Averages, medians and quantiles

Overview

Relevant quantile.

What you need to know

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Questions from this subtopic

Each one with the model answer underneath.

1. 1

   Weekly household incomes in a small sample are £240, £310, £360, £420, £480, £560 and £3,300. What are the mean and the median, and which better describes a typical household?

   Model answer

   The seven incomes total £5,670, so the mean is 5,670 divided by 7, which is £810. With seven ordered values the median is the fourth, £420. Six of the seven households earn less than the mean, because one very high income pulls it upwards. The median is the better description of a typical household, since it depends only on the position of the middle value and not on how large the extremes are.

2. 2

   A firm employs ten production workers on £24,000 a year and one manager on £180,000. What are the mean and median salary, and which figure would each side in a pay negotiation prefer to quote?

   Model answer

   Total pay is 10 multiplied by £24,000 plus £180,000, which is £420,000, so the mean is 420,000 divided by 11, which is £38,182 to the nearest pound. With eleven salaries the median is the sixth in order, which is £24,000. A union arguing that ordinary staff are underpaid would quote the median, since it reflects what almost everyone earns. An employer wanting to show generosity would quote the mean, which one large salary has lifted well above the pay of every production worker.

3. 3

   Why are official statistics on household income usually reported as medians rather than means?

   Model answer

   Income distributions are positively skewed: most households are bunched in the lower and middle ranges while a long thin tail of very high incomes stretches out above them. The mean is pulled up by that tail and is sensitive to a small number of extremely large incomes, so it lies above the income of most households and can move sharply when the top of the distribution changes. The median is the income of the household in the middle and is unaffected by how far above it the top incomes reach, so it tracks the experience of an ordinary household far more reliably.

4. 4

   Eight employees earn hourly wages of £19, £21, £22, £24, £26, £28, £31 and £45. Calculate the median, the quartiles and the interquartile range, and state what the interquartile range shows that the range does not.

   Model answer

   With an even number of values the median is the mean of the fourth and fifth: (24 + 26) divided by 2, which is £25. The lower quartile is the median of the bottom four, (21 + 22) divided by 2, which is £21.50, and the upper quartile is the median of the top four, (28 + 31) divided by 2, which is £29.50. The interquartile range is 29.50 - 21.50 = £8. The full range, £45 - £19 = £26, is determined entirely by the two most extreme wages, whereas the interquartile range measures the spread of the middle half of employees and is not distorted by the single £45 earner.

5. 5

   The five income quintiles in an economy receive 7%, 12%, 17%, 23% and 41% of total income. How many times the bottom quintile's share is the top quintile's, and what would a fall in that ratio signify?

   Model answer

   41 divided by 7 is 5.86, so the richest fifth of households receives nearly six times as much income as the poorest fifth. A fall in the ratio means income is more evenly distributed, which could come from progressive taxes and benefits raising the bottom share, from faster wage growth at the lower end, or from a fall in top incomes. The ratio ignores everything between the two extremes, so two economies with the same ratio can differ considerably in the middle.

6. 6

   Annual earnings at the 90th percentile are £68,000 and at the 10th percentile £12,500. What is the ratio between them, and what does a percentile mean?

   Model answer

   68,000 divided by 12,500 is 5.44, so a worker at the 90th percentile earns nearly five and a half times one at the 10th. A percentile is a cut-point dividing an ordered distribution into hundredths, so the 10th percentile is the value below which a tenth of earners fall and the 90th is the value below which nine-tenths fall. Using these points rather than the very top and bottom avoids letting a handful of extreme observations drive the measure of inequality.

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