In this section we look at random samples and at the difference between a *census* and a *sample*.

*In a census*, information on every member of a population is considered. In the UK, a census is carried out every 10 years. The amount of work required to carry out a census means that it is an expensive process.

*In a sample*, a subset of the population is considered to try to obtain information about a particular problem or issue. Because a sample is normally much smaller than the whole population, it is quicker and easier to take and to analyse a sample than to carry out a census of the entire population. Sampling entails less effort and less expense. In some cases, it is essential to take a sample. For example, imagine a firm that uses quality control to test the light bulbs it manufactures, to see how long they last. (If every item was tested until it stopped working they would have no light bulbs left to sell!) They therefore take samples from the production and test these to see if the quality is up to standard.

In a *random sample*, every member of the population is *equally likely* to be included in the sample. One way of selecting a sample is to use random numbers, as demonstrated in the example below. You can find random numbers in books of statistical tables. You can also generate them using a calculator or a computer.

The diagram shows part of a table of random digits.

Random digits | |||||
---|---|---|---|---|---|

98859 | 09884 | 45275 | 09467 | 93026 | 32912 |

26604 | 95099 | 93751 | 00590 | 93060 | 64776 |

82984 | 65780 | 94428 | 30160 | 86023 | 52284 |

70888 | 14063 | 96700 | 83008 | 17579 | 71321 |

77803 | 61872 | 86245 | 68220 | 66267 | 01379 |

11304 | 01658 | 82404 | 46728 | 35228 | 49673 |

53552 | 51215 | 45611 | 83927 | 00772 | 99295 |

In a class there are 30 pupils. The teacher decides to take a random sample of 5 pupils to estimate the mean height of the pupils in the class. Select a random sample of 5 pupils from the list.

1 Alan | 10 Rachel | 19 Sacha | 28 Salif |

2 Lucy | 11 Ben | 20 Halim | 29 Annie |

3 Tom | 12 Emma | 21 Daniella | 30 Karen |

4 Azar | 13 Hannah | 22 Joseph | |

5 Jayne | 14 Grace | 23 Anna | |

6 Nadima | 15 Miles | 24 Sophie | |

7 Matthew | 16 James | 25 Kathryn | |

8 Sushi | 17 Joshua | 26 Helen | |

9 Mohammed | 18 Lisa | 27 Fatoumata | |

To take a random sample you need to use a list of random digits, as follows:

The digits are taken in pairs to form 2-digit numbers, as shown above. All those numbers greater than 30 are discarded (as there are 30 pupils on the list). The process is continued until 5 different numbers between 1 and 30 have been obtained.

So the sample will be made up of the following pupils:

7 | Matthew |

26 | Helen |

5 | Jayne |

27 | Fatoumata |

15 | Miles |

Discuss whether or not the following situations produce random samples.

(a)

Mark is conducting a survey for a magazine. He stops people at random on a Saturday morning at his local shopping centre.

To produce a random sample, every member of the population must have an equal chance of being selected. In the case of Mark's sample, he is excluding people who are at work on that Saturday, as well as other people who haven't gone to that shopping centre. So although Mark is stopping people at random, he is not producing a random sample.

(b)

Granny Taylor's National Lottery numbers.

Most people use personal reasons when they select their National Lottery numbers. Granny Taylor may, for example, have used the number of grandchildren that she has, their birthdays, the number of her house, etc. If that is the case then she has not selected a random sample. However, if she bought a 'Lucky Dip' from her local shop then the computerised National Lottery till should have produced a random sample.

Note: For small populations it is relatively easy to produce a random sample. Simply number every member of the population, write those numbers on pieces of paper and put them into a hat or tombola. Mix them well and ask someone to pick out as many numbers as you need for your sample, then take the corresponding items from the population. This process clearly becomes unmanageable when we investigate large populations, which is why we tend then to use random number generators.