square sampling

Glyn Davies provided a resource like this in his book (1993) 'Practical Data Handling Book B'

the task illustrates (intends to anyway...) how sampling a field of dandelions
ruled off into squares (or sampled using a quadrat)
can be used to provide an estimate of the size of a population

it is similar to the 'sampling circles' task

in three stages:

• students just pick a sample of 6 squares count the dandelions in these and multiply by 6 to obtain their estimate
• they then select a sample of 6 squares (by chance) using two dice throws to generate the coordinates of the sample (it's fine to have the same square picked twice - sampling with replacement)
• they could also compare (in sub-groups) dice-selected samples of size: 3 (times 12), 9 ( times 4) and even 2 (times 18) to explore how accurate these are

it can be interesting to compare the class results for all of these

Glyn suggests drawing a number line and marking each student's result on it - to gain an appreciation of the data shape

using the mean of the class results can give a close estimate for the actual population size (150)

sampling circles

this task was introduced to me by Mary Rouncefield, Chester University and my understanding is that she and Peter Holmes devised it.

following a discussion about why people sample for statistical work (more economical)
provide students with a copy of this sheet and a ruler

their task is to estimate the mean diameter of all the circles on the sheet by picking a smallish sample (e.g. five circles)

these circles should all have integer diameters, the smallest being 1cm
(it may need to be scaled)

ask students to pick 5 circles at random, measure their diameters and then work out the mean of their 5 circle diameters (so some work on mentally dividing by 5 could take place)

collecting individual's mean results can show some interesting variation

the moral, to be explored in the course of the lesson, and not introduced just yet, is that humans are not very good at picking at random

the next stage is to introduce what the actual mean is, (100 divided by 50 = 2cm)

they can then think about why their estimates based on their samples are too large and consider a better way to select 5 circles at random

this might be, for example, numbering them all and using a random number generator (on a calculator) or in Excel

allowing repeats is important because the sampling procedure would be biased if extreme samples could not occur (the same circle chosen five times)

they can check that a more mechanical method for choosing a sample gives a more accurate estimate of the actual mean and also explore the variation

students should also consider why closing your eyes and picking circles with a pen usually leads to a biased sample (maybe by trying this out)

to emphasise the point that you need a gadget to produce a a set of random numbers, like the lottery, ask students to write down ten digits at random then compare their sets with those generated by e.g. using Excel's random number generator [=randbetween(1,9) dragged down]