Q: At parents’ evening recently I was talking to a parent about helping his daughter learn the difference between the three averages: mean, median and mode. During the conversation I mentioned that, in the media, reporters never say what kind of average they are quoting. He agreed, saying that of course there are three mean averages as well as the mode and median average. I thought there was only one mean average. What are the other two? I didn’t like to ask!
A: The three means that he was probably talking about are the arithmetic, harmonic and geometric mean averages. In fact, the Greeks described 10 different means. The arithmetic-geometric mean has a connection with modular equations and has, in recent times, been used to compute pi to millions of decimal places.
The mean average with which we are most familiar is the “arithmetic mean”, and is the one many students have difficulty remembering. Here is a revision for those readers who are not sure of the calculation. The arithmetic mean is where the values from our data set are added together and the total is divided by the sample size. So the arithmetic mean of a set of heights of a sample of five pupils would be: (1.40 + 1.28 + 1.60 + 1.58 + 1.40)5= 7.265
which is approximately 1.45m.
The harmonic mean (in old texts referred to as the subcontrary mean) is the reciprocal of the sum of the reciprocals of the data values divided by the sample size (what a mouthful - aren’t we glad we have algebra to translate for us). So, for a sample size of four, with data values a, b, c, d, the harmonic mean is calculated using the following formula:
1((1a + 1b + 1c + 1d)4) = 4(1a + 1b + 1c + 1d)
When there are just two data values (x, y) the harmonic mean can be arranged to the more simple formula of
2xy(x+y)
Imagine a path that forms a square with sides of 2 miles each and with a central lake that restricts you to walking along the perimeter. You walk at a different average speeds along each side, say 1mph, 2mph, 4mph, and 1mph.
Your average speed is not the arithmetic mean of (1 + 2 + 4 + 1) 4 = 2mph - as we can see if we look at the time taken for the 8-mile round trip. The times taken along each side are, respectively: 2 hours + 1 hour + 0.5 hour + 2 hours = 5.5 hours. So the average speed for the whole journey is 8 5.5 (distancetime), approximately 1.45mph. As the distances travelled at each speed are the same we can use the harmonic mean to calculate the overall average speed:
4(11+12+14+11) = 42.75 = 1.45mph
The harmonic mean has its origin in music and is used for average rates of change in economic and population sizes, in ecology, resistors connected in parallel in electric circuits, in some cases for calculating students’
final exam and assessment averages, and also in geometric relationships.
An example of an application in a geometric relationship is for the height of a triangle with an inscribed square, with the base of the square sitting on the triangle’s base. Finding half the harmonic mean of the height (h) and base (b) of the triangle gives the length of the side (s) of the inscribed square.
The geometric mean uses multiplication rather than addition of the data values, and is the most useful mean for summarising very skewed data and ratios. As a rule, when you find that a multiplier is involved, such as interest rates, this is the mean to use. However, the geometric mean can’t be used where there are negative values or zero values: it’s calculated by working out the product of the values to the nth root. So, for a sample with five data values, the geometric mean is calculated as: (a x b x c x d x e) 15
An example would be to find the average growth of money invested over three years on variable interest. The first year it earns 4.9%, the second 5.3% and the third year 6.3%. The average rate of return would be:
(1.053 x 1.049 x 1.063)13 = 1.0549% The annual percentage rate of return is about 5.5%.
Other uses of the geometric mean are in wastewater dischargers, and regulators of beach areas for swimming, or the metabolism of drugs.
For more information and activities:
Harmonic mean
* http:microcosmos.uchicago.eduptolemyfeatures.html
* http:homepages.ihug.co.nzlcablesharmonic_means.htm
* www.cse.unsw.edu.auteachadmininfoharmonic3.html
Geometric mean
* www.buzzardsbay.orggeomean.htm
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
www.nesta.org.uk
Email your questions to Mathagony Aunt at teacher@tes.co.uk Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX
