From sample to population

The inference problem

You almost never have data for a whole population — you have a sample.
A sample statistic (like the sample median) is only an estimate of the population value, and it changes from sample to sample — this is sampling variability.
A formal inference makes a justified statement about the population while being honest about that variability.

Why one number is not enough

If group A's sample median is a little higher than group B's, that could be a real population difference — or just the luck of the sample.
To make a call you need a range of plausible values for the population quantity, not a single point — a confidence interval.

The bootstrap idea

The sample is our best picture of the population, so we treat it as if it were the population and resample from it.
Resample with replacement: draw values from the sample, putting each one back, until you have a new sample the same size as the original — a bootstrap sample. Some values appear more than once; some not at all.
Each bootstrap sample gives a slightly different statistic. Doing this many times shows how much the statistic would vary from sample to sample.

Worked ExampleOne bootstrap resample

A sample of six reaction times (ms) is $210, 225, 240, 250, 260, 300$ . One bootstrap resample (with replacement) turns out to be $225, 210, 260, 225, 300, 240$ . Find its median, and say what this shows.

Step 1 — Order the resample

$210, \; 225, \; 225, \; 240, \; 260, \; 300$

Step 2 — Median of six values (mean of the 3rd and 4th)

$\text{median} = \frac{225 + 240}{2} = 232.5 \text{ ms}$

Step 3 — Interpret

The original sample median was $245$ ms; this resample gives $232.5$ ms. Different resamples give different medians — that spread is exactly the sampling variability the bootstrap is measuring.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A bootstrap resample of a sample of five heights (cm) is $162, 170, 170, 181, 162$ . Find its median.

Merit

Explain why a bootstrap resample is taken with replacement rather than without.

Excellence

A student takes one sample and reports only its median as the answer for the population median. Explain why this is not a sound inference, and what should be done instead.

From sample to population

Covers AS91582

The inference problem

You almost never have data for a whole population — you have a sample.
A sample statistic (like the sample median) is only an estimate of the population value, and it changes from sample to sample — this is sampling variability.
A formal inference makes a justified statement about the population while being honest about that variability.

Why one number is not enough

If group A's sample median is a little higher than group B's, that could be a real population difference — or just the luck of the sample.
To make a call you need a range of plausible values for the population quantity, not a single point — a confidence interval.

The bootstrap idea

The sample is our best picture of the population, so we treat it as if it were the population and resample from it.
Resample with replacement: draw values from the sample, putting each one back, until you have a new sample the same size as the original — a bootstrap sample. Some values appear more than once; some not at all.
Each bootstrap sample gives a slightly different statistic. Doing this many times shows how much the statistic would vary from sample to sample.

Worked ExampleOne bootstrap resample

Step 1 — Order the resample

$210, \; 225, \; 225, \; 240, \; 260, \; 300$

Step 2 — Median of six values (mean of the 3rd and 4th)

$\text{median} = \frac{225 + 240}{2} = 232.5 \text{ ms}$

Step 3 — Interpret

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A bootstrap resample of a sample of five heights (cm) is $162, 170, 170, 181, 162$ . Find its median.

Merit

Explain why a bootstrap resample is taken with replacement rather than without.

Excellence

A student takes one sample and reports only its median as the answer for the population median. Explain why this is not a sound inference, and what should be done instead.