A parameter describes a population and is usually unknown. A statistic describes a sample and is what you compute.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The sampling distribution
It is the distribution of a statistic across every possible sample of a given size. It is a distribution of statistics, not of individuals.
Unbiased estimators
A statistic is unbiased when its sampling distribution is centred on the true parameter. The sample mean is unbiased for the population mean.
The standard error
The spread of the sampling distribution is the standard error. For a mean it is sigma ÷ √n.
The square root
Quadrupling the sample size halves the standard error. Precision improves with the square root, which is why it is expensive.
Three distributions to keep apart
The population, one sample, and the sampling distribution are different things. Confusing them is the deepest error in the course.
How a statistic varies from sample to sample
Take many samples and compute the statistic each time; the distribution of those values is the sampling distribution. It is the bridge between one sample and a claim about the population.
Three distributions to keep apart
The population distribution, the distribution of one sample, and the sampling distribution of a statistic. Confusing them is the core misunderstanding this unit exists to prevent.
Unbiased means centred correctly
A statistic is unbiased if its sampling distribution is centred at the parameter. The sample mean and sample proportion are both unbiased, which is what makes them the natural estimators.
Larger samples vary less
The standard deviation of the sampling distribution shrinks with the square root of n. That is why bigger samples give more precise estimates — and why the improvement slows as samples grow.
Step 2: Try It Yourself
Tap and try it out.
- Minimum9
- Lower quartile9.50
- Median10
- Upper quartile11.50
- Maximum12
- Interquartile range2
Each of the four sections holds a quarter of the values, however wide it looks. A narrow box means the middle half of the data is packed close together.
Step 3: Watch an Example
One step at a time.
Watch Kofi Compute a Standard Error
A population has standard deviation 20, and Kofi takes samples of size 100.
- Step 1
The standard error of the mean is sigma ÷ √n.
Step 4: Your Turn
Practice makes it stick.
The Standard Error
Problem 1 of 2
Population standard deviation 20, sample size 100. What is the standard error of the mean?
The Larger Sample
Problem 2 of 2
Same population, sample size 400. What is the standard error?
Centre and Spread of a Statistic
1 of 8
Sigma 30, n = 9. Standard error of the mean?
2 of 8
Sigma 12, n = 36. Standard error?
3 of 8
Quadrupling n does what to the standard error? 1 halves it, 2 quarters it.
4 of 8
A value describing a population. 1 parameter, 2 statistic.
5 of 8
Is the sample mean unbiased for the population mean? 1 yes, 0 no.
6 of 8
Sigma 50, n = 25. Standard error?
7 of 8
Sort each description by which distribution it refers to.
Tap something to move it.
- Empty
- Empty
8 of 8
Sigma 8, n = 64. Standard error?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Sigma 15, n = 25. What is the standard error of the mean?
Question 2 of 2
What does a sampling distribution describe?
What You Learned
- A sampling distribution describes how a statistic varies across samples.
- An unbiased statistic centres on the true parameter.
- The standard error shrinks with the square root of the sample size.