Mathematics and statistics: TestAS practice testlets

Incomes are one of the most important and most frequently misunderstood economic statistics. The two common measures of the centre of a distribution, the arithmetic mean and the median, can differ strongly for incomes. The mean is the sum of all values divided by their number. The median is the middle value when all values are sorted: half of the observations lie below it and half above. For an even number of values, the median is the mean of the two middle values.

Income distributions are skewed to the right: most people have moderate incomes, while a few have very high ones. These few high values pull the mean upwards, but they hardly affect the median. The mean is therefore usually well above the median, and the majority of people earn less than the average. For this reason, statistical offices often report the median income, and the median is described as robust against outliers. When a billionaire moves into a small town, the town's mean income rises dramatically, while the median barely changes.

A measure of the centre alone does not describe a distribution. Two towns can have the same median income although incomes in one are all close to it and in the other they vary widely. Measures of spread capture this difference. The range is the difference between the largest and the smallest value; it depends only on two observations and is very sensitive to outliers. The variance is the mean of the squared deviations from the mean, and the standard deviation is its square root, which has the same unit as the data. The coefficient of variation divides the standard deviation by the mean and makes the spread of data with different units or levels comparable.

mean = Σx ÷ n variance = Σ(x − mean)² ÷ n standard deviation = √variance coefficient of variation = standard deviation ÷ mean

The formula above divides by n and describes the data themselves. When a sample is used to estimate the variance of a larger population, statisticians divide by n − 1 instead, which gives a slightly larger value.

The table shows the monthly net incomes of the five employees of a small architecture office, including the owner.

Economic news constantly reports changes, and two different ideas are easily confused. A percentage change describes how much a value has changed relative to its starting level. A change in percentage points describes the simple difference between two values that are themselves percentages. If a central bank raises its interest rate from 2% to 3%, the rate has risen by one percentage point, but by 50 percent, because the new rate is one and a half times the old one. Both statements are correct; only the words must match the calculation.

percentage change = (new value − old value) ÷ old value × 100

Percentage changes are not symmetric. If a price rises by 20% and then falls by 20%, it does not return to its starting level: 100 becomes 120, and 20% of 120 is 24, so the price ends at 96, 4% lower than at the start. The reason is that the second change is calculated on a larger base. Likewise, a share price that falls by 50% must rise by 100% to recover.

To reverse a change, one must therefore use the new value as the base. A price of 119 euros that includes 19% value added tax (VAT) contains a net price of 100 euros, since 119 ÷ 1.19 = 100. The tax is 19 euros, which is 19% of the net price but only about 16% of the gross price. A common error is to subtract 19% of the gross price, which gives a net price that is too low.

Changes should also be read against their base. A party whose vote share rises from 2% to 4% has doubled its support, a 100% increase, but it has gained only two percentage points. A tax whose revenue grows by 50% may still be small in absolute terms. Headlines often choose whichever form sounds more dramatic, so careful readers convert between them.

Percentage changes can also be combined. Two successive increases of 10% raise a value by 21%, not 20%, because the second increase is applied to the already higher value. For the same reason, a value that doubles has risen by 100%, and a value that falls to half has fallen by 50%, although both describe a change by a factor of two. When comparing changes, it is therefore always important to ask which base each percentage refers to.