AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/1/1-1)
Unit 1 · Topic 1.1
1.1 Introducing Statistics: What Can We Learn from Data?
Statistics is how you learn about a big group when you can only look at part of it. Every study starts with an investigative question about a population, and you answer it with data from a sample. Getting the vocabulary straight here (population, sample, N, n, in context) pays off in every unit after this one.
Key terms
- investigative question
- population (size N)
- sample (size n)
- data set
- in context
What a statistical study is
A statistical study collects data from some members of a group to answer a question about the whole group. You do this because checking everyone is usually impossible, too slow or too expensive. A school can't test the battery life of every phone a company makes, and a news outlet can't call every voter in the country.
A single piece of information about one person or item is a datum. A collection of data is a data set. If you record the height of 30 students, each height is a datum and the 30 heights together are the data set.
Population and sample
The population is every person or item you want to learn about. Its size is written N. The sample is the smaller group you actually collect data from. Its size is written n.
Be precise when you name these. "Students" is too vague. "All 1,850 students enrolled at Lincoln High School this year" is a population. "The 120 Lincoln students who answered the survey" is a sample.
| Term | What it means | Symbol |
|---|---|---|
| Population | Everyone or everything you want to learn about | Size N |
| Sample | The part of the population you collect data from | Size n |
| Data set | All the data you collected | none |
Investigative questions
An investigative question is the question your study is designed to answer. It is about the population, not just the sample. "What proportion of Lincoln students get at least 8 hours of sleep on school nights?" is an investigative question. "How much did Maya sleep last night?" is not, because it's about one person and needs no statistics.
A good investigative question has three traits:
- It has a clear purpose that you decide before you see any data.
- It stays fixed. You don't rewrite it after the results come in to match what you found. Changing the question to fit the data makes the conclusion misleading.
- It can actually be answered with data you're able to collect. "Are students happy?" is hard to measure. "What proportion of students rate their school day 4 or 5 on a 1–5 scale?" can be measured.
Working in context
"In context" means connecting every number and every part of a study back to the real situation. A sample mean of 6.8 means nothing alone. "The 120 surveyed students slept an average of 6.8 hours on school nights" is in context.
Free-response answers lose credit for statements without context more often than for almost any other reason. Make it a habit from day one: name the units, the variable and the group every time you report a result.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Identify the parts of a study
A city wants to know what proportion of its 42,000 households would use a new curbside composting program. It mails a short survey to 600 randomly chosen households, and 410 reply. Identify the population, the sample and the investigative question, and give N and n.
Show the solutionHide the solution
- Step 1: The population is the group the city wants to learn about: all 42,000 households in the city. So N = 42,000.
- Step 2: The sample is the group the city actually got data from. Surveys went to 600 households, but only the 410 that replied gave data, so the sample is those 410 households and n = 410. (The 190 households that didn't reply are a nonresponse problem, covered in topic 1.12.)
- Step 3: The investigative question is about the population: What proportion of the city's households would use a curbside composting program?
Answer: Population: all 42,000 city households (N = 42,000). Sample: the 410 households that returned the survey (n = 410). Question: What proportion of the city's households would use curbside composting?
- Example 2Calculator allowed
Trap: a question rewritten after the results
A team plans to ask, "Do most students at our school support a later start time?" After the survey, only 41% support it, so they rewrite the question as "Do at least 40% of students support a later start time?" and report "Yes." What's wrong?
Show the solutionHide the solution
- Step 1: An investigative question should be set before the analysis and left alone.
- Step 2: The team changed the question to fit the answer they got. That makes the result look like support when the original question's answer was no: 41% is not a majority.
- Step 3: An honest report answers the original question: the sample doesn't show majority support.
Answer: They changed the investigative question after seeing the data. They should answer the original question, which the data don't support.
Common mistakes
- Naming the sample as the population. The population is everyone you want to learn about, not just the people who answered.
- Using n for the number of people contacted instead of the number who actually provided data.
- Writing an investigative question about one individual, or one that can't be measured with data you could collect.
- Reporting results without context: give the variable, the units and the group.
On the exam
- Free-response questions on study design often start by asking you to identify the population, the sample or the investigative question. Be specific about who and where.
- When asked whether a question is a good investigative question, check that it's about a population, has a clear purpose and can be answered with collectable data.
Connected topics
Videos
Check yourself
3 questions on 1.1 Introducing Statistics: What Can We Learn from Data?. Pick an answer to see if you got it, and why.
A student council wants to use data to plan the school's spring events. Which of the following is the best investigative question for a statistical study?
A principal wants to know what percent of the 1,850 students at her high school visited the school library at least once last month. She randomly selects 120 students from the enrollment list and asks each one. Of these, 46% say they visited the library at least once last month.
Described scenario
What is the population in this study?
Which of the following best explains why the principal surveyed 120 students instead of all 1,850?
0 of 3 answered