Population, sample, parameter and statistic

What you will be able to do

• name the population, the sample, the parameter and the statistic in a study;
• give reasons for taking a sample instead of asking everyone;
• explain why two fair samples give different answers, and how the size of a sample changes that.

One spoonful of soup

A cook stirs a large pot of soup and tastes one spoonful. Too salty? Then the whole pot is too salty. She does not drink the pot to find out. One well-stirred spoonful is enough.

You met this idea in First Steps: the population is everyone (or everything) we want to know about, and the sample is the part we really look at. Now we add two more words, because the numbers matter as much as the people.

Parameter and statistic

• A parameter is a number that describes the whole population. It is fixed, but we usually do not know it.
• A statistic is a number worked out from a sample. We know it, but it changes from one sample to the next.

An easy way to remember: population goes with parameter, sample goes with statistic.

Statisticians use different letters for the two. The mean of a population is $\mu$ (the Greek letter mu), and the mean of a sample is $\bar x$ (read "x bar"). The proportion of a population is $p$, and the proportion in a sample is $\hat p$ (read "p hat"). The hat says "worked out from a sample".

Here is a made-up example. A school has 1200 students, and 540 of them walk to school. The share of walkers, $p = 540/1200 = 45\%$, is a parameter. A random sample of 100 students has 41 walkers. The share $\hat p = 41\%$ is a statistic. We use it as our estimate of $p$.

Why not ask everyone?

Asking every member of the population is called a census. A country does one every few years. Most studies take a sample instead.

• Cost and time. Asking 100 people is quicker and cheaper than asking $100\,000$.
• Testing destroys the item. To find how long a battery lasts, you run it until it is flat. Test every battery, and none is left to sell.
• The population is too large or keeps changing. You cannot catch every fish in the sea, and the fish change every day.
• A careful sample can beat a careless census. A small team can check a sample with great care, and a census is often rushed.

Two fair samples, two answers

Take a second random sample of 100 students from the same school. This time there are 48 walkers. Has someone made a mistake? No. Each sample holds different students, so each gives a different statistic. This natural scatter is called sampling variability, and the gap between a statistic and the parameter is the sampling error. It is luck, not a mistake.

Luck has a size, and the size depends on $n$, the number in the sample. The simulation shows a band where nearly every random sample lands. A large sample gives a narrow band. Chapter 6 of this book explains where the band comes from. For now, watch it shrink.

Notice one pattern: make the sample four times as large, and the band becomes half as wide. Large samples cost more, and each extra person helps less than the one before.

Practise with new numbers

Try it in Python

The program builds the made-up school of 1200 students, with 540 walkers. Then it takes 20 random samples of 25, of 100 and of 400 students, and prints the lowest and the highest statistic for each size.

Your turn.

The usual mistakes

• Mixing up parameter and statistic. "The mean height of all the students" is a parameter. "The mean height of the 30 we measured" is a statistic.
• Thinking a statistic is wrong because it differs from another sample. Two fair samples always differ a little. That is sampling variability.
• Thinking a sample must be a large part of the population. A random sample of 1000 people tells you almost as much about a city as about a country. The size of the sample matters, not the share of the population.
• Calling a statistic the truth. A statistic is an estimate. Say "about 41%", not "41% of the school walks".

Remember these

• Population and parameter go together. Sample and statistic go together.
• $\mu$ and $p$ describe a population. $\bar x$ and $\hat p$ describe a sample.
• Statistics vary from sample to sample. A sample four times as large halves the usual error.
• We sample to save time and money, and because some tests destroy what they test.

Check yourself

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

1. A farmer weighs 30 of her 900 melons, and their mean mass is $1.8\ \text{kg}$. Name the population, the sample and the statistic.

2. A city has records for every taxi, and the mean age of all its taxis is 6.2 years. Is 6.2 years a parameter or a statistic?

3. In a random sample of 40 students, 14 own a bicycle. Find $\hat p$ as a percentage.

4. Give one reason why a fireworks maker tests a sample from each batch, and never the whole batch.

5. Two random samples from the same school give 38% and 44% for the share of students who walk. Ravi says that one sample must be wrong. Is he right?

References

• OpenStax, Introductory Statistics, Chapter 1, sections on data, sampling and variation in data and sampling.