Math Core

Once you have a random sample, you can use it to make an inference: an educated estimate about the whole population. In this lesson you'll use proportions to scale a sample up to a population, and you'll see why different samples give slightly different answers.

From a sample to a population

A random sample should look like a small copy of the population. So if $30\%$ of the sample has some feature, a good estimate is that about $30\%$ of the population has it too.

Estimating with a sample

If a random sample is representative, then

\dfrac{\text{part of sample}}{\text{whole sample}} \approx \dfrac{\text{part of population}}{\text{whole population}}.

Use the sample's fraction (or percent, or mean) as your estimate for the population.

Worked example: Estimating a count

A middle school has $720$ students. In a random sample of $60$ students, $14$ said they play a musical instrument. Estimate how many students in the school play an instrument.

Solution. In the sample, $\dfrac{14}{60}$ of the students play an instrument. Apply that fraction to the whole school:

\dfrac{14}{60} \times 720 = 14 \times 12 = 168.

About $168$ students play an instrument. You can also set up a proportion, $\dfrac{14}{60} = \dfrac{x}{720}$ , and solve to get the same answer.

The word "about" matters. A sample gives an estimate, not an exact count. If you took a different random sample, you might get $12$ or $16$ instrument players instead of $14$ .

Estimating a mean

You can estimate a population's average the same way: the mean of a random sample is a good estimate of the mean of the population.

Worked example: Estimating an average

A random sample of $8$ students at a school recorded the number of minutes they spent reading last night:

20,\ 45,\ 30,\ 0,\ 35,\ 25,\ 40,\ 45

Estimate the mean reading time for all students at the school.

Solution. Add the values and divide by $8$ :

\dfrac{20 + 45 + 30 + 0 + 35 + 25 + 40 + 45}{8} = \dfrac{240}{8} = 30.

A good estimate is that students at the school read for about $30$ minutes a night, on average.

Samples vary

If several people each take a random sample from the same population, they will usually get different results. This is called sampling variability. It's normal, and it doesn't mean anyone made a mistake.

Suppose $10$ students each randomly sample $20$ people from a town of $5{,}000$ and record what fraction own a cat. Their sample proportions are shown in the dot plot below.

Proportion of cat owners in 10 random samples of 20 people

The results range from $0.25$ to $0.45$ , but most cluster around $0.35$ . That clustering gives you confidence: the true proportion of cat owners in town is probably close to $0.35$ , so a reasonable estimate is $0.35 \times 5{,}000 = 1{,}750$ cat owners.

What sampling variability tells you

Different random samples give different estimates.
The estimates tend to cluster around the true population value.
Larger random samples vary less, so their estimates are usually closer to the truth.

Worked example: Combining several samples

Four students each took a random sample of $25$ seventh graders at a school with $400$ seventh graders, asking whether they have a pet. They found $10$ , $12$ , $9$ and $13$ pet owners. Use all four samples to estimate the number of seventh graders who have a pet.

Solution. Together the samples include $4 \times 25 = 100$ students, and $10 + 12 + 9 + 13 = 44$ of them have a pet. That's $\dfrac{44}{100} = 44\%$ .

0.44 \times 400 = 176.

About $176$ seventh graders have a pet. The individual samples would have given estimates from $\dfrac{9}{25} \times 400 = 144$ to $\dfrac{13}{25} \times 400 = 208$ , so combining them into one larger sample gives a steadier estimate.

Common mistake

An inference is only as good as the sample. If the sample is biased (for example, you asked only your friends), scaling it up just produces a bigger wrong answer. Always check that the sample was random before trusting an estimate.

Tip

Check that your estimate is reasonable: the population estimate should be the same fraction of the population as the part is of the sample. For example, $14$ out of $60$ is a bit less than one quarter, and $168$ out of $720$ should also be a bit less than one quarter. It is, since $720 \div 4 = 180$ .

Practice

Practice 1

A school has $450$ students. In a random sample of $30$ students, $9$ said pizza is their favorite lunch. Estimate the number of students in the school whose favorite lunch is pizza.