (9640) specification, version 1.2. These skills could be assessed in any assessment. Sections 3.1 and 3.2 are International AS content and 3.3 and 3.4 are the extra International A2 content. Only quantiles carry an A2 note: for A2 assessments you also need to understand and use relevant quantiles. Full explanations are in the study guide. Then use the practice questions. The OxfordAQA Economics hub, the printable checklist and the free diagnostics help you plan. All figures are invented.

Definitions

Term	Meaning
Ratio	Compares parts, such as exports : imports = 4 : 5
Fraction	One part out of the whole, such as 3/8 of a workforce
Percentage	A fraction out of 100
Percentage change	(new − old) / old × 100
Percentage point	The unit for the gap between two percentages or rates
Mean	Sum of values divided by the number of values
Median	Middle value of ordered data; halfway between the two middle values if the count is even
Quantile	A cut point that splits ordered data into equal-sized groups
Quartile	Cut points for four groups: Q1, median, Q3
Decile, quintile	Cut points for ten and for five groups
Percentile	Position out of 100
Interquartile range (IQR)	Q3 − Q1

Formulas and rules

Task	Rule
Share in a ratio a : b : c	Add parts; amount / total parts = one part; multiply by each figure
Fraction of a total	Total × numerator / denominator
Percentage of a total	Part / total × 100
Percentage change	(new − old) / old × 100, with old as the base
Multiplier	1 + (rate / 100): +8% is 1.08, −15% is 0.85
Successive changes	Multiply the multipliers; do not add the rates
Reverse percentage	Original = new value / multiplier
Mean from a frequency table	Sum of (value × frequency) / sum of frequencies
Quantile position (convention in the study guide)	(n + 1) × fraction; interpolate between neighbours
90:10 ratio	90th percentile / 10th percentile

Method boxes

Sharing in a ratio. Add parts. Divide. Multiply. Check the shares sum to the total.

Percentage change. Subtract old from new. Divide by old. Multiply by 100. Keep the sign.

Median. Order the data first. Odd count: the middle one. Even count: average the two middle values.

Quartiles. Order the data. Find the position (n + 1) × 0.25, 0.5 and 0.75. When the position falls between two places, add that fraction of the gap between the neighbouring values to the lower one. State the rule you used.

Small worked reminders

Share 200 in the ratio 1 : 4      ->  parts 5, one part 40  ->  40 and 160
Price 50 -> 58                    ->  8 / 50 = 0.16, so +16%
Rate 3% -> 5%                     ->  +2 percentage points, or +66.7% relative
Price incl. 20% tax = 120         ->  120 / 1.20 = 100 before tax; tax = 20
Output +10%, -5%, +2%             ->  1.10 x 0.95 x 1.02 = 1.0659, about +6.59%
Values 4, 5, 5, 9, 12             ->  mean 35 / 5 = 7; median (3rd) = 5
Pay 10 (3 workers), 20 (2), 50 (1) -> (30 + 40 + 50) / 6 = 20

Must-know distinctions

• Ratio versus fraction. 3:5 has 8 parts; the first part is 3/8, not 3/5.
• Percentage point versus per cent. 8% to 6% is a fall of 2 percentage points and a fall of 25%.
• Adding versus multiplying changes. +10% then −10% is −1%, not 0.
• Mean versus median. The mean uses every value and is pulled by extremes. The median is robust but ignores how far the top values go.
• Range versus IQR. The range uses the two extremes. The IQR uses the middle half.
• Share versus level. A quintile share says how income is divided, not how large it is.
• Quartile versus quintile. Quartiles cut into four groups, quintiles into five.

Worked chain for a data question

A table shows monthly pay in Country D’s capital region and in its rural region (invented). The capital has a mean of 900 and a median of 700. The rural region has a mean of 520 and a median of 500.

Capital: mean - median = 900 - 700 = 200; mean / median = 1.29
Rural:   mean - median = 520 - 500 = 20;  mean / median = 1.04

Step 1: the capital’s mean is 29% above its median, so a few high earners are raising it. Step 2: the rural mean is only 4% above its median, so pay there is spread more evenly. Step 3: the mean pay gap between regions is (900 − 520) / 520 = 73%, while the median gap is (700 − 500) / 500 = 40%. Step 4: the mean exaggerates the gap for the typical worker. Step 5: neither figure shows the very lowest paid, which needs a low percentile.

Using the measures in economics

1. Skew and the mean. If the mean is well above the median, a few high values are pulling the mean up. The typical case is nearer the median.
2. Before and after. To judge a policy’s effect on the poorest, compare the lowest quintile share or a low percentile. To judge the middle, compare the median.
3. Spread. A smaller 90:10 ratio or a smaller IQR means the middle of the distribution has become more equal. It does not show what happened at the very top.
4. Chain for a data question. Identify the measure. Calculate it with working. Compare it with a second figure. State what it shows. State what it hides.

Presenting calculations

• Write the formula or the substitution before the number, so the method is visible.
• Keep full figures through the working and round only at the end. Say how you rounded, such as “to 1 decimal place”.
• Put units on every answer: cu, per cent, percentage points, or a ratio such as 4 : 5.
• Give the sign of a change. A fall of 25% is not +25%.
• Add one sentence on what the answer means for the market, firm or country in the question.

Quick self-test

1. Share 270 cu in the ratio 2 : 3 : 4.
2. A firm has 28 full-time and 12 part-time staff. Write full-time : part-time in simplest form and give part-time as a fraction of all staff.
3. A price rises from 250 to 285. Give the percentage change.
4. A growth rate falls from 4% to 2.5%. Give the change in percentage points and the percentage change.
5. A good costs 207 after a 15% rise. Find the original price.
6. Output grows by 5% in year 1 and falls by 4% in year 2. Give the overall percentage change.
7. Find the mean and median of 14, 16, 16, 18, 21, 25, 70.
8. Find the median of 3, 8, 9, 12, 20, 35.
9. Eleven values are 8, 10, 11, 13, 15, 18, 20, 24, 30, 45, 120. Find Q1, Q3 and the IQR using position (n + 1) × fraction. (A2)
10. Quintile shares of income for a country are 6%, 10%, 15%, 23% and 46%. Find the top-to-bottom ratio and the share of the bottom two quintiles. (A2)

Answers

1. Parts = 9, one part = 30; shares are 60, 90 and 120.
2. 28 : 12 = 7 : 3; part-time = 12/40 = 3/10.
3. 35/250 = 0.14, so +14%.
4. −1.5 percentage points; −1.5/4 = −37.5%.
5. 207 / 1.15 = 180.
6. 1.05 × 0.96 = 1.008, so +0.8%.
7. Sum = 180, mean = 180/7 = 25.71 (2 d.p.); median is the 4th value, 18.
8. (9 + 12) / 2 = 10.5.
9. Q1 = 11, Q3 = 30, IQR = 19.
10. 46 / 6 = 7.67 (2 d.p.), so about 7.7 : 1; bottom two quintiles = 6 + 10 = 16%.

Where marks are usually lost

• Dividing by the final figure rather than the starting figure when finding a percentage change.
• Calling a gap between two rates a percentage change when the question asks for percentage points, or the reverse.
• Adding successive growth rates and reporting the sum as the total change.
• Picking the median from unordered data.
• Averaging the wrong pair of values for an even-sized data set.
• Using a frequency as if it were a data value in a mean.
• Quoting a ratio the wrong way round, such as imports : exports when asked for exports : imports.
• Giving an answer with no working, so no method is on show.
• Calculating well but not saying what the figure means for the economic question.
• Using the mean to describe “typical” income when the data are skewed, with no comment on the skew.

Next steps

• Re-read the study guide.
• Try the practice questions.
• Take a free 10-minute diagnostic.

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.