Understand · explore · practise
Define the population you want to study, distinguish a sample from a census, and see why sample estimates vary. Original examples, a manual model and worked practice.
Before you startCalculate a mean and a percentage.
01 / Population and sample
Population = the whole set of units of interest
A sample contains selected units from that population.
A depot wants the mean processing time for the six parcels handled on one particular morning. Those six parcels form its population. Measuring only three gives a sample. A different question—such as the mean for every parcel handled this month—has a different population.
The model uses invented times. Try a low-valued and a high-valued sample, then choose the census.
Sample: A, B, C (3 parcels).
(2 + 4 + 6) ÷ 3 = 4 minutes.
Estimate minus population mean: 4 − 7 = −3 minutes.
The population mean is 42 ÷ 6 = 7 minutes. All times here are invented and known exactly. Non-integer outputs are rounded to six decimal places. You choose each subset manually; this control does not draw a random sample.
02 / Define the population
Population: all full-time students enrolled at this college this term
Include students who are absent on survey day.
Sampling unit: one eligible student
This is the individual selected.
Variable: that student’s usual one-way journey time in minutes
Specify what is measured, including the direction of the journey.
A library wants to estimate the proportion of its current members who use its online catalogue. State a suitable population and sampling unit.
Do not limit the population to people physically visiting today.
Population: all current members of that library at the survey date. Sampling unit: one current member. Today’s visitors are only part of that population.
A manufacturer investigates the mass of bags filled by one machine during Monday’s morning shift. Identify the sampling unit and the variable.
Distinguish the object from its measurement.
The sampling unit is one bag filled by that machine in that shift. The variable is its filled mass, measured in a stated unit such as grams.
03 / Census or sample?
A complete census measures every member. A sample survey measures a selected subset and uses it to learn about the population. Sending a questionnaire to everybody is an attempt at a census; if only some reply, the collected responses are incomplete.
A sampling frame is the list or other record used to select units. It may not perfectly match the intended population. The next lesson examines that problem in detail.
A club has 36 members. Its treasurer records the membership fee paid by every member. Is this a census or a sample of this club’s members?
Compare the observed units with the defined population.
A census: all 36 members are included.
A survey is emailed to all 500 staff, but 180 respond. Can the 180 replies be described as a complete census?
An invitation is not a measurement.
No. It is an attempted census with only 180 responses. Nonrespondents may have different views, so the replies may give a biased picture.
04 / Use a sample to estimate
Observed times: 2, 6 and 12 minutes
These three parcels are a sample of the six.
Sample mean = (2 + 6 + 12) ÷ 3 = 20/3 minutes
This is approximately 6.667 minutes.
Population mean = (2 + 4 + 6 + 8 + 10 + 12) ÷ 6 = 7 minutes
We know it here because every value is given.
Estimate − true mean = 20/3 − 7 = −1/3 minute
This particular estimate is slightly low.
Find the mean of sample B, D, F: 4, 8 and 12 minutes. Compare it with the population mean 7.
Add the sample’s values and divide by its size.
The sample mean is 24 ÷ 3 = 8 minutes, one minute above the population mean.
In a sample of 25 boxes, 7 have damaged labels. Estimate the percentage of all boxes with damaged labels. Does the sample prove that percentage holds exactly?
Use 7/25, but distinguish an estimate from a known population value.
The estimate is 28%. It does not prove that exactly 28% of the whole population have damaged labels.
05 / Why estimates vary
Sampling variation
Even a properly randomised method can produce different sample estimates on different draws.
Samples A, B, C and D, E, F in the model give means 4 and 10. Both are possible selections of three distinct parcels. An unlucky sample does not by itself prove a random selection method was biased. Bias is a systematic tendency in a method; sampling variation is the change caused by which units happen to be selected.
The animation compares these two deliberately chosen extreme samples. It does not claim they are typical.
Pause, replay or seek freely. The notes explain the same idea and stay in view.
Two students use the same sound random method to select 30 records, but get different means. Must one have made an arithmetic error?
They may have selected different records.
No. Different samples can produce different means even when the method and arithmetic are correct.
06 / Choose between a census and a sample
A census can be manageable for a small accessible population and gives information about every unit, including small groups. It can be expensive or slow for a large population. A sample usually costs less and places less burden on respondents, but leaves uncertainty about unobserved units.
Testing can also destroy or alter the item. Measuring the breaking strength of every item in a delivery could leave nothing usable.
A factory must estimate the mean breaking strength of a batch of 2,400 ceramic tiles. Why might it test a sample?
Consider what measuring the breaking strength does to a tile.
The test breaks each tile. Testing the whole batch would destroy all 2,400; a suitably selected sample leaves most usable. The estimate still has sampling uncertainty.
A team needs each of its 14 players’ shirt sizes before ordering kit. Give a reason to use a census.
An average or estimated distribution cannot tell the team everyone’s size.
Fourteen players are practical to ask, and an individual size is needed for every player. A sample could miss someone’s required size.
07 / A census can still go wrong
All units are measured
There is no uncertainty from selecting only a subset.
Every recorded mass is systematically too high
This is a measurement problem.
The recorded mean is also 5 g too high
A census does not remove measurement error.
Other problems include incomplete replies, unclear questions, missing records and data-entry mistakes. “Census” describes coverage; it does not guarantee perfect data.
“We asked everyone, so our survey result must be correct.” Give two possible problems.
Think about both replies and the way answers are measured.
Some people may not reply. A leading or ambiguous question may distort answers. Data-entry errors are another valid concern.
08 / Does a larger sample help?
Under comparable sound random sampling, a larger sample generally reduces sampling variability. It does not guarantee that one realised sample is closer to the truth. More observations from an unrepresentative group do not automatically fix the exclusion of other groups.
Choose a sample size with the required precision, population variation, cost and time in mind. More detailed study of uncertainty comes later in statistics.
A college estimates all students’ travel times by asking only students in its bicycle club. It increases the sample from 20 to 60 club members. Explain the remaining problem.
Which students still have no chance of being included?
Students outside the bicycle club are still excluded. Their journey times may differ. Increasing this sample does not repair the coverage bias.
09 / Plan before collecting
A depot randomly selects 40 of the 800 parcels it processed last Tuesday. Their mean processing time is 6.2 minutes. Identify the population, unit and estimate, and explain why “all parcels always take 6.2 minutes” is wrong.
Separate the population’s average from each individual observation, and Tuesday from other days.
The population is the 800 parcels processed at that depot last Tuesday; a unit is one of those parcels. The sample mean 6.2 minutes estimates that population’s mean. Individual times vary, the estimate need not equal the true mean, and this sample alone does not justify a claim about all other days.
Section 1 of 9 · Population and sample