Populations and Samples

Populations, Samples, and Censuses

Every time you watch a television show, broadcast networks decide whether to renew or cancel it by tracking the viewing habits of just a few thousand homes, rather than the whole country.

• The population is the entire group of people, objects, or items that are being studied or from which data could be collected.
• A sample is a selection of items taken from the population that is used to represent the whole.
• A census is a survey or investigation that includes every single member of the population.

A census provides 100% accurate results but is often impractical due to high costs, excessive time requirements, or situations where testing destroys the item (such as measuring the lifespan of batteries). Because of this, we use samples. For a sample to be reliable, it must be a representative sample, meaning it shares the exact characteristics and proportions of the population. Larger samples are always more reliable; OCR mark schemes generally consider samples of fewer than 30 items to be unreliable due to natural variation.

Simple Random Sampling

If a teacher asks for volunteers to answer a question, the eager students who raise their hands do not represent the whole class's level of understanding.

• Simple random sampling is a sampling method where every member of the population has an equal probability of being chosen.
• This method is specifically designed to prevent selection bias, where the person picking the sample might subconsciously or purposefully choose certain types of members.

To carry out a simple random sample, you must use a rigorous methodology:

1. Obtain a sampling frame, which is a complete, numbered list of every member of the population.
2. Assign a unique number to every member on the list (for example, from $1$ to $N$).
3. Use a random number generator (such as the $\text{RanInt}$ function on a scientific calculator) or a random number table to generate the required amount of numbers.
4. Crucially, ignore any duplicate numbers generated and continue until your full sample size is reached.
5. Match the randomly selected numbers back to the names or items in your sampling frame to form your sample.

Understanding Sampling Bias

A restaurant asking lunchtime diners to rate their new evening menu will quickly find that their feedback is heavily skewed and not useful.

• Sampling bias is a systematic error where the sampling method results in certain members of a population being more or less likely to be selected than others.

When evaluating or criticising a data collection method, look for these common sources of bias:

• Location bias: Sampling in only one place, which excludes people who never go there (e.g., asking about fitness habits outside a gym).
• Time bias: Sampling at only one specific time, missing people who work or shop on different schedules (e.g., interviewing people at 10 AM on a Tuesday).
• Self-selection bias: Using volunteers for an online poll or radio phone-in, which disproportionately attracts people with extreme or very strong opinions.
• Convenience bias: Choosing the easiest people to reach, such as the first 10 people to walk past a shop, which does not give every member of the population an equal chance of selection.

Inferring Population Properties

Knowing that 6 out of 30 surveyed teenagers prefer a specific clothing brand allows retailers to predict how many shirts to ship to a city of a million people.

• If a sample is representative, the proportions found within it can be scaled up to estimate properties of the whole population. This process is known as statistical inference.
• You can use a scaling factor to estimate population totals or means.

$\text{Estimated Population Frequency} = \frac{\text{Frequency in Sample}}{\text{Sample Size}} \times \text{Total Population Size}$

Worked Example

In a random sample of 60 gym members, 18 said they use the swimming pool. The gym has 1,500 members in total. Estimate the total number of members who use the swimming pool.

Step 1: Identify the sample frequency, sample size, and total population.

• Frequency in sample = $18$
• Sample size = $60$
• Total population = $1500$

Step 2: Substitute the values into the scaling formula.

• $\text{Estimated Total} = \frac{18}{60} \times 1500$

Step 3: Calculate the final answer.

• $\text{Estimated Total} = 0.3 \times 1500 = 450$ members

(Note: If your final estimated frequency is a decimal, you must round it to the nearest whole number, as you cannot have a fraction of a person or item).