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In Further Mathematics - Exam Revision

Population, samples and simple random sampling


A population consists of all members (people, objects or events) of a specified group about which information is sought.


  • all the teenagers in a country;
  • all the students in a school;
  • all the kangaroos in Australia.

    A sample is a representative subset of the population and it consists of some members (people, objects or events) of the population that we collect to make inferences.


  • a group of teenagers in a country;
  • a group of students in a school;
  • a group of kangaroos in Australia.

    If the sample is random and large enough, the information collected from the sample can be used to make inferences about the population.

    Note: Populations consist of very large numbers of members; the example with the cats above is used for visualisation purposes.

    Population parameters Sample statistics

    population mean = μ

    standard deviation = σ

    population size = N (large size)

    population proportion = p

    sample mean = 

    standard deviation = s

    sample size = n (manageable size)

    sample proportion = 

    * hard and costly to collect the data * easy and cost effective to collect the data


    Population proportion, p, and Sample proportion

    The population proportion is constant; it's value does not change.


    If the group of children shown above represents the population of children in a tutoring class, for example, then the proportion of girls is


    where X is the number of girls in the population and N is the population size.


    The sample proportion is variable; it's value changes from sample to sample.

    In the table below there are some examples of sample proportions corresponding to the size of the sample.

    Sample proportion


    x = 3 girls

    n = 5 children


    x = 1 girl

    n = 5 children


    Simple random sampling

    A simple random sample is a subset of a population with the following properties:

    * every member of the population is chosen entirely by chance;

    every member of the population has an equal chance of being selected;

    * all samples have the same size, n;

    * every possible sample of size n has an equal chance of being selected.


    Simple random sampling has the advantage that is easy to perform. It works well for a small sample size.

    Simple random sampling can be done using

    * the lottery type of selection. Each number is written an a ball and put in a lottery machine and selected one by one.

    Keno goose
    * the old "pick a number from a hat/container/box"
    * a random number generator which can be a calculator, a software or an online number generator.

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