Weighted and Geometric Mean
A weighted mean gives different values in a data set different levels of importance, or weights, rather than treating them equally, found as the sum of value x weight divided by the sum of the weights. A geometric mean multiplies n values together and takes the nth root, and is used to correctly average ratios or growth rates. Both appear on GCSE Statistics Higher tier.
Before you start
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Method
- For a weighted mean, multiply each value by its weight, add up all the value x weight products, then divide by the total of the weights.
- Weighted mean = sum of (value x weight) / sum of weights.
- For a geometric mean of n values, multiply all n values together to get one product.
- Take the nth root of that product: geometric mean = product^(1/n).
- Use a geometric mean instead of an arithmetic mean when averaging ratios, growth rates, or index numbers over several time periods, since it correctly averages multiplicative change.
- Check the answer is sensible: for the same positive data, the geometric mean is always less than or equal to the arithmetic mean.
Worked example
A student's end-of-year grade is made up of three assessments: coursework (weight 2) scoring 65, a mock exam (weight 3) scoring 72, and a final exam (weight 5) scoring 80. Calculate the weighted mean grade.
- Multiply each score by its weight: coursework 65 x 2 = 130, mock exam 72 x 3 = 216, final exam 80 x 5 = 400.
- Add the products together: 130 + 216 + 400 = 746.
- Add the weights together: 2 + 3 + 5 = 10.
- Divide the total of the products by the total of the weights: weighted mean = 746 / 10 = 74.6.
- Final answer: the weighted mean grade is 74.6.
Practice questions
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Q1Find the weighted mean of 4 (weight 1) and 10 (weight 3).Show answer
Answer: 8.5 ((4 x 1 + 10 x 3)/4 = 34/4 = 8.5)
Q2A cafe's average daily takings are weighted by the number of days open at each level: 200 pounds (3 days) and 300 pounds (5 days). Find the weighted mean daily takings.Show answer
Answer: 262.50 pounds ((200 x 3 + 300 x 5)/8 = 2100/8 = 262.5)
Q3Find the geometric mean of 4 and 9.Show answer
Answer: 6 (sqrt(4 x 9) = sqrt(36) = 6)
Q4Find the geometric mean of 2, 4 and 8.Show answer
Answer: 4 (product = 2 x 4 x 8 = 64; 64^(1/3) = 4, since 4 x 4 x 4 = 64)
Q5In a survey, exam scores are weighted by number of candidates: Grade A average 88 (120 candidates), Grade B average 74 (200 candidates), Grade C average 60 (80 candidates). Calculate the weighted mean score across all candidates.Show answer
Answer: 75.4 ((88 x 120 + 74 x 200 + 60 x 80)/400 = 30160/400 = 75.4)
Q6A company's sales grew by multipliers of 1.20, 1.50 and 1.10 over three years. Calculate the geometric mean growth multiplier, to 3 significant figures, and state the approximate average percentage growth rate per year.Show answer
Answer: About 1.26 (product = 1.20 x 1.50 x 1.10 = 1.98; 1.98^(1/3) = 1.2557..., which rounds to 1.26); average growth of about 25.6% per year
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A student's coursework score is 56 (weight 1) and exam score is 82 (weight 4). Calculate the weighted mean score.
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Find the geometric mean of 3, 9 and 27, giving your answer as an exact value.
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A portfolio's returns over three years were multipliers of 1.10, 0.95 and 1.20. (a) Calculate the product of the three multipliers. (b) Calculate the geometric mean multiplier, to 3 significant figures. (c) State the average annual percentage return this implies.
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Free printable worksheet
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This topic is chapter 6 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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