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Maths

Sampling - Higher

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Stratified sampling

'Strata' means 'layer'. A stratified sample is made up of different 'layers' of the population, for example, selecting samples from different age groups. The sample size for each layer is proportional to the size of the 'layer'. This is shown in the following equation:

Sample size for each layer = size of whole sample/size of population ×  size of layer

So we look at what fraction of the whole population do we want. Then take that fraction of each layer.

Example

To show how this works, go back to the survey in which 50 pupils in a school of a 1000 pupils were asked what music they liked.

To make sure the survey is accurate you will need a range of pupils across the year groups - different layers. Pupils in year 7 may like different music to those in year 11.

To work out the sample size for year 7:

  • There are 180 students in year 7 - this is the size of the layer.
  • There are 1000 pupils in the school - this the size of the whole population.
  • You want answers from 50 people in total - this is the size of the whole sample.

So we want 50/1000 of the population, so for year 7 we want 50/1000 of 180=9

50/1000 × 180 = 9

You could write a similar calculation for each year in the table below:

 

YearNumber of pupils
7180
8200
9240
10220
11160

The proportion of pupils from each of the other year groups would be as follows:

 

YearNumber of pupils
750/1000 × 180 = 9
850/1000 × 200 = 10
950/1000 × 240 = 12
1050/1000 × 220 = 11
1150/1000 × 160 = 8
Question

In a different school 10 year 8 pupils are questionned for a survey. The total sample size of the survey is 70. There are 840 pupils in the school. How many pupils are there in year 8?

toggle answer

Answer

10 pupils is 10 / 70 = 1 / 7 of the total sample. So if there are 840 pupils, 1 / 7 of them are in year 8.

1 / 7 x 840 = 120 pupils in Year 8

Sampling

Once the sample sizes for each layer have ben calculated random samples are then taken from each layer of the 'population'.

It would be possible to number the pupils from 000 to 199, use 3-digit random numbers, and disregard all numbers from 200 to 999. However, this would be very time consuming, and you would waste more numbers than you use. A better way is to allocate the numbers 000 to 199, as before, then allot an equal number of random numbers to each pupil.

For example:

  • pupil 000 is allotted the numbers 000, 200, 400, 600 and 800
  • pupil 001 is allotted 001, 201, 401, 601 and 801
  • pupil 134 is allotted 134, 334, 534, 734 and 934
  • pupil 199 is allotted 199, 399, 599, 799, 999.

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