Weighted and Chain Base Index Numbers - Worksheets, Questions and Revision

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HIGHER

H24 Weighted and Chain Base Index Numbers

EDEXCEL 1ST0 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A campsite tracks the price of a standard pitch fee each year, using 2020 as the base year. The index number for the pitch fee in a given year is defined as:

index number = (price in that year / price in the base year) x 100

Year	Price of pitch fee (£)
2020 (base year)	25.00
2021	27.50
2022	30.00

(a)Show that the index number for the pitch fee in 2021 is 110.(2)

(b)Work out the index number for the pitch fee in 2022.(2)

(c)Write down what the index number of 120 tells you about the price of the pitch fee in 2022, compared with the base year.(1)

(Total for Question 1 is 5 marks)

For each statement about index numbers, write down whether it is True or False.

(i)The base year (or base period) of an index number always has an index value of 100.(1)

(ii)If a quantity has decreased since the base year, its index number will be a negative number.(1)

(iii)If a price index rises from 100 to 250, the price has increased by 250%.(1)

(iv)Index numbers allow you to compare how a quantity has changed over time without needing to know its actual original value.(1)

(Total for Question 2 is 4 marks)

A furniture shop uses 2019 as the base year for the price of a particular sofa, when it cost £480. In 2023, the index number for the sofa's price is 137.5.

(a)Work out the price of the sofa in 2023.(2)

(b)Write down the percentage increase in the price of the sofa from 2019 to 2023.(1)

(c)A newspaper reports 'the cost of living has increased by 37.5% since 2019', using only this sofa's price index as evidence. Comment on whether this claim is fully justified.(2)

(Total for Question 3 is 5 marks)

A gym records its total number of members at the end of each month:

Month	Members
January	240
February	258
March	300
April	276

A chain-base index number compares each month with the month immediately before it:

chain-base index number = (this month's members / previous month's members) x 100

(a)Show that the chain-base index number for February (compared with January) is 107.5.(2)

(b)Work out the chain-base index number for March (compared with February), giving your answer correct to 1 decimal place.(2)

(c)Work out the chain-base index number for April (compared with March).(2)

(d)Write down which month had fewer members than the month before it.(1)

(Total for Question 4 is 7 marks)

Two students are working with index numbers.

Student A is finding a chain-base index number for a shop's sales, using Year 3 sales of £84,000 and Year 2 sales of £80,000:
Step 1: 84,000 / 80,000 = 1.05
Step 2: chain-base index number = 1.05

Student B is given a fixed-base index number (Year 1 = 100) for the same shop's sales. The fixed-base index number for Year 4 is 118. Student B says: 'Since the index is 118, sales have increased by 118% since Year 1.'

(a)Identify Student A's error, and give the correct chain-base index number.(2)

(b)Identify Student B's error, and state the correct percentage increase in sales from Year 1 to Year 4.(2)

(Total for Question 5 is 4 marks)

A statistician builds a simple household 'cost of living index' from three categories of spending. Each category has its own price sub-index for this year (base year = 100), and a weight reflecting how much of a typical household's budget goes on that category:

Category	Weight	Sub-index this year
Housing	5	130
Food	3	110
Transport	2	120

The weighted index number is defined as:

weighted index = (sum of (weight x sub-index)) / (sum of weights)

(a)Work out the sum of (weight x sub-index) for the three categories.(2)

(b)Hence work out the weighted cost of living index for this year.(2)

(c)Write down what the weighted index number of 122 tells you about the cost of living for a typical household.(1)

(d)The statistician originally forgot to use the weights, and instead worked out the simple (unweighted) mean of the three sub-indices. Work out this unweighted mean, and explain why the weighted index in part (b) is more appropriate for measuring a typical household's cost of living.(3)

(Total for Question 6 is 8 marks)

The pie chart shows how a typical household splits its monthly spending across four categories. These percentages are used as the weights for a weighted price index. The table shows the price sub-index for each category this year (base year = 100).

Category	Sub-index this year
Housing	112
Food	106
Transport	118
Leisure	102

(a)Use the pie chart to write down the weight (as a percentage) for Transport.(1)

(b)Use the percentages from the pie chart as weights to work out the weighted price index for this household's typical spending, giving your answer correct to 1 decimal place.(3)

(c)A different household spends a much larger share of its budget on transport than shown in the pie chart (and correspondingly less on the other categories). Given that transport has the highest sub-index (118) of the four categories, explain, without recalculating, whether this household's overall weighted index is likely to be higher or lower than 110.2.(2)

(Total for Question 7 is 6 marks)

A cinema records the total number of tickets sold in three consecutive weeks:

Week	Tickets sold
Week 1	320
Week 2	368
Week 3	345

Week 1 is used as the fixed base period for a fixed-base index number (index = 100).

(a)Show that the fixed-base index number for Week 2 is 115.(2)

(b)Work out the fixed-base index number for Week 3, giving your answer correct to 1 decimal place.(2)

(c)Use the raw ticket figures (not your fixed-base index numbers) to work out the chain-base index number for Week 3, compared with Week 2.(2)

(d)Show that combining the chain-base index number for Week 2 (115, from part a) with the chain-base index number for Week 3 (93.75, from part c) gives the same fixed-base index number for Week 3 as in part (b), without returning to the raw figures.(3)

(Total for Question 8 is 9 marks)

A company publishes fixed-base index numbers for its total annual revenue (2018 = 100):

Year	Fixed-base index (2018 = 100)
2018	100
2019	120
2020	108
2021	135

(a)Work out the chain-base index number for 2019, compared with 2018.(2)

(b)Work out the chain-base index number for 2020, compared with 2019.(2)

(c)Work out the chain-base index number for 2021, compared with 2020.(2)

(d)Write down what the chain-base index number found in part (b) tells you about the company's revenue.(1)

(Total for Question 9 is 7 marks)

A local council wants to build its own weighted 'Local High Street Price Index' to track the cost of living for residents, rather than relying only on the national Consumer Price Index (CPI). It plans to follow the statistical enquiry cycle.

(a)(Plan) State two decisions the council must make before it can construct its weighted local price index.(2)

(b)(Collect) The council decides to collect price data quarterly from a sample of local shops. Give one reason why the council should collect prices from more than one shop for each category, rather than from just one shop.(1)

(c)(Process) The council collects the following weights and sub-indices for its first year:

Category	Weight	Sub-index
Housing	6	108
Food	3	115
Leisure	1	122

Calculate the local weighted price index.(3)

(d)(Interpret) The national CPI for the same period is 106. Compare the council's local index to the national CPI, and suggest one reason why the local figures might differ.(2)

(Total for Question 10 is 8 marks)

The line graph shows the fixed-base index number (2020 = 100) for the average monthly rent of a one-bedroom flat in a town, for the years 2020 to 2023.

(a)Use the graph to write down the fixed-base index number for 2022.(1)

(b)Use the graph to work out the percentage increase in rent from 2020 to 2023.(2)

(c)Use the graph to work out the chain-base index number for 2023, compared with 2022.(2)

(d)A tenant notices that the index rose by 4 points from 2020 to 2021 (100 to 104), but by 14 points from 2022 to 2023 (112 to 126). The tenant claims: 'Rent rose about three and a half times as fast between 2022 and 2023 as it did between 2020 and 2021, since 14 is three and a half times 4.' Use your chain-base result from part (c), together with the percentage increase from 2020 to 2021, to comment on whether comparing raw index-point rises like this gives an accurate measure of how much faster rent was rising.(3)

(Total for Question 11 is 8 marks)

The price of a unit of energy was £40 in 2015, £52 in 2020, and £78 in 2024.

(a)Work out the index number for 2024, using 2015 as the base year (2015 = 100).(2)

(b)Work out the index number for 2024, using 2020 as the base year (2020 = 100).(2)

(c)The actual prices have not changed. Explain why the two index numbers in parts (a) and (b) are different.(2)

(d)A journalist argues: 'The choice of base year for an index number doesn't matter, since the trend is the same either way.' Evaluate this claim, using your answers to parts (a) and (b), and suggest one situation where the choice of base year could matter in practice.(2)

(Total for Question 12 is 8 marks)

A weighted price index has four categories:

Category	Weight	Sub-index
A	2	105
B	3	118
C	4	w
D	1	130

The overall weighted index for all four categories is 117.

(a)Show that the combined weighted contribution of categories A, B and D is 694.(2)

(b)Given that the overall weighted index for all four categories is 117, find the value of w, the missing sub-index for category C.(3)

(c)Explain why knowing that category C's sub-index is 119 does not, by itself, tell you the actual price of category C's goods this year.(1)

(Total for Question 13 is 6 marks)

A weighted index has two categories. Category 1 has a sub-index of 108 and a weight of w. Category 2 has a sub-index of 124 and a weight of (10-w). The overall weighted index for the two categories combined is 118.

(a)Show that 108w + 124(10-w) = 1180.(2)

(b)Solve the equation to find w, and hence state the weight given to Category 2.(3)

(Total for Question 14 is 5 marks)

A national statistics office publishes a fixed-base Consumer Price Index (CPI), with 2019 = 100:

Year	Fixed-base CPI (2019 = 100)
2019	100.0
2020	105.0
2021	115.5
2022	121.3
2023	127.4

(a)Work out the chain-base index number for 2021, compared with 2020, giving your answer correct to 1 decimal place.(2)

(b)Work out the chain-base index number for 2022, compared with 2021, giving your answer correct to 1 decimal place.(2)

(c)Work out the chain-base index number for 2023, compared with 2022, giving your answer correct to 1 decimal place.(2)

(d)A commentator claims: 'The fixed-base index rose by a broadly similar number of points every year (100.0 to 105.0, 105.0 to 115.5, 115.5 to 121.3, 121.3 to 127.4), so inflation (the year-on-year percentage rate) must have been roughly the same every year.' Work out the chain-base index number for 2020, compared with 2019, and use it together with your answers to parts (a) to (c) to evaluate the commentator's claim.(3)

(Total for Question 15 is 9 marks)

Mark scheme · H24 Weighted and Chain Base Index Numbers

Question 1

(a) M1 27.50 / 25.00, oe (correct ratio)
(a) A1 110 cso
(a) Answer: 110
(b) M1 30.00 / 25.00, oe
(b) A1 120 cao
(b) Answer: 120
(c) B1 the price in 2022 is 20% higher than the price in the base year (2020), oe; accept 'the price has increased by 20% since 2020'
(c) Answer: The pitch fee in 2022 was 20% higher than in 2020 (the base year).

Question 2

(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 False cao; correct increase is 150%
(iii) Answer: False
(iv) B1 True cao
(iv) Answer: True

Question 3

(a) M1 480 x (137.5/100), oe
(a) A1 660 cao
(a) Answer: £660
(b) B1 37.5 (%) cao
(b) Answer: 37.5%
(c) B1 the claim is not fully justified because it is based on the price of a single item (one sofa), oe
(c) B1 the general cost of living covers many different goods and services, which may have changed in price by different amounts, so a single item's index cannot represent overall cost-of-living change; a weighted index across many goods would be needed instead, oe
(c) Answer: The claim is not fully justified: it is based on the price of only one item, but the cost of living covers many goods and services that may have changed by different amounts. An overall cost-of-living index would need to combine (weight) many items, not rely on a single sofa's price.

Question 4

(a) M1 258/240, oe
(a) A1 107.5 cso
(a) Answer: 107.5
(b) M1 300/258, oe
(b) A1 116.3 awrt (from 116.279...)
(b) Answer: 116.3
(c) M1 276/300, oe
(c) A1 92 cao
(c) Answer: 92
(d) B1 April cao, ft from (c) (chain-base index below 100)
(d) Answer: April

Question 5

(a) B1 Student A has not multiplied the ratio by 100 (an index number is expressed as a value around 100, not as a decimal ratio), oe
(a) B1 105 cao
(a) Answer: Student A forgot to multiply by 100; the correct chain-base index number is 105.
(b) B1 an index of 118 means sales are 118% of the Year 1 value (a growth multiplier of 1.18), so the increase is 118-100=18%, not 118%, oe
(b) B1 18 (%) cao
(b) Answer: Student B should have subtracted 100 from the index; the correct percentage increase is 18%.

Question 6

(a) M1 5x130, 3x110, 2x120 (=650, 330, 240), oe
(a) A1 1220 cao
(a) Answer: 1220
(b) M1 1220 / (5+3+2), ft from (a)
(b) A1 122 cao
(b) Answer: 122
(c) B1 the overall cost of living has risen by 22% since the base year, taking into account how much of the budget goes on each category, oe
(c) Answer: The overall cost of living has risen by 22% since the base year, once the different amounts spent on housing, food and transport are taken into account.
(d) M1 (130+110+120)/3, oe
(d) A1 120 cao
(d) B1 housing takes up the largest share of a typical household's spending (weight 5, the highest) and also rose the most (sub-index 130, the highest), so the weighted index (122) correctly reflects its greater importance, unlike the unweighted mean (120) which treats all three categories as equally important, oe
(d) Answer: Unweighted mean = 120. The weighted index (122) is more appropriate because it reflects that housing takes up the largest share of typical spending and rose the most, whereas the unweighted mean treats housing, food and transport as equally important, which does not match real spending patterns.

Question 7

(a) B1 20 (%) cao
(a) Answer: 20%
(b) M1 40x112, 25x106, 20x118, 15x102 (=4480, 2650, 2360, 1530), oe
(b) M1 (4480+2650+2360+1530)/100, dep
(b) A1 110.2 cao
(b) Answer: 110.2
(c) B1 higher, oe
(c) B1 because more weight is being placed on transport, which has the highest sub-index (118) of the four categories, this pulls the weighted average up above 110.2, oe
(c) Answer: Higher. Giving transport a greater weight pulls the weighted index towards transport's sub-index (118), which is the highest of the four, so the overall weighted index would rise above 110.2.

Question 8

(a) M1 368/320, oe
(a) A1 115 cso
(a) Answer: 115
(b) M1 345/320, oe
(b) A1 107.8 awrt (from 107.8125)
(b) Answer: 107.8
(c) M1 345/368, oe
(c) A1 93.75 cao
(c) Answer: 93.75
(d) M1 (115/100) x (93.75/100), ft from (a) and (c)
(d) M1 ...x100, dep
(d) A1 107.8125 (=107.8, 1 dp) cso, matching part (b)
(d) Answer: 1.15 x 0.9375 x 100 = 107.8125 (107.8 to 1 dp), matching part (b).

Question 9

(a) M1 120/100, oe
(a) A1 120 cao
(a) Answer: 120
(b) M1 108/120, oe
(b) A1 90 cao
(b) Answer: 90
(c) M1 135/108, oe
(c) A1 125 cao
(c) Answer: 125
(d) B1 revenue fell by 10% from 2019 to 2020, ft from (b)
(d) Answer: The company's revenue decreased by 10% from 2019 to 2020.

Question 10

(a) B1 which categories of goods and services to include (the 'basket'), oe
(a) B1 what weight to give each category, based on typical local spending patterns, oe; accept 'which base period to use'
(a) Answer: Which categories of goods and services to include in the basket, and what weight to give each category based on typical local spending.
(b) B1 to get a more representative typical price for each category, reducing the effect of any one shop's unusually high or low price, oe
(b) Answer: Using more than one shop gives a more representative average price for each category, rather than being skewed by one shop's unusually high or low prices.
(c) M1 6x108, 3x115, 1x122 (=648, 345, 122), oe
(c) M1 (648+345+122)/(6+3+1), dep
(c) A1 111.5 cao
(c) Answer: 111.5
(d) B1 the local cost of living has risen by more (11.5%) than the national average (6%), oe
(d) B1 a plausible reason, e.g. local housing costs (which carry the largest local weight, 6 out of 10) may have risen faster than the national average, or the local basket/weights differ from the national CPI's, oe
(d) Answer: The local index (111.5) shows a bigger rise than the national CPI (106): 11.5% locally compared with 6% nationally. This could be because local housing costs, which carry the largest weight (6 out of 10) in the local index, have risen faster than the national average, or because the local basket and weights differ from the national CPI's.

Question 11

(a) B1 112 cao
(a) Answer: 112
(b) M1 (126-100)/100, oe
(b) A1 26 (%) cao
(b) Answer: 26%
(c) M1 126/112, oe
(c) A1 112.5 cao
(c) Answer: 112.5
(d) M1 (104-100)/100, oe, to find the 2020-2021 percentage increase (=4%)
(d) B1 12.5 (%), ft from (c) (percentage increase 2022 to 2023)
(d) B1 comparing the percentage increases (12.5%/4% = 3.125) instead of the raw point rises (14/4=3.5) gives a different, more accurate ratio, because the same number of index points represents a smaller percentage change when the index level itself is already higher, so raw point differences should not be used to compare speeds of change; percentage/chain-base changes should be used instead, oe
(d) Answer: 2020 to 2021: 4% increase. 2022 to 2023: 12.5% increase (from part c). The true ratio is 12.5/4=3.125, not 3.5, so comparing raw index-point rises overstates the comparison; percentage (chain-base) changes give the accurate measure, since the same point-rise represents a smaller percentage change when it is measured from a higher index level.

Question 12

(a) M1 78/40, oe
(a) A1 195 cao
(a) Answer: 195
(b) M1 78/52, oe
(b) A1 150 cao
(b) Answer: 150
(c) B1 the two index numbers are each expressed relative to a different base period (2015 in (a), 2020 in (b)), oe
(c) B1 an index number only measures change relative to whichever period is set as the base (=100), so the same real prices give different numerical index values when a different base period is chosen, oe
(c) Answer: The index numbers are calculated relative to different base years (2015 and 2020), and an index number only shows change relative to its own base period; since the base prices differ (£40 vs £52), the resulting index values differ even though the actual prices have not changed.
(d) B1 the direction of the trend (an increase) is the same either way, but the reported size of the change is very different (a 95% increase vs a 50% increase), which could give a misleading impression of the scale of change, oe
(d) B1 a valid situation, e.g. if the chosen base period itself had an unusually high or low price (such as a one-off price spike or crash), the resulting index numbers could distort how 'normal' subsequent changes appear, oe
(d) Answer: The claim is only partly true: both index numbers show an increase, but the reported percentage increase is very different (95% vs 50%), which could mislead readers about the scale of the change. The choice of base year matters in practice, for example if the base period itself had an unusually high or low price (such as a one-off shortage or price crash), which would distort how later changes appear.

Question 13

(a) M1 2x105, 3x118, 1x130 (=210, 354, 130), oe
(a) A1 694 cso
(a) Answer: 694
(b) M1 117 x (2+3+4+1) (=1170), ft from (a)
(b) M1 1170 - 694 (=476), dep
(b) A1 w=119 cao
(b) Answer: w = 119
(c) B1 a sub-index only shows the price relative to the base year (=100); you would also need to know the actual base-year price of category C to work out the actual current price, oe
(c) Answer: A sub-index of 119 only tells you the price is 19% higher than in the base year; without knowing the actual base-year price, you cannot work out the actual current price.

Question 14

(a) M1 weighted index = (108w+124(10-w)) / 10 = 118, oe
(a) A1 108w+124(10-w) = 1180 cso, from multiplying both sides by 10
(a) Answer: 108w + 124(10-w) = 1180
(b) M1 108w+1240-124w=1180, oe (expands the bracket)
(b) M1 -16w=-60, dep (collects terms)
(b) A1 w=3.75, so Category 2's weight = 6.25 cao
(b) Answer: w = 3.75 (Category 1's weight); Category 2's weight = 6.25