central tendency

students are quickly (i.e. not given enough time to count them) shown three flocks of birds

a powerpoint is here

they record how many they think there are in each flock

they are told there are not a neat number of tens (otherwise they just guess decade numbers) and it is a competition to see who can guess the best...

student's guesses are recorded on the board (maybe they write these down as well)
then ask them to consider what an 'average' value might be for this set - a representative number, standing for the whole set
" if I wanted to tell someone one number that was, mostly, around what people guessed - what do you think I could do?"

these (student chosen) values can be compared with the actual numbers



there are several more 'averages' than are normally considered

there may be some value in introducing this wider variety before homing in on the more regularly used values

some discussion of which value is 'best' (advantages/disadvantages) might follow the calculations

the geometric mean, as with the median, is less affected by outlier values

for estimating values (e.g. the size of a crowd) it is supposedly helpful to multiply the lowest and highest estimates (that people guess for the size) and find the geometric mean of the two

for two numbers it can be established algebraically that the mean (arithmetical) is more than (or equal to) the geometric mean

start with (a - b)^2  >  0