Introducing statistics: what can we learn from data?
Statistics is the science of learning about a large group from a smaller group you can actually observe.
A school principal wants to know how much sleep her 1,800 students get on school nights, but she cannot follow every student home.
A food company wants to know how long its granola bars stay fresh, but testing a bar means opening it, so it cannot test every bar it sells.
In both cases somebody collects data from some of the individuals and uses the data to answer a question about all of them.
Everything in this course, from drawing a dotplot to carrying out a hypothesis test, is a tool for doing that well.
On the exam, this section is tested in the multiple-choice section (why a study is needed, which group is the sample and which the population) and it sets up free-response Question 1, which asks you to work with a study described in a paragraph and with the investigative question behind it.
The pieces of a statistical study
You need to know the vocabulary for the parts of a study, because a free-response part often begins "Identify the population" or "Identify the sample".
A statistical study collects data from a sample to answer an investigative question about a larger population.
The population is everyone or everything you want to know about; the sample is the part you actually measure.
The population size is written and the sample size is written .
A capital letter for the big group and a lower-case letter for the small one is an easy way to remember which is which.
A single recorded fact about one individual, such as "Maya sleeps 7.5 hours", is a datum.
A whole collection of them, such as the 150 sleep times the principal collects, is a data set.
The word "data" is plural, so careful writers say "the data are" and "the data show".
Here is the sleep study taken apart.
Components of the Sleep Study
Component
In this study
Population
all 1,800 students at the high school
Population size
Sample
the 150 students who answered the survey
Sample size
One datum
the number of hours one surveyed student sleeps on a typical school night
Data set
the 150 recorded sleep times
Investigative question
How many hours do students at this school typically sleep on a school night?
Why not just measure everyone?
Measuring every member of the population is called a census, and you will meet it properly in the unit on collecting data.
A census is often impossible or pointless, which is why statistical studies are needed.
There are two main reasons:
- The population is too large. A state cannot interview all of its several million adult residents every month to track how many of them are working.
- It is too difficult to collect data from every individual. Some measurements destroy what they measure (a crash test destroys the car, a battery-life test drains the battery), some individuals cannot be found or will not answer, and some measurements are expensive or slow.
A well-chosen sample answers the question at a fraction of the cost.
Later units show how to say how close the sample's answer is likely to be to the population's answer.
For now, the key idea is that a sample is useful only because it can tell you about the population.
A sample studied for its own sake is just a list of numbers.
What "in context" means
The exam asks for answers "in context" again and again, and students lose points for writing statistics without the story attached.
Every piece of a study corresponds to something real: the population is a real group of people or things, the variable is a real characteristic with real units, and every calculated number describes something real.
Answering in context means saying what that real thing is.
Compare two answers about the sleep study:
- "The mean is 6.8." This is a bare number. It could be anything.
- "The mean number of hours of sleep on a school night for the 150 surveyed students is 6.8 hours." This names the statistic, the variable, its units and the group it describes.
Only the second answer is in context.
A good habit is to check every sentence you write on a free-response question for three things: the variable (with units), the individuals it describes, and whether you are talking about the sample or the population.
Copying the words of the question is a start, but a few words lifted from the stem are often not enough on their own: say which group and which measurement you mean.
Investigative questions
Every study starts from an investigative question: the question about the population that the study is meant to answer.
Two rules make a question a good one.
It has a defined purpose and is not changed after the analysis. You decide what you want to know before you look at the data.
Why does this matter?
Suppose a coach collects data on 20 different habits of her runners and then reports only the one habit that happened to look connected to faster times.
With 20 habits, one is likely to look connected by chance alone.
Changing the question to fit the data makes a chance pattern look like a discovery.
Fixing the question first protects you from fooling yourself.
It can be answered with data you can actually collect and analyze. "Do students at this school like science?"
is vague: like it how much, measured how?
"What proportion of students at this school plan to take a science course next year?"
names something you can record for each student (yes or no) and then summarize.
A question about something no one can observe, or that has no clear measurement, cannot be answered by a study however large.
In the unit on collecting data you will learn the three components the exam expects a full investigative question to name (the variables, the parameter with its direction or goal of estimation, and the population).
Free-response Question 1 can award a point for each component, so treat this section as the start of that skill.
- Each free-response question is worth 10 points, and each point is scored 0 or 1, earned on its own.
- Show work where a calculation is asked: the formula with the values substituted, then the answer. A bare value, or calculator syntax with unlabeled numbers, earns nothing.
- Later parts are judged against your own earlier answers, so a wrong value carried forward and used correctly can still earn.
- An arithmetic slip is usually forgiven when correct work is shown.
- If you give two answers or two methods, the weaker one is scored, so cross out the one you do not want.
- A correct answer followed by a wrong extra statement can lose the point, so do not pad answers with claims you are unsure of.
- Answers need context: the variable with its units, the individuals or population, and the groups being compared.
- Each multiple-choice question is worth 1 point, with no penalty for a wrong answer.
- Statistical study: A statistical study is a study in which data are collected from a sample to answer an investigative question about a larger population.
- Why studies are needed: Statistical studies are necessary when the population is too large or it is too difficult to collect data from every item or individual in the population.
- Datum and data set: A datum (singular form of data) is a piece of information about an item or individual. A collection of data is called a data set.
- Population: A population consists of all items or individuals of interest. The population size is represented by the symbol N.
- Sample: A sample selected for study is a subset of the population from which data are obtained. The number of items in the sample, called the sample size, is represented by the symbol n.
- In context: Each component of a statistical study and the resulting calculations can be related to an aspect of the corresponding real-world context from which the components were derived. This identification of a statistical result with the corresponding contextual component is what is meant by “in context.”
- Investigative question (purpose): An investigative question for a specific study should have a defined purpose and should not be changed based on the data analysis or results.
- Investigative question (data): An investigative question should be posed so that the required data can be collected and analyzed.
You need to know these definitions by heart; none of them is on the formula sheet.
A county public health office in Wisconsin wants to know how many of the county's 46,000 households have a working carbon monoxide detector.
Staff visit 320 randomly selected households and record whether each has a working detector.
A. Identify the population and the sample, and give the value of and of .
B. Identify one datum and the data set.
A. The population is all 46,000 households in the county, so .
The sample is the 320 households visited, so .
Point 1 (population with its size) Point 2 (sample with its size)
B. One datum is the record for a single visited household, for example "the household on Elm Street has a working detector".
The data set is the 320 yes or no records collected by the staff.
Point 3 (a datum is one piece of information about one individual; the data set is the whole collection)
What it means: the office will use the 320 records to say something about all 46,000 households.
If the population and sample are swapped in part A, both points are lost, so read the stem for "all" (population) and the number actually measured (sample).
For each situation, explain why a statistical study is needed and a measurement of the whole population is not practical.
A. A company makes 2 million light bulbs a month and wants to know how long its bulbs last before burning out.
B. A national parks agency wants to know how many hours visitors spend in a large park during a summer.
A. The only way to find a bulb's lifetime is to run it until it burns out, which destroys it.
Testing every bulb would leave nothing to sell, so the company tests a sample.
Point 1 (too difficult: the measurement destroys the item)
B. The park has a very large number of visitors arriving and leaving all summer through many entrances, so recording every visitor's time is not possible with the agency's staff.
A sample of visitors can be surveyed instead.
Point 2 (population too large and too difficult to reach)
What it means: both reasons appear in the definition of why studies are needed: the population is too large, or collecting from every individual is too difficult.
A high school student council plans a study about the 1,200 students at its school.
Determine whether each proposed investigative question is a valid one for a statistical study, and justify your answer.
Q1. What is the mean number of minutes students at this school spend getting to school on a typical morning?
Q2. Are students at this school good people?
Q3. The council collects data on how students rate the cafeteria, then, after seeing that ratings of the salad bar were surprisingly high, changes its question to "Do students prefer the salad bar to the hot lunch line?"
Q1 is valid.
Each student's travel time can be measured in minutes, and the data can be summarized with a mean.
Point 1 (decision with a reason tied to measurable data)
Q2 is not valid.
"Good people" has no clear measurement, so the required data cannot be collected and analyzed.
Point 2 (decision with the reason: the data cannot be collected)
Q3 is not valid as a question for that study.
The question was changed after looking at the results, so it no longer has the defined purpose it started with; the salad-bar pattern could be a chance result in this one data set.
A new study, with this question fixed in advance, could investigate it.
Point 3 (decision with the reason: changed based on the results)
What a weaker answer scores: writing "Q2 is too vague" with no link to collecting data earns the decision but may not earn the reasoning.
Say what makes the data impossible to collect or analyze.
The athletic director at a high school in Raleigh, North Carolina, wants to know the typical amount of money families of the school's 650 student athletes spend on sports equipment in a season.
She emails a survey to all athlete families and receives 140 replies, each giving the amount spent in dollars.
Use the given information to respond to parts A, B and C.
A. Identify the population and the sample.
B. Explain why a statistical study is used here, not a measurement of every family.
C. The director computes that the 140 families spent a mean of $212.
Interpret this value in context.
A. Population: the families of all 650 student athletes at the school.
Sample: the 140 families who replied to the survey.
Point 1 (population) Point 2 (sample, including that it is the families who replied, not all who were emailed)
B. Getting an exact spending figure from every one of the 650 families is too difficult: many families do not answer emails, and the director cannot force a reply.
A study of the families who respond is what she can carry out.
Point 3 (too difficult to collect from every individual, in context)
C. The 140 families who replied spent a mean of $212 on sports equipment for the season.
Point 4 (the statistic named, the variable with its units, and the group it describes: the sample, not all 650 families)
Looking ahead: families who choose to reply may spend differently from those who do not, so $212 might not describe all 650 families well.
The unit on collecting data teaches you to spot this kind of problem.
"In context" means naming the real variable, its units and the real individuals, every time.
"The mean is 212" earns nothing; "the mean amount spent on sports equipment this season by the 140 responding families is $212" earns the point.
The words of the question alone are often not enough: always say which group and which measurement.
- Swapping population and sample. It is tempting to call the people you measured "the population" because they are the only people you have data on. Fix: the population is the whole group the question is about ("all", "every"); the sample is the subset you measured.
- Mixing up and . Both are counts, so they are easy to confuse. Fix: capital is the population size, lower-case is the sample size, and is the one you use in calculations from data.
- Changing the question after seeing the data. Finding an interesting pattern and then asking about it feels natural. The problem is that any large data set contains some patterns by chance. Fix: fix the investigative question first; a pattern found by looking is a reason for a new study, not an answer.
- Asking a question no data can answer. Words like "good", "better" or "popular" sound clear but have no measurement attached. Fix: say what will be recorded for each individual and how it will be summarized.
- Writing statistics without context. A bare number feels complete because you computed it. Fix: attach the variable, units and group to every number you report.
Check your understanding
1. A city transportation department wants to know the mean number of miles driven per week by the 92,000 registered car owners in the city. It mails a questionnaire to a random selection of 600 registered car owners, and 410 return it. Which of the following is the population? (A) The 410 car owners who returned the questionnaire (B) The 600 car owners who were mailed the questionnaire (C) All 92,000 registered car owners in the city (D) The mean number of miles driven per week Total [1]
2. A soft-drink bottler wants to check that its bottles contain the amount printed on the label. Which of the following is the best reason for measuring a sample of bottles instead of every bottle? (A) Measuring the contents means opening the bottle, so measuring every bottle would leave none to sell. (B) A sample always gives the same answer as measuring every bottle. (C) The sample size is always larger than the population size . (D) Bottles do not vary, so one bottle is enough. Total [1]
3. A school counselor wants to plan a study about the 900 students at her school. Which of the following is a valid investigative question for a statistical study? (A) Is homework a good idea? (B) What proportion of students at this school have a part-time job during the school year? (C) Which student at this school is the hardest worker? (D) After seeing that the juniors in a survey reported less stress than expected, are juniors less stressed than anyone thought? Total [1]
4. A state forestry agency wants to know the mean height of the 30,000 young pine trees planted on a restoration site last year. A ranger measures the heights, in feet, of 75 trees chosen at random across the site. Use the given information to respond to parts A, B and C. A. Identify the population and give its size. B. Identify the sample and give its size. C. Explain why a statistical study is needed here. Total [3]
5. Jordan, a student researcher, planned to study whether the number of hours students at his school spend on video games each week is related to their grades. After collecting the data, he noticed that students who listed a particular snack as their favorite had higher grades, and he reported that as his main finding. Identify the problem with Jordan's approach, and explain why it is a problem. Total [2]
6. A survey of 200 adults in a town records the time, in hours, each spends outdoors on a typical weekend day. A student summarizes the result as "the mean is 2.4". Interpret the value 2.4 in context, using all the information given. Total [1]
7. A grocery chain asks: "What is the mean amount, in dollars, that a customer at our Austin stores spends on one visit?" Determine whether this is a valid investigative question, and justify your answer using both features of a valid investigative question. Total [2]
Answers
1. (C). The population is the whole group the question is about. (A) is the sample (those who actually provided data), (B) is the group contacted, and (D) is the quantity being estimated, a parameter, not a group.
2. (A). Measuring destroys the item, which is one way collecting data from every individual is too difficult. (B) is false: samples vary. (C) reverses the sizes. (D) is false: the whole point of checking is that bottles vary.
3. (B). Each student either has a job or not, so the data can be collected and summarized as a proportion. (A) and (C) have no clear measurement; (D) was formed after looking at the data.
4. A. All 30,000 young pine trees planted on the site last year; . Point 1 B. The 75 trees whose heights the ranger measured; . Point 2 C. The population is too large to measure every tree in a reasonable time (30,000 trees spread across the whole site), so a sample of 75 trees is measured instead. Point 3 (too large or too difficult, in context)
5. Jordan changed his investigative question after looking at the results. Point 1 (the problem named) A question chosen because of a pattern in the data may describe a chance result in this one data set; the question should have a defined purpose fixed before the analysis, and the snack pattern would need a new study planned in advance. Point 2 (why it matters: chance patterns and a defined purpose)
6. The mean time spent outdoors on a typical weekend day by the 200 surveyed adults in the town is 2.4 hours. Point 1 (variable, units and the group, identified as the sample; "the mean for all adults in the town is 2.4 hours" does not earn, because 2.4 describes only the 200 surveyed)
7. Valid. Point 1 (decision) It has a defined purpose (the typical spending per visit at the Austin stores), and the data can be collected and analyzed: each visit's total in dollars is recorded at the register and can be averaged. Point 2 (both features linked to this question)