Statistical questions and variability
One student goes to bed at one time, but a whole class goes to bed at many different times. What kind of question asks about the class, and how is it answered?
Where you are 8 in Data and variables
Why this exists
Ask Bodil what time she went to bed last night and she gives one answer, say 9{:}30 pm. That settles it.
Now ask every student in Bodil’s class. The answers are not all the same: some went to bed at 8{:}30, some at 10{:}00, one at 11{:}00. No single time is “the answer”. The answer is the whole collection of times, and the job becomes describing it: where the bedtimes bunch up, how far they run, which times are rare.
Questions of the second kind are what statistics is built for. They come up whenever the thing asked about differs from one person, object or day to the next. This lesson is about recognizing such a question, turning a one-answer question into one, and taking a first look at the answers that come back.
Definition
A statistical question is a question that expects its answers to vary, and is answered by collecting data and looking at all the answers together.
Two tests tell whether a question is statistical.
- Can the answer differ from one individual to the next? If the question is about a group of individuals (people, pairs of shoes, nights, tries) and different individuals can give different answers, it is statistical. If there is one fixed answer, it is not.
- Is it settled by collecting many values? A statistical question is answered by gathering one value from each individual and describing them all. A question settled by one look, one count or one measurement is not statistical.
The individuals, and the thing recorded about each one, are the subject of Individuals, variables and units.
Variability is the fact that the values differ from one another. A statistical question anticipates variability: it is asked knowing the answers will not all agree. Variability has several sources. Different individuals are different. The same individual changes from one occasion to the next (a person skips more on some days than on others). And no measurement is perfectly exact, so measuring one thing twice can give two values.
The whole collection of answers, with how often each one turns up, is called a distribution. Answering a statistical question means describing the distribution, in three parts:
- its center: roughly where the middle of the values sits, a typical value;
- its spread: how far the values range and how tightly they bunch together;
- its shape: where the values pile up and how they trail off, for instance one peak with the values falling away evenly on both sides, or a long tail on one side.
Intuition
Figure 1 shows the bedtimes of the students in Bodil’s class as a dot plot: each dot is one student, stacked above the time that student gave.
The plot answers “what time do the students in this class go to bed on a school night?” without picking out any one dot.
- Center. The tallest stack is at 9{:}30, and about as many dots sit before it as after it, so 9{:}30 is a fair description of a typical bedtime.
- Spread. The bedtimes run from 8{:}00 to 11{:}00, a stretch of 3 hours, but most of them, 5 + 7 + 5 = 17 of the 24, fall from 9{:}00 to 10{:}00.
- Shape. One peak, with the stacks shrinking on both sides at about the same rate, so the plot is close to symmetric.
Each of those words will later get a number: the center from the mean (Mean as balance point) and the median (Finding the median), the spread from the Mean absolute deviation and the Quartiles and the five number summary, and the shapes their names in Shape of a distribution. Here the descriptions are by eye.
What the plot shows, and a single answer cannot, is the variation itself. “Students go to bed at 9{:}30” hides the fact that some go to bed an hour and a half later. Describing the variability is part of the answer.
Results
A question is statistical exactly when the values collected to answer it can differ from one individual to the next. Then no single value settles it, and the answer is a description of many values.
A one-answer question about one individual becomes statistical when it is asked of a group of individuals, or of one individual on many occasions.
A distribution is described by its center, its spread and its shape. Two distributions can share a center and still differ in spread, so the center alone is not the whole answer.
Why it is true
Suppose every individual in the group would give the same answer. Then asking one of them tells you everything; there is nothing left to collect and nothing to describe. That is a one-answer question, whatever its wording.
Now suppose the answers differ. Asking one individual gives one value, but that value could have come from anywhere in the spread, the crowded middle or a rare end, and nothing in it says which. So the only honest answer is to gather values from many individuals and describe the whole collection. Variability is what forces the collecting of data, and it is also what the answer has to report. The same argument covers one individual on many occasions, where the individuals are the occasions.
Worked examples
Sorting bedtime questions
Which of these questions are statistical? Give the reason for each.
(a) What time did Zelig go to bed last night?
(b) What time do the students in Zelig’s class usually go to bed on a school night?
(c) What time does the last bus leave the stop outside the school?
(d) How many hours did Zelig sleep on each night of last month?
Solution(a) is not statistical. It asks about one student on one night, so it has exactly one answer.
(b) is statistical. It asks about a group, the students in the class, and different students go to bed at different times. It is answered by collecting a bedtime from each student and describing them all, for instance “mostly between 9{:}00 and 10{:}00”.
(c) is not statistical. The schedule gives one time, and it does not vary from person to person.
(d) is statistical, even though it is about one person. Here the individuals are the nights, about 30 of them, and Zelig will not sleep the same number of hours on all of them.
The test was never whether the question mentions a group of people. It was whether the answer can differ from one individual, night or try to the next.
Rewriting sneaker questions
Each question has one answer. Rewrite it as a statistical question, and name the group the data would come from.
(a) What size sneakers does Albin wear?
(b) How much did Dunja’s new sneakers cost?
(c) How many pairs of sneakers does Fabienne own?
SolutionA one-answer question is about a single individual. To make it statistical, keep what is being asked but ask it of many individuals whose answers will differ.
(a) “What sizes of sneakers do the players on Albin’s soccer team wear?” The group is the players; their feet differ, so the sizes will vary.
(b) “How much do the pairs of sneakers on sale at the town’s sports shop cost?” The group is the pairs in the shop, and prices differ from pair to pair.
(c) “How many pairs of sneakers does each student in Fabienne’s class own?” The group is the students in the class.
Each has more than one good rewrite. What every good rewrite shares is a named group and an answer expected to vary across it.
Words that look like clues
Decide whether each question is statistical.
(a) How many legs does each insect in the class’s bug jar have?
(b) What is the typical price of a pair of sneakers at a market stall?
(c) How many students are in Albin’s class?
(d) What is the most common bedtime among the students in Albin’s class?
Solution(a) is not statistical, even though it says “each”. Every insect has 6 legs, so asking about every insect in the jar would give 6 every time.
(b) is statistical. Prices differ from pair to pair, and “typical” asks for the center of the distribution of prices, found only by looking at many prices.
(c) is not statistical. It is about a group, but it asks for one number about the whole group: count the students once and you are done.
(d) is statistical, even though its answer is a single time. The most common bedtime can only be found by collecting a bedtime from each student and seeing which turns up most often. A statistical question may ask for one summary; what makes it statistical is that the data behind the summary vary.
So “each”, “typical” and “most common” are hints, not the test. The test is whether the values collected would differ.
Fabienne’s skips
Fabienne counted how many skips she could do with a skipping rope in one minute, once a day for ten days:
Is “How many skips can Fabienne do in one minute?” a statistical question? Describe what her counts say.
SolutionIt is statistical when it is asked about her skipping in general rather than about a single try. The individuals are the tries, and the counts vary from try to try even though the skipper is the same. A single try could have given her 49 or her 61.
In order, the counts are easier to read:
There are 10 counts. The smallest is 49 and the largest is 61, so they stretch over 61 - 49 = 12 skips. The two counts in the middle of the ordered list are 55 and 57, so a typical minute is about 56 skips. Nine of the ten counts lie from 52 to 61, spread fairly evenly, with the 49 a little below the rest.
A fair answer: Fabienne usually does about 56 skips in a minute, and on any one day it can be anything from about 50 to about 60.
Shoe sizes at the shop
On Saturday a sports shop sold some pairs of sneakers to children. Figure 2 shows the size of each pair. How many pairs were sold, and how many were size 7 or larger? Describe the center, spread and shape, and say which sizes the shop should stock most of.
Each dot is one pair, so the number sold is the number of dots: 1 + 2 + 4 + 5 + 3 + 2 + 1 = 18 pairs. Sizes 7, 8 and 11 have 3, 2 and 1 dots, so 3 + 2 + 1 = 6 pairs were size 7 or larger.
The tallest stack is at size 6, with 5 and 7 close behind, so the center is about size 6. The sizes run from 3 to 11, but every pair except one lies from 3 to 8. The shape has one peak at 6, falling away on both sides, then a gap at 9 and 10 before a single pair of size 11 on its own.
If the shop orders by Saturday’s sales, it wants most of its stock in sizes 5 to 7, which made up 4 + 5 + 3 = 12 of the 18 pairs, and size 11 only as an occasional order. The distribution, not any one sale, answers the shop’s question.
Same center, different spread
Bodil and Zelig each counted their skips in one minute on eight days, shown in Figure 3:
The class team needs a skipper who will do at least 52 skips in a minute on the day. Who should be picked?
SolutionBoth lists are already in order. The two middle counts are 56 and 58 for Bodil and 57 and 57 for Zelig, so both are centered at about 57 skips. If only the center were reported, the two would look like equal choices.
The spreads are very different. Bodil’s counts run from 44 to 70, a stretch of 70 - 44 = 26 skips. Zelig’s run from 54 to 60, a stretch of only 60 - 54 = 6 skips.
All of Zelig’s counts are 54 or more, so on all eight days he cleared 52. Two of Bodil’s counts, 44 and 50, fell short. Zelig is the safer pick. The question was about the low end of each distribution, so the spread, not the center, decided it.
Boundaries
Whether a question is statistical depends on what it is asked about, not on its topic. “How tall is Dunja?” is not statistical; “How tall are the students in Dunja’s class?” is.
A question about a fixed rule is not statistical even when asked of many individuals. Every insect has 6 legs and every week has 7 days, so collecting answers gives the same value each time.
Measuring can sit on the line. “How long is the classroom?” has one true length, so as asked it is not statistical. But if ten students each measure it with a tape, their numbers will differ a little, and “what lengths do ten careful measurers get?” is statistical: the variability comes from the measuring, not from the room.
Center, spread and shape, as described here, are for answers that are numbers. A statistical question can also have answers that are words (“What color are the students’ sneakers?”); those answers are sorted and counted instead, as in Types of data: categorical, numerical, discrete, continuous.
Finally, the data describe the group they came from. Figure 1 describes Bodil’s class. Saying something about all children of that age needs a group chosen with care, the work of Populations, samples and census.
Where this goes next
A statistical question is where every piece of statistics starts. The next lessons say what is recorded about each individual (Individuals, variables and units), sort data into numbers and categories (Types of data: categorical, numerical, discrete, continuous), and gather the answers by asking many people (Collecting data with a survey).
Then the eyeballed words get numbers: the mean as a balance point (Mean as balance point), the median (Finding the median), the Mean absolute deviation and the Quartiles and the five number summary, together the measures of Center and spread. After that come displays that show a distribution at a glance, dot plots and Histograms, and the full written report of a distribution in Describing a distribution.
Exercises
Statistical or not
Decide whether each question is statistical, and give the reason.
- How tall is the flagpole outside the school?
- How many minutes did each student in Dunja’s class spend reading before bed last night?
- How many wheels does each bicycle in the school’s bike rack have?
- What is the most common shoe size among the children at a summer camp?
- What was the temperature at noon outside the town library on each day of January?
- How many students are on the school’s chess team?
Rewrite as a statistical question
Rewrite each question as a statistical question, and name the group the data would come from.
- How long are the laces in Zelig’s sneakers?
- What time did Dunja go to bed on Sunday?
- How many skips did Albin do on his first try today?
- How far is it from Fabienne’s home to school?
Reading a dot plot
Every student in Bodil’s class answered “How many pairs of sneakers do you own?” Figure 4 shows the answers.
- How many students answered?
- Which answer was given most often, and by how many students?
- How many students own 3 or more pairs?
- What are the smallest and largest answers, and how far apart are they?
- Describe the center and the spread of the answers in words.
- Describe the shape. Which answer sits apart from the rest?
Reading a list
Twelve students in Dunja’s class said how many minutes they read before bed last night: 15,\ 20,\ 10,\ 30,\ 20,\ 25,\ 20,\ 5,\ 15,\ 20,\ 45,\ 20.
- Which answer was given most often, and how many times?
- What are the smallest and largest answers, and how far apart are they?
- Which answer sits apart from the others, and by how much is it above the next largest? Describe the center in words.
Explain
- “How old are you?” asked of Albin has one answer. The same words asked of every guest at a family party make a statistical question. Explain how the same words can be both.
- Explain why a statistical question cannot be answered by asking only one individual.
- Two classes answered the bedtime question. Both distributions are centered near 9{:}30, but in class A every bedtime is from 9{:}00 to 10{:}00, while in class B they run from 8{:}00 to 11{:}00. One student is picked without looking. In which class is 9{:}30 the safer guess for that student’s bedtime? Explain.
- Is “What is the average height of the students in our class?” a statistical question, even though its answer is one number? Explain.
- Write a statistical question about the sneakers of the students in your class whose answers are numbers. Say what the individuals are, and why the answers will vary.
Answers
- Not statistical. The flagpole has one height.
- Statistical. The individuals are the students, and their reading times will differ.
- Not statistical. A bicycle has 2 wheels, so every answer would be 2.
- Statistical. Sizes differ from child to child, and “most common” needs a size from each child.
- Statistical. The individuals are the 31 days, and the noon temperature changes from day to day.
- Not statistical. It is one count of the team.
- For example: “How long are the laces in the sneakers of the students in Zelig’s class?” The group is the students in the class.
- For example: “What time did the students in Dunja’s class go to bed on Sunday?”, with the students as the group; or “What time did Dunja go to bed on each night of this month?”, with the nights as the group.
- For example: “How many skips does Albin do on each try this week?” The group is the tries.
- For example: “How far do the students in Fabienne’s class live from school?” The group is the students in the class.
- 20 students: one per dot, 3 + 7 + 5 + 3 + 1 + 1 = 20.
- 2 pairs, given by 7 students: the tallest stack.
- 10 students: 5 + 3 + 1 + 1 = 10, the dots at 3, 4, 5 and 8.
- Smallest 1, largest 8; they are 8 - 1 = 7 pairs apart.
- The center is about 2 or 3 pairs: the tallest stack is at 2 with 3 next, and half the students own 1 or 2 pairs while half own 3 or more. The answers run from 1 to 8, but all except one lie from 1 to 5.
- One peak at 2, with the stacks shrinking more slowly to the right, so the values trail off to the right. The answer 8 sits apart, after a gap at 6 and 7.
- 20 minutes, given 5 times.
- Smallest 5, largest 45; they are 45 - 5 = 40 minutes apart.
- 45 sits apart: the next largest is 30, and 45 - 30 = 15. In order the answers are 5, 10, 15, 15, 20, 20, 20, 20, 20, 25, 30, 45; the middle two are both 20, which is also the most common answer, so a typical student read for about 20 minutes.
- Asked of Albin, the question names one individual, so it has one answer. Asked of every guest, it names a group whose ages differ, so it is answered by collecting all the ages and describing them.
- When answers vary, one individual’s answer could come from anywhere in the distribution, the middle or a rare end, and nothing in that one value shows which.
- Class A. Its bedtimes are all within half an hour of 9{:}30, while a class B student could be as much as an hour and a half away. The smaller the spread, the more the center tells you about one individual.
- Yes. Heights differ from student to student, so the average needs every student’s height. What makes it statistical is that the data behind the one number vary.
- For example: “What sizes of sneakers do the students in our class wear?” The individuals are the students. Feet differ in length, so the sizes will vary.