harmonic mean

if one piece of data is an extreme outlier it is recommended to use the harmonic mean to more appropriately represent an 'average' for the data set (but there are problems if one of the items in the data set is zero...)

try this for some simple data sets: compare the arithmetical mean with the harmonic mean where one of the numbers is large or small compared with the rest

for just two numbers, the arithmetical, geometric and harmonic means (along with the root mean square) can be represented by the following lengths:

establish that these lengths are the various means
















the harmonic mean of two lengths occurs in the crossed ladders problem - for the height at which two crossed ladders 'meet' (i.e. 'h' from 'A' and 'B')

decreasingly speedy

two part journeys (legs)
average speed over both legs

trapezium property

construct any trapezium (a trapezoid in the US) and draw the two diagonals

at the point where the diagonals meet, construct a line parallel to the two parallel sides of the trapezium

what do you notice?
can you prove it?




this is an old problem, utilised in Michael de Villiers' admirable work on proof and geometry

use similar triangles to show that the length is the harmonic mean of 'a' and 'b'