Populations, Samples, and Bias | Grade 7 Math | LevelKite

Your school has $600$ students, and you want to know how many hours a typical student sleeps on a school night. Asking all $600$ would take a long time. Could you ask only some of them and still learn about everyone?

You can, if you choose them with care. What you learn will be approximate, not exact. You already know how to describe a data set with a mean or a median. This lesson is about where the data come from: which group you want to learn about, which members you actually ask, and whether their answers can tell you about the whole group.

Two words name the groups in a question like this one.

The population is the whole group you want to learn about. It can be a group of people or a group of things.
A sample is the part of the population that you actually collect data from.

The question decides the population. If you ask about all the students at your school, the population is all $600$ students. If you ask only about Grade 7 students, the population is only the Grade 7 students. Every member of a sample is also a member of the population, because a sample is a part of it.

The question decides the population

Question	Population	Sample
Which lunch do the $600$ students at a school like best?	All $600$ students at the school	$60$ of those students, who are asked
What is the mean mass of the $2{,}400$ granola bars a factory made today?	All $2{,}400$ bars made today	$120$ of those bars, which are weighed
Do the $900$ households in a neighborhood want a new park?	All $900$ households in the neighborhood	$75$ of those households, which are called

Worked example

Name the population and the sample

A school has $600$ students, and $200$ of them are in Grade 7. The Grade 7 teachers want to know how many minutes Grade 7 students spend on homework each night. They ask $40$ Grade 7 students. Identify the population and the sample. Then find what percent of the population is in the sample.

Start with the question, because the population is the whole group the question is about. The teachers' question is about Grade 7 students, not about every student in the school. So the population is the 200 Grade 7 students, and the 600 is extra information.
$\text{population size} = 200$
Find who actually gives data. Only the 40 Grade 7 students who are asked give answers, so they are the sample. Check the definition: each of the 40 is one of the 200 Grade 7 students, so the sample is a part of the population.
$\text{sample size} = 40$
Write the sample size over the population size to see what part of the population was asked. Dividing the numerator and the denominator by the same nonzero number, here 2, keeps the value of the fraction. It gives a denominator of 100, which is what percent means.
$\frac{40}{200} = \frac{40 \div 2}{200 \div 2} = \frac{20}{100} = 20\%$

Answer

The population is the $200$ Grade 7 students, the sample is the $40$ Grade 7 students who were asked, and the sample contains $20\%$ of the population.

Naming the population and the sample is the first step. Knowing that $40$ out of $200$ students were asked does not yet show whether their answers describe all $200$ students well.

A conclusion about a whole population that you draw from a sample is called an inference. Whether you can trust an inference depends on the sample, and the rest of this lesson shows how.

A representative sample is like the population in the ways that matter for your question. Each kind of member makes up about the same share, or fraction of the whole, in the sample as in the population. Suppose $40\%$ of a school's students ride the bus, and your question is about travel to school. A sample in which close to $40\%$ of the students ride the bus matches the population in this way. A sample in which $90\%$ ride the bus does not.

How a sample is chosen decides whether you can expect it to be representative.

A random sample is chosen by a method that gives every member of the population the same chance of being picked. For example, you could write every member's name on identical slips, mix them, and draw some without looking.
A biased sample is chosen by a method that favors some members of the population. Some members are more likely to be picked, and others are less likely or cannot be picked at all.

Random sampling tends to produce representative samples, because chance does not favor any group. It does not promise a perfect match: a random sample can differ a little from the population just by chance. So a result from a random sample is an estimate, a value you expect to be near the true value for the population but usually not exactly equal to it. A biased method tends to pull the sample toward the favored group, so its results can mislead you.

Key concept

When a sample can tell you about a population

The population is the whole group you want to learn about. A sample is the part of it you collect data from.

An inference about the population is valid (well supported) only when the sample is both:

random: every member of the population has the same chance of being chosen, and
representative: it is like the population in the ways that matter for the question.

Random sampling tends to produce representative samples, but it does not guarantee a perfect match. A biased sample comes from a method that favors some members, so it does not support an inference about the population.

Even a random, representative sample gives only an estimate. Results vary from sample to sample, so keep your conclusion approximate and say "about."

Worked example

Random or biased?

A school has $600$ students. You want to estimate the mean number of hours the students sleep on a school night, using a sample of $30$ students. Method A: put the names of all $600$ students on identical slips in a box, mix them, and draw $30$ without looking. Method B: do the same, but use only the names of the $100$ students in the school's early-morning breakfast club. Which method gives a random sample of the population, and which gives a biased sample?

Name the population first, because a method is random only if it gives every member of the population the same chance. The question is about the students at the school, so the population is all 600 students. Each method produces a sample of 30 students.
$\text{population size} = 600, \quad \text{sample size} = 30$
Test Method A against the definition of a random sample. All 600 names are in the box, and identical, mixed slips favor no one. So every student has the same chance of being picked: 30 out of 600. Dividing the numerator and the denominator by 6 keeps the value of the fraction and gives a denominator of 100.
$\frac{30}{600} = \frac{30 \div 6}{600 \div 6} = \frac{5}{100} = 5\%$
Test Method B in the same way. Only the 100 club members have a slip in the box, so each of them has a chance of 30 out of 100. That is 30 percent, because percent means out of 100.
$\frac{30}{100} = 30\%$
Count the students that Method B leaves out by subtracting the club members from the population. These 500 students have no slip in the box, so their chance of being picked is 0 percent. Method B does not give every member of the population the same chance.
$600 - 100 = 500, \quad 30\% \ne 0\%$
Decide whether the favored group matters for the question. Club members come to school early, so they may wake up earlier and sleep a different number of hours than the other 500 students. A sample from Method B leans toward the club members' answers.
State what each method supports. Method A is random, so you can expect its sample to be representative, and it supports an inference about all 600 students. That inference is an estimate, because another random sample of 30 would probably give a somewhat different mean. Method B could tell you about the 100 club members, but not about all 600 students.

Answer

Method A gives a random sample of the population, because all $600$ students have the same $5\%$ chance of being picked. Method B gives a biased sample, because club members have a $30\%$ chance while the other $500$ students have no chance.

Sometimes you know how the population is made up. Then you can check a sample with numbers.

Pick a feature that is linked to your question, such as how students travel to school. Use the rule below to find the share of each group, once for the population and once for the sample. The total is the size of the population or of the sample, whichever one you are describing. Then write each share as a percent. You cannot compare the counts themselves, because the population and the sample have different sizes. Percents put every share on the same scale, out of $100$ .

If the percents are close, the sample is representative for that feature. This check has limits. It tells you only about the feature you compared. Choosing the sample at random still matters, because random sampling tends to match the population on the features you did not check. And "close" is a judgment, because even random samples vary. A random sample that passes supports an approximate conclusion, not an exact one.

\text{share of a group} = \frac{\text{number of members in the group}}{\text{total number of members}}

Worked example

Check whether a sample is representative

A school has $600$ students: $240$ ride the bus, $210$ walk and $150$ come by car. You want to estimate the mean number of minutes students take to get to school. Sample P is $50$ students picked at random from a list of all $600$ : $21$ ride the bus, $17$ walk and $12$ come by car. Sample Q is the first $50$ students you meet at the school's bus stop: $45$ ride the bus, $4$ walk and $1$ comes by car. Which sample is representative of the population?

Choose what to compare. Travel time is linked to how a student travels, so compare the shares of bus riders, walkers and car riders. Start with the population of 600. Dividing the numerator and the denominator of each fraction by 6 keeps its value and gives a denominator of 100.
$\frac{240}{600}=\frac{40}{100}=40\%,\quad \frac{210}{600}=\frac{35}{100}=35\%,\quad \frac{150}{600}=\frac{25}{100}=25\%$
Find the shares for Sample P, which has 50 students. Multiplying the numerator and the denominator of each fraction by 2 keeps its value and gives a denominator of 100.
$\frac{21}{50}=\frac{42}{100}=42\%,\quad \frac{17}{50}=\frac{34}{100}=34\%,\quad \frac{12}{50}=\frac{24}{100}=24\%$
Compare Sample P with the population, group by group, by subtracting the smaller percent from the larger one. The percents are only 2, 1 and 1 apart, so each group has about the same share in Sample P as in the population.
$42 - 40 = 2,\quad 35 - 34 = 1,\quad 25 - 24 = 1$
Find the shares for Sample Q, which also has 50 students. As before, multiplying the numerator and the denominator by 2 keeps each value and gives a denominator of 100.
$\frac{45}{50}=\frac{90}{100}=90\%,\quad \frac{4}{50}=\frac{8}{100}=8\%,\quad \frac{1}{50}=\frac{2}{100}=2\%$
Compare Sample Q with the population in the same way. Bus riders are 90 percent of the sample but 40 percent of the population, so those percents are 50 apart. Walkers and car riders have much smaller shares than in the population. Sample Q is biased toward bus riders, whose travel times may differ from those of other students.
$90 - 40 = 50,\quad 35 - 8 = 27,\quad 25 - 2 = 23$
State what each sample supports. Sample P is random and close to the population on travel type, so its mean travel time is a reasonable estimate for all 600 students. It stays approximate, because another random sample of 50 would probably give somewhat different numbers. Sample Q is neither random nor representative, so it supports no conclusion about all 600 students.

Answer

Sample P is representative for this question, because its shares of $42\%$ , $34\%$ and $24\%$ are close to the population's $40\%$ , $35\%$ and $25\%$ . Sample Q is not representative, because $90\%$ of its students ride the bus but only $40\%$ of the population does.

Here is the idea to keep. The population is the whole group your question is about, and the sample is the part you collect data from. A sample can tell you about its population only when it is random, so that every member has the same chance of being chosen, and representative, so that it is like the population in the ways that matter. A biased sample favors some members and does not support a conclusion about the whole group. Even a random, representative sample gives an estimate, because results vary from sample to sample.

Next, you will use sample data to make estimates about a population, and you will compare many random samples to see how much an estimate can vary.