Statistics

Sampling

Take a small sample. Compare repeated estimates.

Predict, test and explain

Step 1 of 3

Predict

The Even spread population contains 1 to 20. If a sample uses all 20 members, what is its mean?

population mean=10.5\text{population mean}=10.5
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Population values; highlighted members form your sample\text{Population values; highlighted members form your sample}
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sample mean (bins rounded to whole numbers)\text{sample mean (bins rounded to whole numbers)}

Scroll the diagram sideways to see every label.

samples=0\text{samples}=0

Take a sample, then repeat to explore how estimates vary.

InvestigateCan two samples of the same size give different estimates?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

A population is 2, 4, 6, 8 and 10. A random sample contains 2 and 4. Find both means.

population:2,4,6,8,10sample:2,4\text{population}:2,4,6,8,10\qquad \text{sample}:2,4

Hint

Use each group's own total and number of values.

Reveal answer

The sample mean is 3; the population mean is 6. A random sample can differ from its population.

xˉsample=62=3,μ=305=6\bar{x}_{\text{sample}}=\frac{6}{2}=3,\quad \mu=\frac{30}{5}=6

Build an explanation

Hint 1

Compare the sample mean with the mean of the whole population.

Hint 2

Repeat samples of the same size and watch the estimates vary.

Hint 3

Larger random samples generally vary less; sampling the entire population gives its exact mean.

Worked example

This example uses fixed values, separate from the diagram controls.

A population contains 1 to 20. A sample contains 1, 5, 10 and 20. Compare their means.

  1. The population mean is (1 + 20)/2 = 10.5.
  2. The sample total is 36; divide by 4.

Answer: Sample mean 9; population mean 10.5.

Connect this idea

Watch out: Different random samples can give different estimates without either calculation being wrong.