Populations, sampling frames and simple random sampling

Key words

• Population: the whole set of items of interest.
• Census: data from every member of the population. Accurate, but slow, costly, and impossible if testing destroys the item.
• Sample: a subset of the population, used to estimate population features.
• Sampling unit: one item that can be chosen.
• Sampling frame: a list of all the sampling units, numbered, from which the sample is drawn.

Simple random sampling

A simple random sample of size n is one in which every possible sample of size n has an equal chance of being chosen (so every member has an equal chance too).

Methods:

• Lottery: write each member's number on a ticket, mix, draw n without replacement.
• Random numbers: number the frame (for 80 members use 00 to 79), take digits from a random number table or calculator in groups of the right length, ignore numbers outside the range and repeats, until n are chosen.

Worked digits

Population of 80 numbered 00 to 79. Random digits: 5793 1457 8002 6634 9171.

Pairs: 57, 93, 14, 57, 80, 02, 66, 34, 91, 71. Ignore 93 and 80 (too large) and the second 57 (a repeat). The first four selected are 57, 14, 02, 66.

Advantages and disadvantages

• Advantages: free from selection bias; easy to use for small populations; each member has a known equal chance.
• Disadvantages: needs a complete sampling frame; slow and impractical for large populations; may by chance under-represent a group.

Worked example

Describe how to take a simple random sample of 20 students from a college of 650 students.

1. Obtain a list of all 650 students (the sampling frame) and number them 001 to 650.
2. Use a random number table or generator to choose 3-digit numbers.
3. Ignore numbers above 650 and repeats until 20 different numbers are chosen.
4. The students with those numbers form the sample.

Answer: Frame, number, random numbers, ignore invalid and repeats, select 20.

Check yourself

1. Which statement defines a simple random sample of size n?
   • Every nth member of a list is chosen
   • Every possible sample of size n is equally likely to be chosen
   • Members are chosen until quotas for each group are filled
   • The sample has the same proportions of each group as the population
   Show answer

   Every possible sample of size n is equally likely to be chosen. That is the definition; it implies every member also has an equal chance.

2. A population of 80 is numbered 00 to 79. Using the random digits 5793 1457 8002 6634 9171 in pairs from the left, which number is the third member selected? Type it as a whole number (for example 5 for 05).
   Show answer

   2. Pairs: 57, 93, 14, 57, 80, 02... Valid and new: 57, 14, then 02 (93 and 80 are too big, the second 57 is a repeat).

3. A sampling frame lists 650 items. How many digits should each random number have?
   Show answer

   3. The items are numbered 001 to 650, so 3-digit random numbers are needed.

4. What is a sampling frame?
   Show answer

   A list of all the sampling units in the population, from which the sample is chosen.. A sampling frame is a list (often numbered) of the members of the population.

5. A factory tests light bulbs by running them until they fail. Why is a census unsuitable?
   • Testing destroys the bulbs, so none would be left to sell
   • A census is always less accurate than a sample
   • There is no sampling frame for bulbs
   • Bulbs are not a population
   Show answer

   Testing destroys the bulbs, so none would be left to sell. Destructive testing makes a census pointless.

6. Which is a disadvantage of simple random sampling for a national survey?
   • It is biased towards larger groups
   • The interviewer chooses who to ask
   • A full list of the population is needed and hard to obtain
   • It cannot be used with random numbers
   Show answer

   A full list of the population is needed and hard to obtain. Simple random sampling needs a complete sampling frame, which is impractical for very large populations.

7. Which is an advantage of simple random sampling?
   • It is free from selection bias
   • It guarantees every group is represented
   • It needs no sampling frame
   • It is the quickest method for large populations
   Show answer

   It is free from selection bias. Every sample of size n has the same chance, so the person sampling cannot bias the choice.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S3

Last updated 8 November 2026. Spotted a mistake? Tell us.