Averages: means and medians of incomes

Press, 26 April 1979, Page 20

Averages: means and medians of incomes

"Ihe reference to the “average wage’’ in an article in “The Press” on April 20 has prompted a query from a reader, John Christie. The article, on “Wages and Prices.” by Norman Macbeth, was the first of a series of four on this subject.

Mr Christie writes: “Norman Macbeth talks about an ‘average wage’ of $155. Many people would be quite surprised at this high figure, and would ask themselves where they can go to get that wage. "No wonder, because the average person in New Zealand does not receive $155, but something less than that. A simple example: Suppose 10 people receive $lOO a week, six people $2OO, and one person $5OO. The average wage here is $2700 divided by 17, which is nearly $159. “But the average person is the ninth person in the list, who receives only $lOO. The average person does not receive the average wage. “The same applies in the real world. My guess would be that the average person takes home about $llO. Does Mr Macbeth know the actual figure, which I believe is called the median wage?” Mr Macbeth replies: “No. I don’t know what the median wage is; nor does the Department of Labour, whose six-monthly survey of earnings I quoted. The method used by the department is designed for the calculation of the average, but not the median, wage.

“The point made by Mr Christie is valid, but he is predicating in his hypothetical example a distribution of incomes much more ‘skewed’ than is apparent in contemporary data. “The ‘average income,’ as Mr Christie points out, is the total of incomes divided by the number of income earners. The statisticians call this measure the ‘arithmetic mean.’ The arithmetic mean is normally close to the median, which is the income received by the person halfway up (or down) the list of all income earners.

“One series which permits the (approximate) calculation of the median as well as the arithmetic mean is provided by the Department of Statistics, on information derived from tax returns. Headed ‘Total income from all sources’ (and hence different from the Labour Department’s survey, based on what employers pay their staffs), the Statistics Department’s tabulation shows the number of taxpayers in each income bracket. “In 1976-77 (latest year available), 1,710,280 taxpayers filed returns showing total incomes of $9,379,400,000. The arithmetic mean (average) income was therefore $5484 — which may be compared with the following figures for average w'eekly earnings as revealed by the Labour

Department’s six-monthly surveys:—

“April, 1976: $102.84 a week (equals $5348 per annum). “October, 1976: $109.75 ($5707). “April, 1977: $118.16 ($6144). “From the Statistics Department’s tabulation we can calculate (roughly) the income of the ‘median’ taxpayer (the 855,1395 th out of 1,710.280): in the $4BOO-$5199 bracket — probably slightly under $5OOO p.a. The income of the median taxpayer in that year was therefore within 10 per cent of the arithmetic mean ($5484). “In another series, the Government Statistician calculates each year both the arithmetic mean and the median income for 14 categories of self-employed people (farmers, entrepreneurs, professional people). In 1976-77 the median income of builders was $7300, or 25 per cent below the arithmetic mean for builders of $9700. But the median income of a road transport operator ($5900) was within 3.3 per cent of the arithmetic mean ($6100). The average (I mean arithmetic mean) by which the median fell short of the arithmetic mean in all 14 categories was just under 16 per cent.

“Take 10 per cent off the ‘average’ $155 and you get $139.50 for the median; take 16 per cent off and you get $130.20. Somewhere between $l3O and $l4O, I suggest, is a more plausible median than Mr Christie’s $110.”