Simple random samples

What you will be able to do

• say what makes a sample a simple random sample;
• choose one by hand and with random.sample in Python;
• find the chance that a person is chosen, and spot a frame that leaves people out.

Names in a hat

A class of 30 students must send 6 to a meeting. The teacher writes every name on a slip of paper, folds the slips, shakes them in a hat and draws 6 without looking. Nobody can complain. Every student had the same chance.

In First Steps you met this idea as "choosing by chance". Now we give it its proper name and look at it more closely.

The definition

A simple random sample (SRS) of size $n$ is chosen so that every possible group of $n$ members of the population has the same chance of being the sample. Then each member has the same chance too: $\dfrac{n}{N}$, where $N$ is the size of the population.

The second fact does not give the first. Suppose a teacher picks 3 boys and 3 girls from a class of 15 boys and 15 girls. Each student has the chance $\dfrac{3}{15} = \dfrac{1}{5}$, the same as in a simple random sample of 6 from 30. But a group of 6 boys can never be chosen. So this is not a simple random sample. (It is a stratified sample, which the next lesson meets.)

Three ways to choose one

1. Slips in a hat. Good for a small class. The slips must be the same size and well mixed.
2. Random digits. Number the members 01 to 30. Read pairs of digits from a list of random digits: 73, 05, 41, 18, 05, 26, ... Skip numbers above 30 and repeats. Here the sample starts 05, 18, 26.
3. A computer. In Python, random.sample(class_list, 6) picks 6 different members, and every group of 6 is equally likely.

Be careful with random.choices(class_list, k=6). It picks with replacement, so the same student can come up twice. That is not a simple random sample.

The sampling frame

You can only choose from a list. The list you choose from is the sampling frame. If the frame leaves people out, so does the sample, however random the choice. A sample of a town drawn from a list of house owners can never include people who rent. This is the undercoverage of the first course, now seen from the inside.

A worked example

A school has 1000 students, and 450 of them are girls. A simple random sample of 40 is chosen.

1. One student's chance. $\dfrac{40}{1000} = 4\%$.
2. Girls in the sample, on average. $40 \times \dfrac{450}{1000} = 18$.
3. One real sample. It might hold 15 or 21 girls. The average of 18 holds over many samples, not in each one.

Practise with new numbers

Try it in Python

The program takes one simple random sample of 6 from a class of 30. Then it takes $10\,000$ samples and counts how often roll number 1 is chosen.

Your turn.

The usual mistakes

• Thinking that equal chances for each person are enough. A simple random sample needs every group to be equally likely.
• Choosing with replacement by mistake. random.choices can repeat a name. Use random.sample.
• Trusting a random choice from a poor list. If the frame leaves people out, the sample does too.
• "Random" meaning "without a plan". Picking people who "look random" in a street is a convenience sample. Random means chosen by a chance device.

Remember these

• A simple random sample gives every group of $n$ the same chance. Each member then has the chance $\dfrac{n}{N}$.
• Choose with slips in a hat, random digits or random.sample, never "by eye".
• The sampling frame is the list you choose from. It must cover the whole population.
• random.seed(...) makes a program pick the same "random" sample every time, which helps when you check work.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A simple random sample of 25 is chosen from 500 workers. What is the chance that a particular worker is chosen?

2. Number 80 students from 01 to 80. You read the random digits 9 4 2 7 0 6 2 7 5 3 in pairs. Which three students are chosen first?

3. A teacher picks the first 5 students who arrive in the morning. Is this a simple random sample?

4. What is the difference between random.sample(names, 4) and random.choices(names, k=4)?

5. A town picks a random sample of adults from its list of people who pay for water. Which adults might be missed?

References

• OpenStax, Introductory Statistics, Chapter 1, section on data, sampling and variation, which describes the sampling methods.