The sampling distribution and its standard error

What you will be able to do

• describe the sampling distribution of a sample mean;
• find the standard error $\dfrac{\sigma}{\sqrt{n}}$ of a sample mean;
• say why sample means are mound-shaped for larger samples, even from a flat or skewed population.

Dice as a population

A fair die is a population we know completely. Its faces $1$ to $6$ are equally likely, so the population mean is $\mu = 3.5$. Its population standard deviation is $\sigma = \sqrt{\dfrac{35}{12}} \approx 1.708$. Here we divide by $n = 6$, because the six faces are the whole population.

Now take a sample: throw $2$ dice and work out the mean of the two faces. There are $36$ equally likely outcomes. A mean of $1$ needs a double one, so it happens in only $1$ outcome of $36$. A mean of $3.5$ happens in $6$ outcomes, the most of any value.

The sampling distribution

The sampling distribution of a statistic is the pattern of its values over all possible samples of the same size. For the mean of $2$ dice it is a triangle: low at the two ends and highest at $3.5$.

The picture shows the exact sampling distribution of the mean of $1$, $2$ and $4$ dice, worked out by listing every outcome. Look at three things.

1. Centre. All three are centred at $3.5$, the population mean. A sample mean does not lean up or down. We say it is unbiased.
2. Spread. The pattern gets narrower as the sample grows. The share of means between $3$ and $4$ is $\dfrac{1}{3}$ for one die, $\dfrac{4}{9}$ for two dice and $\dfrac{169}{324} \approx 0.52$ for four dice.
3. Shape. One die is flat. Two dice make a triangle. Four dice already make a mound.

The standard error

The standard deviation of a sampling distribution has its own name, the standard error (SE). For a sample mean,

For two dice, $SE = \dfrac{1.708}{\sqrt{2}} \approx 1.208$. Listing all $36$ outcomes gives the same value. For four dice, $SE = \dfrac{1.708}{\sqrt{4}} = 0.854$.

The rule assumes that each value in the sample is drawn independently, as with dice. For a sample without replacement, it is very close when the sample is a small part of the population.

The shape of sample means

Why do the means make a mound? A mean near the middle can be made in many ways. A mean at an end needs every value in the sample to be at that end, and that is rare.

This happens even when the population itself is skewed, such as travel times with a long tail of slow journeys. For samples of about $30$ or more, the sample means are close to a normal curve. This result is called the central limit theorem. You met the normal curve in the chapter on random variables and distributions.

Practise with new numbers

Try it in Python

The program throws $4$ dice $2000$ times. It prints the mean and the spread of the $2000$ sample means, and draws their histogram.

Common mistakes

• Mixing up the two spreads. $\sigma$ is the spread of single values. The standard error is the spread of sample means, and it is smaller.
• Dividing by $n$. The standard error divides by $\sqrt{n}$, not by $n$.
• Saying the population becomes normal. The population keeps its shape. Only the sample means become mound-shaped.
• Forgetting the conditions. The rule needs a random sample. No formula can rescue a biased one.

Remember these

• The sampling distribution of $\bar x$ is centred at $\mu$.
• Its spread is the standard error $\dfrac{\sigma}{\sqrt{n}}$.
• For larger samples it is close to a normal curve, whatever the shape of the population.

Check yourself

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

1. The heights of a made-up group of plants have standard deviation $12$ cm. Find the standard error of the mean height of a random sample of $16$ plants.

2. A population of bags has mean mass $41$ kg. What is the mean of the sampling distribution of $\bar x$ for samples of $10$ bags?

3. Two dice are thrown. What is the probability that the mean of the two faces is exactly $6$?

4. A sample of $9$ is replaced by a sample of $36$ from the same population. What happens to the standard error?

5. A population of waiting times is strongly skewed to the right. Describe the shape of the sampling distribution of the mean for samples of $2$ and for samples of $50$.

References

• OpenStax, Introductory Statistics, Chapter 7, sections on the central limit theorem for sample means.