Weighted and Chain Base Index Numbers
An index number expresses a quantity in one time period as a percentage of its value in a chosen base period, which is always given the value 100. A weighted index number combines several items' index numbers using weights reflecting relative importance, while a chain-base index number compares each period only to the one before it, not a fixed base year. Both are assessed on GCSE Statistics Higher tier.
Before you start
Make sure you're comfortable with these topics first:
Method
- For a simple index number: index = (value in current period / value in base period) x 100.
- For a weighted index number covering several items, calculate each item's simple index number, multiply each by its weight, add the products, then divide by the sum of the weights.
- Weighted index = sum of (index x weight) / sum of weights.
- For a chain-base index number, express each period's value as a percentage of the PREVIOUS period's value: index = (value this period / value previous period) x 100.
- To convert linked chain-base indices into a fixed-base index across several periods, multiply the chain-base indices together as decimals in sequence, then rescale to 100.
- Compare index numbers to describe change: an index above 100 means an increase, below 100 means a decrease, and the percentage change is the index value minus 100.
Worked example
A household's spending on three items in 2024 (base year, index 100) and 2025 is shown, along with each item's weight (share of total spending): food, 80 pounds rising to 88 pounds, weight 5; transport, 40 pounds rising to 46 pounds, weight 3; leisure, 20 pounds falling to 19 pounds, weight 2. Calculate the weighted index number for 2025, with 2024 = 100.
- Find each item's simple index number: food = 88/80 x 100 = 110, transport = 46/40 x 100 = 115, leisure = 19/20 x 100 = 95.
- Multiply each index by its weight: food 110 x 5 = 550, transport 115 x 3 = 345, leisure 95 x 2 = 190.
- Add the products together: 550 + 345 + 190 = 1085.
- Add the weights together: 5 + 3 + 2 = 10.
- Divide to find the weighted index: 1085 / 10 = 108.5.
- Final answer: the weighted index number for 2025 is 108.5, showing spending on these items rose by 8.5% overall since 2024.
Practice questions
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Q1The price of a bus ticket was 2.00 pounds in 2020 and 2.50 pounds in 2023. Calculate the simple index number for 2023, using 2020 = 100.Show answer
Answer: 125 (2.50/2.00 x 100 = 125)
Q2An item cost 40 pounds in the base year and has an index number of 115 this year. Find its price this year.Show answer
Answer: 46 pounds (115/100 x 40 = 46)
Q3Two items have index numbers, base year = 100, of 108 (weight 3) and 96 (weight 7). Calculate the weighted index number.Show answer
Answer: 99.6 ((108 x 3 + 96 x 7)/10 = 996/10 = 99.6)
Q4A shop's total sales were 5000 pounds in Month 1, 5400 pounds in Month 2, and 5832 pounds in Month 3. Calculate the chain-base index number for Month 2 (based on Month 1) and for Month 3 (based on Month 2).Show answer
Answer: Both 108 (Month 2: 5400/5000 x 100 = 108; Month 3: 5832/5400 x 100 = 108)
Q5A quantity has chain-base index numbers of 110 (Year 2 based on Year 1) and 120 (Year 3 based on Year 2). Calculate the fixed-base index number for Year 3, using Year 1 as the base (= 100).Show answer
Answer: 132 ((110/100) x (120/100) x 100 = 1.1 x 1.2 x 100 = 132)
Q6A cost-of-living index combines three categories with weights: housing 6, food 3, leisure 1. This year's simple index numbers, last year = 100, are housing 104, food 112, leisure 90. (a) Calculate the weighted index number. (b) State, with a reason, whether the overall cost of living has risen or fallen, and by roughly what percentage.Show answer
Answer: (a) 105 ((104 x 6 + 112 x 3 + 90 x 1)/10 = 1050/10 = 105); (b) risen, since 105 is greater than 100, by about 5%
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
The cost of a school trip was 60 pounds in 2022 and 69 pounds in 2024. (a) Calculate the simple index number for 2024, using 2022 = 100. (b) State the percentage change in cost.
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Three items have this year's simple index numbers, last year = 100, of 106, 118 and 95, with weights 4, 1 and 5. Calculate the weighted index number for the three items combined, and interpret your answer.
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A quantity's value was 200 in Year 1, 220 in Year 2, and 209 in Year 3. (a) Calculate the chain-base index number for Year 2 (based on Year 1) and for Year 3 (based on Year 2). (b) Use these to calculate the fixed-base index number for Year 3, using Year 1 as the base (= 100).
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the weighted and chain base index numbers worksheet pack - 18 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 29 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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