Sampling

• When it is important to understand the characteristics of the population and the population can be divided into clear groups (also called strata) then stratified random sampling is applicable. For example, our population of 1000 students can be divided into girls and boys, different age groups, and so forth. If we are interested in understanding the differences amongst these groups on whatever we may be investigation, whether that is exam results or class attendance, for example, then we need to ensure that each group is represented in our sample of 200 students. To achieve this, we first identify the stratum (groups) that we are interested in; let’s say boys and girls). Then we count the number of boys and girls amongst the 1000 students and state their relative frequency. For example, if there were 600 boys and 400 girls, this would give a frequency of 0.6 and 0.4 respectively. Since we need 200 students in our sample, we simply multiply this figure by the frequencies to arrive at the required number of boys and girls that must be included in our sample. In this instance, that would be 120 boys (0.6 x 200 = 120) and 80 girls (0.4 x 200 = 80). Nonetheless, these 120 boys and 80 girls should still be selected at random from their respective populations.
• For the purpose of cluster random sampling are example of the 1000 students is no longer applicable. This is because the cluster random sample is useful when the population being studied is spread out geographically, perhaps across counties, states, regions or countries. For example, when a general election is near, opinion-poll organizations need to assess the general way that the population of a country will vote. However, it would be unfeasible and unpractical to sample people from every state or county, which is where cluster sampling helps. First, every state/county is assigned a number. Then, a random sample of these states/countries is selected. The researcher can then chose to perform another probability-based sampling method at the state/county level to select those individuals to be polled.
  

In many research settings researchers draw on a variety of probability-based sampling techniques in what becomes multi-stage sampling.

Non-Probability Sampling:

There are a wide variety of non-probability sampling techniques that can be used. These techniques tend to be popular in student’s research because they are less costly and time consuming. Two of the main techniques include (1) quota sampling and (2) convenience sampling. Again, in order to discuss these two sampling methods we use the example of 1000 students in a school from which a researcher needs to survey 200 of them.

• Quota sampling is similar to stratified random sampling in the sense that our population of students would also be divided into groups and a number from each group would be sampled based on their relative frequency. However, it differs significantly from stratified random sampling by not involving a random means of choosing which students in each group should be sampled. Instead, the choice of which students from each group should be selected is left to the researcher. Whilst this inevitably saves considerable data collection time, it does result in a number of potential biases, which may mean that the sample selected is not representative of the population being studied.
• Convenience sampling involves picking a sample that is simply available; that is, convenient. Where researchers have limited funds they may choose to collect data from the most accessible and cheapest source. In the case of our 200 students, it may be easier for the researcher to access those students that are 16 years old and above because parental consent to be involved in the research is not necessary, which would otherwise result in the study taking longer to complete, as well as require the purchase of 200 letters and their associated postage cost. However, whilst convenient it would not be possible to make generalizations about the 1000 students from the sample of 200 students with any acceptable degree of accuracy.

In the sections that follow, we return to the statistics of sampling.

Sample Statistics: The Sample Mean and Sample Standard Deviation

Whilst it is important to know various statistical sampling techniques, it is vital to understand how sampling statistics help us to more accurately analyse our population. Let’s use an example.

The Government sets a countrywide exam for 16 year olds to assess levels of maths ability. Maths ability is assessed using a single exam in which students can score between 0 and 100. The nationwide mean (that is, the population mean) is 55 marks out of 100 and the population standard deviation is 15 marks. The question arises: If a Government statistician who was responsible for examining the results looked at the marks of 100 students from a single school, would the marks achieved be representative of the whole country; in other words, would the sample be representative of the population?

Most likely, the answer is no. Neither the sample mean, X, nor sample standard deviation, s (what we call sample statistics), is likely to be representative of the population mean, µ, or population standard deviation, σ (what we call population parameters). The reason lies in the distinction we are making between the mean and standard deviation of the “population”, and the mean and standard deviation of the “sample”. After all, if we have all the scores for the population we are investigating then there cannot be any error when we calculate the mean and standard deviation (apart from human error when adding up the scores, for example). As such, if we apply the formulas below (these should be familiar from the statistical guides on Measures of Central Tendency and Measures of Spread) then we will accurately have the mean and standard deviation values for our population.