A random sample is like a small copy of the population. So the fraction of the sample with some feature is a good estimate for the population. Turning that into a number is solving a proportion.
Key idea
Use the sample's proportion (or its mean) for the whole population: $$\frac{\text{part of sample}}{\text{sample size}} = \frac{\text{part of population}}{\text{population size}}$$ Different random samples give somewhat different results. Larger samples vary less.
Why it works
A random sample gives every member the same chance, so on average it has the same mix as the population. Any single sample can be a bit off, the way 10 coin flips rarely give exactly 5 heads. Taking several samples shows how much the estimates wobble.
Worked examples
Worked example: Estimating a count
In a random sample of 50 students, 18 prefer an afternoon recess. The school has 600 students. About how many prefer it?
- Sample proportion: $\frac{18}{50}$.
- $\frac{18}{50} \times 600 = 18 \times 12 = 216$.
Answer: About 216 students
Worked example: Combining samples
Three random samples of 40 students had 12, 15 and 9 students who play an instrument. Estimate the percent of all students who do.
- The samples differ, which is normal. Combine them: $12 + 15 + 9 = 36$ out of $120$.
- $\frac{36}{120} = 0.3 = 30\%$.
Answer: About 30%
Worked example: Using a mean
A random sample of families in a town has a mean of 1.6 pets per family. There are 450 families. Estimate the total number of pets.
- About 1.6 pets for each family.
- $1.6 \times 450 = 720$ pets.
Answer: About 720 pets
Common mistake
Expecting every sample to give the same answer, and deciding a sample is "wrong" when it differs from another. Random samples vary. Their results cluster around the true value, and with more or bigger samples the estimate gets steadier.