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Parameters against statistics, the sampling distribution of a mean and of a proportion, the standard error and its square root, and the central limit theorem.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you can say what a sampling distribution is — the distribution of a statistic across every sample that might have been taken — and give its centre, its spread and its shape for a sample mean and a sample proportion. You can compute a standard error, explain why it falls like the square root of the sample size rather than in proportion to it, and state the central limit theorem in terms of the right one of the three distributions in play.
You know that the variances of independent quantities add, and that adding variances makes the spread of a sum grow like $\sqrt{n}$ rather than like $n$. This lesson turns that one algebraic fact into the whole machinery of inference.
Parameter: a number describing the population — $\mu$, $\sigma$, $p$. Fixed, and unknown.
Statistic: a number computed from a sample — $\bar{x}$, $s$, $\hat{p}$. Known, and different every time.
Sampling distribution: the distribution of a statistic across all the samples that could have been taken.
Standard error: the standard deviation of that sampling distribution.
Unbiased: the sampling distribution is centred on the parameter.
Imagine a population whose shape is badly skewed — household incomes, say, most of them modest and a few enormous. Now do something no real study ever does: take a sample of $n$, write down its mean, put it back, and repeat ten thousand times. Then draw a histogram of the ten thousand means.
With $n = 1$ the histogram is the population itself: skewed, long-tailed.
With $n = 4$ it is already narrower and less skewed. Averaging four values makes an extreme result need four extreme values at once.
With $n = 25$ it is close to symmetric and clearly bell-shaped, and it is five times narrower than the population, not twenty-five times.
With $n = 100$ it is a tight, unmistakable normal curve, centred exactly where the population is centred, one tenth as wide.
Three things happen at once, and all three matter. The centre never moves — the sample mean is unbiased. The spread shrinks like $\sqrt{n}$ — quadrupling the sample halves the error, and no faster. And the shape becomes normal whatever the population looked like, which is the central limit theorem.
The histogram you have just imagined is a sampling distribution, and it is the single most slippery object in statistics: nobody ever sees one. A real study takes exactly one sample. Everything that follows in this course is the business of reasoning about that invisible histogram from the one dot of it you actually have.
A parameter describes the population and is fixed and unknown; a statistic is computed from a sample and is different every time a sample is taken. Because it is different every time, a statistic has a distribution of its own — its sampling distribution — and inference is entirely a matter of knowing that distribution's shape, centre and spread.
For a sample mean, all three are known:
$$\mu_{\bar{x}} = \mu, \qquad \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}, \qquad \bar{x} \text{ is approximately normal for large } n$$
The first says the sample mean is unbiased — it does not systematically miss high or low. The second, the standard error, follows directly from variances adding: the variance of a sum of $n$ independent observations is $n\sigma^2$, and dividing by $n$ to make an average divides the variance by $n^2$, leaving $\sigma^2/n$ and a standard deviation of $\sigma/\sqrt{n}$. The third is the central limit theorem, and it is what lets a normal calculation be used on a population whose shape nobody knows.
For a sample proportion the same three facts hold with $\hat{p}$ in place of $\bar{x}$: it is centred on $p$, its standard error is $\sqrt{\dfrac{p(1-p)}{n}}$, and it is approximately normal when $np$ and $n(1-p)$ are both at least about $10$.
Another way: picture
Picture two histograms side by side, drawn from the same skewed population. The left one is the data from a single sample: skewed, wide. The right one is ten thousand sample means: symmetric, bell-shaped and far narrower, sitting over the same centre. They are pictures of different things, and confusing them is the classic error.
Another way: steps
| Sample size | Standard error, $\sigma = 20$ | Cost relative to $n=25$ |
|---|---|---|
| 25 | 4.00 | 1× |
| 100 | 2.00 | 4× |
| 400 | 1.00 | 16× |
| 1,600 | 0.50 | 64× |
Each halving of the error costs four times the data. This is why surveys settle at around a thousand respondents: the next meaningful improvement costs four thousand.
"The central limit theorem says large samples are normal." It says the distribution of the sample mean is approximately normal. The data in a large sample from a skewed population is just as skewed as the population — more visibly so, in fact.
"The standard error is the standard deviation of the data." It is the standard deviation of the statistic. The data has spread $\sigma$; the mean of $n$ observations has spread $\sigma/\sqrt{n}$, which is smaller and gets smaller still.
"A bigger population needs a bigger sample." The population size does not appear in $\sigma/\sqrt{n}$ at all. A sample of 1,000 measures a country as precisely as it measures a town — a fact that sounds wrong and is one of the most useful in the subject.
$\sigma = 12$, $n = 36$.
Given.
$\sqrt{36} = 6$.
Root of the sample size.
$SE = 12 / 6 = 2$: the sample mean varies about a sixth as much as a single observation.
$\hat{p} = 0.4$ in a sample of $100$.
Given.
$p(1-p) = 0.4 \times 0.6 = 0.24$.
Multiply first.
$0.24 / 100 = 0.0024$, and $\sqrt{0.0024} \approx 0.049$ — about five percentage points.
Root last.
A population has standard deviation $12$. Samples of size $9$ are taken. What is the standard deviation of the sample mean?
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A population is strongly skewed. Samples of size $64$ are taken and each sample's mean is recorded. What does the central limit theorem say about those means?
In a sample of $400$, a proportion $\hat{p} = 0.25$ said yes. What is the standard error of that proportion, to four decimal places?
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A survey's standard error must be divided by $5$. By what factor must the sample size be multiplied?
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A researcher takes one sample of $16$ and draws its histogram. Is that the sampling distribution of the mean?
A quantity has population standard deviation $12$. A study needs the standard error of its sample mean to be no more than $3/2$. What is the smallest sample size that achieves it?
answer
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A study is planned with $\sigma = 60$ and $n = 4$, giving a standard error of $30$. Using the controls, reach a standard error of $15$ — and note what it costs.
This task has no paper form; do it on a device.
You can describe and compute with a sampling distribution. Without looking: what does the central limit theorem say is approximately normal, and by what factor must a sample grow to halve its standard error?