SAT® · Math: Problem-Solving and Data Analysis
Sample Statistics as Estimates of a Population
A random sample's mean or proportion is a close estimate of the same value for the whole population, so multiplying that proportion by the population size estimates a count. The estimate only holds up if the sample was chosen randomly.
Nobody can ask two million people a question, but a random sample lets you skip that. If every person in the city had an equal, independent chance of landing in the sample, the people who did get picked look like the city in miniature: same mix of ages, incomes, opinions, left and right hands. So whatever fraction of the sample has a trait is a good stand-in for the fraction of the whole population that has it, and the sample's average value stands in for the population's average value. Neither is exact, but with random selection the sample isn't systematically tilted in any direction.
Turning a sample proportion into a population count is one multiplication. Find the fraction in the sample, then apply that same fraction to the full population size, since the sample fraction is your best estimate of the population fraction. For a mean, the logic is identical: the sample's average stands in for the population's average, and multiplying that average by the number of people or units gives an estimated total.
This idea is the quiet partner behind margin of error and causal claims elsewhere in this unit: an estimate from a sample is only as trustworthy as the randomness behind how that sample was chosen, and the margin of error tells you how much that single estimate might drift from the true population value.
Key terms
- Sample proportion
- The fraction of a sample that has some trait, used to estimate the same fraction in the full population.
- Population
- The entire group a study wants to describe, which is usually too large to measure completely.
- Random sample
- A subset chosen so every member of the population has an equal, independent chance of being included.
- Self-selected sample
- A sample made of people who chose on their own to participate, which can misrepresent the population regardless of size.
Watch it explained
- How it worksHow does a sample mean stand in for a population mean you could never actually measure?
- How it worksWalk me through turning a sample proportion into an estimate of a whole population's count.
- Worked exampleFind the estimated number of lefties in a town of 24000 when 37 of 400 sampled are lefties.
- Common mistakeWhy isn't an enormous sample useful when the people in it were not chosen at random?
Check yourself
End of topic.