Random sampling

It is easy to misuse the word 'random'. You might say that you carried out your survey by questioning people 'at random', but what does it really mean to choose someone at random?

Random sampling means that members of a 'population' have equal chances of being selected.

To carry out this type of sampling, you will need to use a table of random numbers. Random numbers can also be generated using a calculator or computer. These can then be listed.

For example, if these random numbers are generated by a calculator:

017, 029, 300, 914, 037, 849, 111, 559, 333, 400, 598, 255.

Use these to make up the list of numbers to select a sample:

0 1 7 0 2 9 3 0 0 9 1 4 0 3 7 8 4 9 1 1 1 5 5 9 3 3 3 4 0 0 5 9 8 2 5 5.

Example

Of 1000 pupils in a school, 50 are to be questioned about their favourite pop group. How should the pupils be chosen?

Solution

The pupils should be numbered 000, 001, 002, 003, ....., 999.

You can use a calculator to generate random numbers. Three-digit random numbers can then be used to choose 50 pupils.

Remember: when carrying out a random sample, you must ensure that all possible samples are equally likely to be chosen.

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 is proportional to the size of the 'layer'. This is shown in the following equation:

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 this can be written as:

¹⁸⁰/₁₀₀₀ × 50 = 9

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

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

Random samples are then taken from each section (or 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.

Systematic sampling

If a sample of size s is to be taken from a population of size n, then every ⁿ/_s member of the population is tested. The starting point is chosen at random.

If we want to test a 100-strong sample from a population of 2000, we test every ²⁰⁰⁰/₁₀₀ = every 20^th member. We use random numbers to determine the starting point.

For example, if we obtained the random number 7, we would test 7, 27, 47, 67...

You will have to make sure that each member of the population is arranged randomly. If they are grouped together before the sample is taken, the sample could become biased.

Quota sampling

This is where a sample as far as possible reflects the whole population. It is a method that is often used by market research companies. The interviewer will normally be given some instructions (eg, ask approximately the same number of men and women aged between 20 and 60), but they will then be left to choose the interviewees themselves.

Other forms of sampling

Attribute sampling

In this case, you would pick your sample based on an attribute which is totally unrelated to the variables being investigated. For example, if you wanted to investigate the relationship between weight and daily calorie intake, you might choose your sample by asking people what their favourite colour is.

Cluster sampling

First, you would need to divide the population into small groups, or clusters. Then you select random clusters, or a random selection from the random clusters, to form your sample.

These problems may arise from your way of collecting the data, as well as your chosen method of sampling. For example, a survey to find out what your local town's favourite pet is will reveal biased results if you ask passers-by in the park because you are more likely to find people out walking their dogs.

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