1c.05 Simple random sampling Notes — Edexcel Statistics 1ST0

Understand the key features of a simple random sample, primarily ensuring that every single item in the population has an equal likelihood of selection.
Demonstrate how various physical and electronic techniques are used to select random members, including dice, cards, random number lists, and calculator functions.
Select items accurately for a sample while managing practical issues, such as ignoring repeated random numbers or dealing with generated numbers outside the required population range.

What makes a sample random?

A population is the complete group you want to investigate. A sample is the smaller group you select from it. For example, if a company wants to investigate its customers’ opinions, the population is all its customers, while the sample consists of the customers selected for the survey.

In a simple random sample, every member of the population has the same likelihood of being included. Selection is controlled by chance rather than by someone choosing the people they prefer or find easiest to contact. This helps avoid selection bias.

Random selection does not guarantee that the sample perfectly represents the population. By chance, a small sample might contain no members of a particular group. The important feature is that the selection process gives every member an equal opportunity, not that the resulting sample has a particular mix of people.

Connect random numbers to real members

Start with a sampling frame: a list of all the members of the population. Give each member a different number, usually from 1 up to the population size. Each number must identify exactly one member, and no member should be missing or listed twice.

You can then generate random numbers and select the members whose labels match them. For example, with 80 items numbered 1 to 80, the random number 24 selects item 24. The labels are identifiers; they do not measure anything about the items.

The physical or electronic method must give each eligible label the same chance of selection. Whichever method you use, continue until you have enough different members for your sample.

Physical methods: cards and dice

For a card method, write each member’s number on a separate, otherwise identical card. Mix the cards thoroughly and draw without looking. Keep selected cards out of the pack so that a member cannot be selected twice. Numbered slips mixed in a bag work in the same way. This is straightforward for a small population but cumbersome for a large one.

A fair six-sided die can select among six members labelled 1 to 6. For larger populations, fair ten-sided dice labelled 0 to 9 can generate digits. Roll one die twice, or use two distinguishable dice, deciding beforehand which result is the tens digit and which is the units digit. A tens digit of 4 and a units digit of 7 give 47.

This generates labels from 00 to 99. For 80 members numbered 1 to 80, accept 01 to 80 and reject 00 and 81 to 99. Three digit rolls can similarly generate labels from 000 to 999. Each digit position has a fixed role; the scores are not added together.

Electronic methods and random number lists

A calculator’s random-integer function can generate an integer between chosen limits. For 80 members numbered 1 to 80, set the lower limit to 1 and the upper limit to 80. The exact buttons depend on the calculator. A computer random-number generator can also be set to use the required range.

Some calculators instead display random decimals. If a function supplies random three-digit decimals such as 0.541, 0.414 and 0.929, multiplying by 1000 gives labels 541, 414 and 929. These labels still need to be checked against the population’s range, including rejecting 000 when numbering starts at 1.

A random number list or table provides digits that you can read in groups of a suitable length. For labels up to 80, use two-digit groups. For labels up to 832, use three-digit groups. Leading zeros are useful: 068 means member 68, not three separate labels.

Choose a starting point and reading direction before selecting members, or follow those given in the question. Read consistently rather than searching for numbers you like.

Handle out-of-range numbers and repeats

Two problems can arise when selecting labels:

An out-of-range number does not identify a population member. Reject it and continue. For a list numbered 1 to 80, both 00 and 93 must be rejected.
A repeated number identifies someone already selected. Keep its first occurrence, ignore later occurrences and continue.

Rejecting these results does not mean replacing them with nearby numbers. Replacing 93 with 80, for example, would give member 80 an extra route into the sample and make the selection unfair.

Worked selection from a random number table

Florence has 832 customers and wants a simple random sample of 12. She numbers the customer list from 1 to 832. Starting at the top left and reading across, she splits each six-digit entry into two three-digit labels:

855737 648311 989903 068440 922412 748392
445546 862885 418648 010910 148805 533291

The first row becomes:

855 737 648 311 989 903 068 440 922 412 748 392

Reject 855, 989, 903 and 922 because they exceed 832. The eight accepted labels are 737, 648, 311, 68, 440, 412, 748 and 392.

Continue along the second row. Accept 445 and 546, reject 862 and 885, then accept 418. The next label, 648, has already been selected, so ignore it. Accept 010, meaning customer 10, and stop: there are now 12 different valid labels.

Florence selects customers 737, 648, 311, 68, 440, 412, 748, 392, 445, 546, 418 and 10 from her numbered list. The method determines who is selected; Florence does not choose which customers she would prefer to survey.

Essential feature

Every member of the population has the same likelihood of inclusion in a simple random sample. Random selection reduces selection bias but does not guarantee a representative mix.

Selection procedure

Use a complete sampling frame and give each member a unique number.
Generate random labels using fair dice, mixed numbered cards, a random number table or a calculator function.
Select the members with matching labels.
Reject out-of-range labels and later repeats.
Continue until the required number of different members has been selected.

Method reminders

Cards: mix thoroughly; draw without looking and without replacement.
Dice: assign rolls to digit positions in advance.
Number tables: use suitable digit groups and a fixed reading direction; 068 means 68.
Calculator: set random-integer limits to match the population labels.

Describe how the random numbers identify members: number every member of the sampling frame, generate random numbers, then select the matching members.

When using a random number table, follow the specified starting point, direction and digit grouping. Do not choose numbers because they look convenient.

Keep the first occurrence of a valid number, ignore later repeats and reject numbers outside the population’s numbering range.

Continue until you have the required number of different members, not merely the required number of generated numbers.

For two dice representing digits, identify which gives the tens digit and which gives the units digit. Adding their scores is a different method.

The complete group of people or items being investigated.

A group of members selected from a population to investigate.

A list of all the members of the population from which a sample can be selected.

A sample selected by a random method so that every member of the population has the same likelihood of inclusion.

A systematic tendency for a selection method to favour some members or types of member over others.