Weighted and Geometric Mean - Worksheets, Questions and Revision

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H22 Weighted and Geometric Mean

EDEXCEL 1ST0 · Calculator allowed · about 90 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
A talent show has three judges. Each judge gives a contestant a score out of 10, but the judges' scores are combined using weights that reflect each judge's experience:
JudgeScoreWeight
A91
B62
C53

The weighted mean score is defined as:

weighted mean = (sum of (score x weight)) / (sum of weights)
(a)Work out the sum of (score x weight) for the three judges.(2)
(b)Hence work out the weighted mean score.(2)
(Total for Question 1 is 4 marks)
2
For each statement about weighted means and geometric means, write down whether it is True or False.
(i)In a weighted mean, giving a data value a larger weight increases how much it influences the mean.(1)
(ii)The weights used in a weighted mean must always add up to 1 (or 100%).(1)
(iii)For a set of positive numbers that are not all equal, the geometric mean is always less than the arithmetic (simple) mean.(1)
(iv)The geometric mean can be used to average a data set containing negative numbers in exactly the same way as for positive numbers.(1)
(Total for Question 2 is 4 marks)
3
A market trader buys three types of fruit:
FruitPrice per kgQuantity bought (kg)
Apples2.40 pounds3
Plums3.60 pounds5
Pears1.80 pounds2
(a)Work out the total weight of fruit bought.(1)
(b)Work out the total amount spent, in pounds.(2)
(c)Hence work out the weighted mean price per kg paid by the trader, giving your answer to the nearest penny.(2)
(Total for Question 3 is 5 marks)
4
The geometric mean of two positive numbers p and q is defined as:

geometric mean = √p x q
(a)Find the geometric mean of 8 and 32.(2)
(b)Find the geometric mean of 5 and 20.(2)
(Total for Question 4 is 4 marks)
5
The geometric mean of three positive numbers a, b and c is defined as:

geometric mean = cube root of (a x b x c)
(a)Find the geometric mean of 2, 4 and 8.(2)
(b)Find the geometric mean of 5, 10 and 20.(2)
(Total for Question 5 is 4 marks)
6
A college course has three assessed components, each with a different weighting towards the final course percentage:
ComponentWeightingStudent's score (%)
Coursework20%84
Practical30%70
Final exam50%62
(a)Work out the simple (unweighted) mean of the student's three percentage scores.(2)
(b)Work out the student's weighted mean percentage for the course, using the weightings shown.(3)
(c)Explain why the weighted mean in part (b), rather than the simple mean in part (a), should be used as the student's overall course percentage.(1)
(Total for Question 6 is 6 marks)
7
A small business's revenue grew by 12% in Year 1 and by a further 75% in Year 2.
(a)Work out the overall growth multiplier for the two years combined (that is, final revenue divided by starting revenue).(2)
(b)Work out the arithmetic mean of the two percentage growth rates, 12% and 75%.(1)
(c)Use your answer to part (a) to work out the geometric mean annual growth rate: the single constant percentage growth rate per year that would give the same overall growth over the two years.(2)
(d)Explain why the geometric mean growth rate in part (c), not the arithmetic mean in part (b), gives the correct average annual growth rate.(1)
(Total for Question 7 is 6 marks)
8
Over three years, a company's annual profit changed as follows:
YearChange in profit
Year 1increased by 15%
Year 2decreased by 5%
Year 3increased by 25%
(a)Write down the growth multiplier for each of the three years.(1)
(b)Work out the overall growth multiplier for the three years combined, giving your answer correct to 3 significant figures.(2)
(c)Hence work out the geometric mean annual growth rate over the three years, giving your answer as a percentage correct to 3 significant figures.(3)
(d)A colleague suggests using the arithmetic mean of 15%, -5% and 25% instead. Explain why this would give a less accurate measure of the average annual growth rate.(1)
(Total for Question 8 is 7 marks)
9
A student's overall course grade is a weighted mean of four coursework components, weighted 10%, 15%, 25% and 50%. The student scored 68%, 74% and 80% on the first three components. Their overall weighted mean percentage for the whole course is 79.9%.
ComponentWeightingScore (%)
110%68
215%74
325%80
450%?
(a)Show that the total weighted contribution from the first three components is 37.9 (percentage points).(2)
(b)Work out the student's score on the fourth component (weighted 50%).(3)
(Total for Question 9 is 5 marks)
10
A secondary school is following the statistical enquiry cycle to find a single average GCSE point score across Year 11, combining results from three subjects with different numbers of students entered:
SubjectNumber of studentsMean point score
Maths2005.4
English1805.0
Science1504.6
(a)(Plan) Before combining these three subject means into one overall figure, state two pieces of information the school needs in order to calculate a valid combined average.(2)
(b)(Process) Calculate the simple (unweighted) mean of the three subject means.(2)
(c)(Process) Calculate the weighted mean point score across all Year 11 students, weighted by the number of students entered for each subject. Give your answer correct to 3 significant figures.(3)
(d)(Interpret) Explain why the weighted mean in part (c) is a more appropriate overall summary than the simple mean in part (b).(1)
(Total for Question 10 is 8 marks)
11
A restaurant chain has three branches. Each customer leaves a satisfaction rating out of 5. The mean ratings are: Branch X = 4.8, Branch Y = 4.2, Branch Z = 3.0. The bar chart shows the number of customers who left a rating at each branch.

The regional manager states: "The average rating across the three branches is (4.8+4.2+3.0)/3 = 4.0, so the chain is doing well."
0 20 40 60 80 100 120 140 160 180 200 Branch X Branch Y Branch Z Branch Number of customers
(a)Use the bar chart to work out the weighted mean satisfaction rating across all customers, weighting by the number of customers at each branch.(3)
(b)Explain why the regional manager's simple mean of 4.0 is misleading.(2)
(Total for Question 11 is 5 marks)
12
A price index for a basket of goods (Year 0 = 100) is recorded at the end of each of the next three years, as shown on the line graph.
0 20 40 60 80 100 120 140 160 180 200 100 125 150 180 Year 0 Year 1 Year 2 Year 3 Year Price index
(a)Use the graph to work out the percentage increase in the index from Year 0 to Year 1, from Year 1 to Year 2, and from Year 2 to Year 3.(3)
(b)Work out the overall percentage increase in the index from Year 0 to Year 3.(2)
(c)Use the geometric mean of the three yearly growth multipliers (1.25, 1.20 and 1.20) to find the average annual percentage increase in the index, giving your answer correct to 3 significant figures.(3)
(Total for Question 12 is 8 marks)
13
An investor splits money across three funds for one year:
FundProportion investedGrowth this year
Fund A40%+10%
Fund B35%+6%
Fund C25%-4% (a loss)
(a)Work out the weighted mean percentage growth of the whole portfolio for the year, weighting by the proportion invested in each fund.(3)
(b)Suppose the portfolio grows at this same weighted mean rate of 5.1% for 3 consecutive years. Work out the overall growth multiplier for the 3 years combined, giving your answer correct to 3 significant figures.(2)
(c)In part (a) a weighted arithmetic mean was the correct tool for combining the three funds' growth rates. Explain why, if instead you were finding the single average annual growth rate over several unequal years of growth, the geometric mean (not the arithmetic mean) of the yearly growth rates would be the correct tool to use.(2)
(Total for Question 13 is 7 marks)
14
Two students' working is shown below.

Student 1 is finding the geometric mean of 4, 6 and 9:
Step 1: 4 + 6 + 9 = 19
Step 2: 19 / 3 = 6.33
Step 3: geometric mean = 6.33

Student 2 is finding the weighted mean of the values 10, 20 and 30, with weights 2, 3 and 5:
Step 1: (10x2) + (20x3) + (30x5) = 20+60+150 = 230
Step 2: weighted mean = 230 / 3 = 76.7
(a)Identify the error in Student 1's method.(1)
(b)Work out the correct geometric mean of 4, 6 and 9.(2)
(c)Identify the error in Student 2's method.(1)
(d)Work out the correct weighted mean.(2)
(Total for Question 14 is 6 marks)
15
The geometric mean of the three positive numbers 5, 8 and x is 10.
(a)Show that 5 x 8 x x = 1000.(1)
(b)Hence find the value of x.(2)
(c)The geometric mean of two positive numbers, y and 18, is 12. Find the value of y.(2)
(Total for Question 15 is 5 marks)
Mark scheme · H22 Weighted and Geometric Mean

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15