Unit 1 · Topic 1.1 Beta

Introducing Statistics: What Can We Learn from Data?

Statistics answers questions about a population, the whole group of interest, using data from a sample, the part you actually measure.

Practice 1: Formulate QuestionsPractice 2: Collect Data

Part 1 · Hook

Why this matters

A student council at a school of 2,400 students wants to know whether students would use a quiet study room at lunch. It cannot ask everyone, so it asks 120 students chosen at random, and 78 say yes. What can 78 answers out of 120 tell the council about all 2,400 students? That question, how to learn about a big group from a smaller one, is what statistics is for.

Part 2 · See it

See it first

A study of a school. On the left, a population of 2,400 students drawn as gray dots, a few in plum. An arrow labeled choose leads to a sample of 120 students. An arrow labeled ask leads to the data: each answer is yes or no, and 78 of 120 say yes. A curved arrow returns to the population with the conclusion, in context: about 65% of the school's students say yes.
A study moves from a question about a population, to a sample, to data, and back to a conclusion about the population, stated in context. LevlPrep original diagram.

Part 3 · Step by step

How it works, step by step

  1. The question is about a large group, the populationasking everyone would take too long and cost too much
  2. So you collect data from a sample, a smaller part of the populationthe answers vary from person to person, so you summarize them: 78 of 120 is 65%
  3. The summary describes the sampleit is a good estimate for the population only if the sample represents the population
  4. A different sample would give a slightly different resultso a conclusion about the population is an estimate, stated in context, never a certainty

Part 4 · Key ideas

Key ideas

  • Statistics is the science of collecting, summarizing and reading data to answer questions whose answers vary.
  • The population is the whole group a question is about. The sample is the part of it that data come from.
  • A study uses data from a sample to answer a question about a population. It works only if the sample represents the population.
  • State every result in context: who was studied, what was measured, and what the number tells you.
  • Another sample gives a slightly different result, so a sample result is an estimate for the population, not its exact value.

Part 5 · Misconception

A common mistake

The wrong idea: 78 of the 120 students said yes, so exactly 65% of all 2,400 students would use the room.

What actually happens: The 65% describes the sample. It estimates the school’s percent, and another random sample of 120 would give a slightly different number.

Part 6 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

A study room at lunch

A high school has 2,400 students. The student council wants to know whether students would use a quiet study room at lunch. The council chooses 120 students at random from the school list and asks each one, "Would you use a quiet study room at lunch once a week or more?" Of the 120 students, 78 answer yes (invented data).

1. What is the population in this study?

  1. The 120 students asked
  2. All 2,400 students at the school
  3. The 78 students who answered yes to the question
  4. Students who already study in the library at lunch
Show the answer

The population is the whole group the question is about: all 2,400 students at the school.

  • The 120 students asked: These 120 are the sample: the part of the population the council actually asked.
  • Correct: All 2,400 students at the school: Right: the question is about every student at the school, so the whole school is the population.
  • The 78 students who answered yes to the question: The 78 are a count inside the sample, not the group the question is about.
  • Students who already study in the library at lunch: The question is about every student, not just those who already use a study space.

2. What is the sample in this study?

  1. The 2,400 students enrolled at the high school
  2. The 78 who said yes
  3. The members of the student council who ran the study
  4. The 120 students the council asked
Show the answer

The sample is the part of the population that data come from: the 120 students who were asked.

  • The 2,400 students enrolled at the high school: That is the population, the whole group the question is about.
  • The 78 who said yes: They are part of the sample, but the sample is everyone who was asked, yes or no.
  • The members of the student council who ran the study: The council ran the study; they are not the students who were asked.
  • Correct: The 120 students the council asked: Right: the sample is the group data were actually collected from.

3. Which statement correctly uses the data?

  1. 65% of the students asked said yes
  2. Exactly 65% of the school’s 2,400 students would use the room
  3. 78% of the school would use it
  4. Exactly 78 students at the school would use the study room
Show the answer

The 78 yes answers are 78 out of 120, which is 65% of the sample. That is a fact about the sample and an estimate for the school.

  • Correct: 65% of the students asked said yes: Right: 78 out of 120 is 0.65, and it describes the students who were asked.
  • Exactly 65% of the school’s 2,400 students would use the room: The 65% describes the sample. A different sample would give a slightly different number, so it is not exact for the whole school.
  • 78% of the school would use it: 78 is the count of yes answers, not a percent: 78 out of 120 is 65%.
  • Exactly 78 students at the school would use the study room: Just 120 of the 2,400 students were asked; far more than 78 students at the school could want the room.

4. A second group of 120 students is chosen at random from the same school list and asked the same question. What is most likely?

  1. Exactly 78 yes answers again
  2. A percent very far from 65%, since each group is chosen by chance
  3. A percent near 65%, but probably not exactly 65%
  4. Exactly 65% of the 120, because the method of choosing is the same
Show the answer

Different students give different answers, so a new group of 120 gives a result that varies a little from the first one but is usually close to it.

  • Exactly 78 yes answers again: Different students are chosen each time, so the count changes from group to group even when the school stays the same.
  • A percent very far from 65%, since each group is chosen by chance: Groups chosen by chance from the same school tend to give similar results; a huge change would be surprising.
  • Correct: A percent near 65%, but probably not exactly 65%: Right: answers vary from student to student, so each group of 120 gives a somewhat different percent, usually close to the first.
  • Exactly 65% of the 120, because the method of choosing is the same: The method is the same, but the students are different, so the result usually shifts a little.

5. Which question is answered by collecting data that vary from one person to another?

  1. How long do students here spend on homework on a typical night?
  2. How many minutes did I spend on my own homework last night?
  3. How many minutes long is the lunch period at this school?
  4. How many students are enrolled at this school this year?
Show the answer

Statistics answers questions where the answers vary, so the data must come from many individuals.

  • Correct: How long do students here spend on homework on a typical night?: Right: homework time differs from student to student, so it takes data from many students to answer.
  • How many minutes did I spend on my own homework last night?: That is one number about one person; there is no variation to study.
  • How many minutes long is the lunch period at this school?: The lunch period has one fixed length; looking it up answers the question.
  • How many students are enrolled at this school this year?: That is one fixed number from the school records, not data that vary.

6. A park department asks 300 park visitors whether they want more shade trees; 210 say yes. Which conclusion is stated in context?

  1. The answer is 70%
  2. Most of the people who answered the question want more of it
  3. 210 is greater than 300 divided by 2, so the result is a majority
  4. 70% of the 300 visitors asked want more shade trees
Show the answer

A conclusion in context names who was studied, what was measured, and what the number tells you: 210 out of 300 is 70% of the visitors surveyed.

  • The answer is 70%: A number with no context does not say who, or what the 70% measures.
  • Most of the people who answered the question want more of it: This does not name the people, the park or the trees.
  • 210 is greater than 300 divided by 2, so the result is a majority: True arithmetic, but it says nothing about the visitors or the trees.
  • Correct: 70% of the 300 visitors asked want more shade trees: Right: it says who was asked, what about, and the number: 210 out of 300 is 70%.

Part 7 · Summary

Summary

Statistics answers questions about a population, the whole group of interest, using data from a sample, the part you actually measure. Answers vary, so a sample result is an estimate. It is useful only when the sample represents the population, and every result is stated in context.

Part 8 · Up next

What comes next

Part 9 · Connections

Connections

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