The Sampling Distribution of the Mean
Lesson 22 ended on a question it could not answer: sample means vary, but how much ? Four samples cannot tell you. So stop looking at a few samples and look at every sample of that size at once . The collection of all those means is an object in its own right, with a centre and a spread you can measure, and it is the object the central limit theorem is about.
Do this before reading on. It takes two minutes and the rest of the lesson lands better for it.
A population contains just four values: $2, 4, 6, 8$ . List every sample of size $2$ you could draw, find each sample's mean, then find the mean of those means. How does your last number compare with the population mean?
- Change what you are looking at. Stop treating $\bar{x}$ as a number you got, and start treating it as a quantity with a whole distribution behind it. Every sample you did not draw still has a mean, and those means are part of the picture.
- Ask two questions about that distribution. Where is it centred, and how wide is it? Those are the only two things you ever need from it, and the next lesson gives you formulas for both.
- Define, for a given sample size $n$ , the sampling distribution of the mean as the distribution of the sample means of all samples of size $n$ .
- Build one in full for a small population, and read its centre and spread.
- Recognise that $\bar{x}$ is a random variable that estimates $\mu$ once $n$ is large enough.
- Tell the sampling distribution apart from the population distribution and from a single sample.
- Explain what it is for, in politics, finance, agriculture and biology.
Lesson 22 drew four samples and got four different means, then stopped, because four samples cannot tell you how much means vary in general. The fix is to stop sampling and start enumerating : take every sample of that size that could possibly be drawn, and collect all of their means.
Take the population $2, 4, 6, 8$ , so $N = 4$ and $\mu = \dfrac{2+4+6+8}{4} = 5$ . With $n = 2$ there are exactly six possible samples:
| Sample | $\bar{x}$ | Sample | $\bar{x}$ |
|---|---|---|---|
| $\{2, 4\}$ | $3$ | $\{4, 6\}$ | $5$ |
| $\{2, 6\}$ | $4$ | $\{4, 8\}$ | $6$ |
| $\{2, 8\}$ | $5$ | $\{6, 8\}$ | $7$ |
Those six means, $3, 4, 5, 5, 6, 7$ , are the sampling distribution of the mean for $n = 2$ . Plotted:
Two things are worth reading off that picture, and they are the two questions from card 02.
- Where is it centred? The mean of the six sample means is $\dfrac{3+4+5+5+6+7}{6} = \dfrac{30}{6} = 5$ , which is exactly $\mu$ . Not approximately. The sampling distribution is centred on the population mean.
- How wide is it? The population runs from $2$ to $8$ , a spread of $6$ . The sample means run only from $3$ to $7$ , a spread of $4$ . Averaging pulls the extremes in: to get a mean of $2$ you would need to draw the $2$ twice, and you cannot.
For a given sample size n, the sampling distribution of the mean is the distribution of the sample means of all possible samples of size n. It is centred on the population mean μ, and it is narrower than the population itself, because averaging pulls extreme values in.
Pause, copy the definition word for word, then copy the six-sample table and the two readings from it: centred on $\mu$ , narrower than the population.
Quick check: For a given sample size $n$ , the sampling distribution of the mean is the distribution of:
Look again at what card 05 built. Before you draw a sample, $\bar{x}$ does not have a value. It has a set of possible values, each with a chance of occurring , and that is exactly what a random variable is. You have met the idea already in this module: $X \sim B(n, p)$ is a random variable, and so is $\bar{x}$ .
That reframing is the whole payoff of the arc so far:
| Lesson | How $\bar{x}$ looked |
|---|---|
| 21 | a number you compute from a sample |
| 22 | a number that changes when the sample changes |
| 23 | a random variable with a distribution of its own, centred on $\mu$ |
And once $n$ is large enough, in the sense of Lesson 21, $\bar{x}$ is a usable estimator of $\mu$ : centred in the right place, and tight enough that one draw from it is worth acting on.
A bigger population. Take Lesson 22's eight students, $1, 3, 4, 6, 7, 9, 10, 24$ hours, with $\mu = 8$ . For $n = 4$ there are $\binom{8}{4} = 70$ possible samples, so the sampling distribution has $70$ values in it:
- The mean of all $70$ sample means is exactly $8$ , again equal to $\mu$ .
- They run from $3.5$ to $12.5$ . The population ran from $1$ to $24$ . Far narrower , as before.
- Lesson 22's four samples were four of these $70$ . Its lowest, $3.5$ , and its highest, $12.5$ , are the extreme means of the whole distribution.
What it is for. You will never build a sampling distribution in practice, because building it needs the whole population, and if you had that you would not be sampling. Its value is that it tells you how the one sample mean you do get behaves. The syllabus names four fields, and the pattern is the same in each: measuring everything is impossible, too slow, too expensive, or destroys what you are measuring.
| Field | The population | Why you cannot measure all of it |
|---|---|---|
| Politics | every voter | an election is the only full count, and it comes too late to be useful |
| Finance | every transaction, or every future return | future returns have not happened yet |
| Agriculture | every plant in every plot | harvesting everything to weigh it defeats the purpose |
| Biology | every organism in a habitat | most cannot be found, and testing can be destructive |
In every one of them the move is identical: take one random sample, compute $\bar{x}$ , and use what you know about the sampling distribution to say how far off it might be.
The sample mean x̄ is a random variable: before the sample is drawn it has a distribution of possible values, not a value. Once n is large enough it is a usable estimator of μ. You never build the sampling distribution in practice, because that needs the whole population; you use what it tells you about the one sample mean you have.
Pause, copy the sentence "$\bar{x}$ is a random variable, so it has a distribution", and copy one of the four contexts with a note on why the whole population cannot be measured.
True or false: To use the sampling distribution of the mean in a real study, a researcher must first draw every possible sample of size $n$ and record all of their means.
Worked examples · 3 in a row, reveal as you go
A population consists of the values $1, 5, 9$ . (a) Find $\mu$ . (b) List all samples of size $2$ and their means. (c) State the sampling distribution of the mean and find its mean. (4 marks)
For each description, name which of the three distributions is meant: the population distribution, one sample, or the sampling distribution of the mean. (a) The heights of all $4{,}000$ trees in a forest. (b) The heights of the $50$ trees a surveyor measured. (c) The set of average heights you would get from every possible group of $50$ trees. (3 marks)
An agronomist samples $36$ plots at random from a large farm and finds a mean yield of $4.2$ tonnes per hectare. Explain what $4.2$ is, what it estimates, and why the sampling distribution of the mean matters even though only one sample was taken. (3 marks)
Complete: The sampling distribution of the mean is the distribution of the sample means of possible samples of size $n$ , and it is centred on .
Misconceptions to fix · the 3 traps that cost marks
True or false: In card 05 the mean of the six sample means was exactly $5$ , the population mean. It follows that any single sample of size $2$ from that population has a mean of $5$ .
Activities · practice with the ideas
A population is $4, 8, 12$ . List all samples of size $2$ , find their means, and find the mean of those means. Check it against $\mu$ .
How many samples of size $3$ are there from a population of $6$ values? What would the sampling distribution of the mean for $n = 3$ contain?
Explain in one sentence why the sampling distribution of the mean is narrower than the population distribution.
A pollster reports a mean from one sample of $1{,}200$ voters. Name the population, the statistic, and the distribution that tells you how reliable the statistic is.
Give one reason a biologist cannot measure the whole population, and say what they do instead.
Which does NOT belong? True statements about the sampling distribution of the mean:
Earlier you listed every sample of size $2$ from $2, 4, 6, 8$ and averaged the means.
You should have found $3, 4, 5, 5, 6, 7$ , averaging to $5$ , which is exactly the population mean. Those six numbers are the sampling distribution of the mean for $n = 2$ , and you built it before it had a name. Two features of it carry the rest of the topic: it sits centred on $\mu$ , and it is narrower than the population it came from.
The question Lesson 22 left open, how much do sample means vary, now has somewhere to be answered: it is the spread of this distribution. The next lesson measures that spread, and answers the question this lesson raised in turn, which is what shape the distribution takes.
Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. A population consists of the four values $3, 6, 9, 12$ . (a) Find $\mu$ . (b) List all samples of size $2$ and find each sample mean. (c) State the sampling distribution of the mean for $n = 2$ , and find its mean. (4 marks)
Q2. A hospital records the waiting time of every patient over a year. A researcher takes a random sample of $60$ of those times. Describe the three distributions in play, the population distribution, the distribution of the sample, and the sampling distribution of the mean for $n = 60$ , and state which is narrowest and why. (3 marks)
Q3. An analyst samples $50$ trading days at random from the past decade and finds a mean daily return of $0.04\%$ . (a) Explain why $\bar{x}$ is described as a random variable rather than a fixed number. (b) The analyst says "the sampling distribution of the mean is centred on $\mu$ , so my $0.04\%$ must equal $\mu$ ." Identify the error. (c) Explain why the sampling distribution is useful even though the analyst took only one sample. (4 marks)
Comprehensive answers (click to reveal)
Activity answers:
1. $\mu = (4 + 8 + 12) \div 3 = 8$ . The three samples of size $2$ are $\{4,8\}$ with mean $6$ , $\{4,12\}$ with mean $8$ , and $\{8,12\}$ with mean $10$ . The mean of those means is $(6 + 8 + 10) \div 3 = 8$ , exactly $\mu$ .
2. There are $\binom{6}{3} = 20$ samples, so the sampling distribution of the mean for $n = 3$ contains $20$ sample means, one for each possible sample. Some of those means may repeat.
3. Because averaging pulls extreme values in: a single extreme member can sit far out in the population, but to move an average of $n$ values you would need several extreme members in the same sample at once, which is far less likely.
4. Population: all voters in the relevant electorate or country. Statistic: the value computed from the $1{,}200$ sampled voters. The distribution that says how reliable it is: the sampling distribution of the mean for $n = 1{,}200$ .
5. Most organisms in a habitat cannot be found or caught, populations move, and some measurements destroy the specimen. Instead the biologist draws a random sample, computes $\bar{x}$ , and treats it as an estimate of $\mu$ .
Q1 (4 marks): (a) $\mu = (3 + 6 + 9 + 12) \div 4 = 30 \div 4 = 7.5$ [1]. (b) There are $\binom{4}{2} = 6$ samples: $\{3,6\}$ mean $4.5$ ; $\{3,9\}$ mean $6$ ; $\{3,12\}$ mean $7.5$ ; $\{6,9\}$ mean $7.5$ ; $\{6,12\}$ mean $9$ ; $\{9,12\}$ mean $10.5$ [1 for a complete list, 1 for correct means]. (c) The sampling distribution of the mean is the set $4.5, 6, 7.5, 7.5, 9, 10.5$ , each with probability $\tfrac{1}{6}$ . Its mean is $45 \div 6 = 7.5 = \mu$ [1].
Q2 (3 marks): The population distribution is the individual waiting times of every patient over the year [1]. The sample is the $60$ individual waiting times the researcher actually recorded, again individual times but only $60$ of them [1]. The sampling distribution of the mean for $n = 60$ is the set of mean waiting times you would obtain from every possible sample of $60$ patients; its entries are averages, not individual times. It is the narrowest of the three, because averaging $60$ values pulls extremes in: one very long wait can sit far out among individual times, but it moves an average of $60$ only slightly [1].
Q3 (4 marks): (a) Before the sample is drawn, $\bar{x}$ has no determined value: which $50$ days are selected decides it, and each possible sample gives a possibly different value. A quantity with a distribution of possible values rather than one fixed value is a random variable [1]. (b) "Centred on $\mu$ " is a statement about the sampling distribution as a whole, that the mean of all possible sample means equals $\mu$ . It says nothing about any individual sample mean, and most individual sample means do not equal $\mu$ [1 for identifying the confusion between the distribution and one draw from it, 1 for stating what "centred" actually claims]. (c) The analyst has one draw from that distribution. Knowing it is centred on $\mu$ and knowing how wide it is tells them how far from $\mu$ a single draw is likely to fall, which is what turns one sample mean into a defensible statement about the population [1].
A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.
Start the module quiz →Mark lesson as complete
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