Statistical Questions & Distributions — Challenge
A statistical question is one that expects a spread of answers, not a single one.
What these problems cover
These questions use 9 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Range — The set of outputs that actually occur.
- Cluster — A group of dots bunched together.
- Outlier — A dot far away from the rest of the pattern.
- Number line — A line with numbers increasing to the right.
- Height — The perpendicular distance from the base to the opposite corner.
- Statistical question — One that anticipates variability in the answers.
- Distribution — The pattern of how the values are spread out.
- Dot plot — A chart with one dot per value, stacked above a number line.
Problem 1
Which of these is a statistical question?
- How tall is my brother?
- How tall are the students in my class?
- What is my shoe size?
- What day is it today?
- How old am I?
Worked answer
How tall are the students in my class?
A statistical question is one whose answer varies, so it needs data from a group rather than a single fact.
Asking about every student in a class gives many different heights, which is data worth summarising.
The others each have one fixed answer, so no data is collected and nothing varies.
Problem 2
Which question would produce data that varies?
- How many minutes do pupils spend reading each night?
- How many days are in June?
- What is the capital of France?
- How many sides has a hexagon?
- What is 7× 8?
Worked answer
How many minutes do pupils spend reading each night?
Data varies when different members of a group give different answers.
Different pupils read for different lengths of time, so that question produces a spread of values.
Every other option has a single correct answer that does not depend on who is asked.
Problem 3
A pupil wants to know the favourite sport of everyone in a school of 800. They ask their 10 closest friends. What is the main problem with this?
- The sample is too small and not chosen at random
- The question is not a statistical question
- Favourite sport cannot be measured
- The sample is too large to handle
- Nothing is wrong with it
Worked answer
The sample is too small and not chosen at random
Ten pupils out of 800 is a very small fraction of the school.
Worse, close friends are likely to share interests, so the sample is not representative even for its size.
A sample should be chosen at random from the whole group if it is to stand in for that group.
Problem 4
Which measure of centre is pulled furthest by one unusually large value?
- The mean
- The median
- The mode
- Both the median and the mode
- None of them
Worked answer
The mean
The mean uses every value in its sum, so one very large value raises the total and drags the mean up with it.
The median depends only on the position of the middle value, so an extreme value at the end barely moves it.
The mode is the most common value and need not change at all.
Problem 5
A set of house prices contains one extremely expensive house. Which measure of centre best describes a typical price?
- The mean
- The median
- The mode
- The range
- The total
Worked answer
The median
The one very expensive house inflates the mean, so the mean would sit above almost every actual price.
The median is the middle price and is barely affected by a single extreme value, so it describes a typical house better.
The range and the total are not measures of centre at all.
Problem 6
Twenty pupils have a mean score of 15. One more pupil scores 36. What is the mean of all 21 scores?
- 15
- 15.5
- 16
- 17
- 21
Worked answer
16
The first twenty scores total 20· 15 = 300.
Adding the new score gives 300+36 = 336 across 21 pupils.
The new mean is 33621 = 16. One high score lifts the mean by a whole point, which is the effect the previous question describes.
Problem 7
A dot plot of siblings shows 3 students with 0, 7 with 1, 5 with 2, 2 with 3, and 1 with 8. Which description is best?
- Most students have 0 to 2 siblings, with one outlier at 8
- The distribution is symmetric
- The mode is 8
- Every student has at least one sibling
- The range is 3
Worked answer
Most students have 0 to 2 siblings, with one outlier at 8
The dots pile up between 0 and 2, where 15 of the 18 students sit. The single student with 8 siblings is far from everyone else, which makes it an outlier. The mode is 1, not 8; three students have none; and the range is 8, not 3. Only the first description fits.
Problem 8
Which question would produce a distribution rather than a single value?
- How many minutes do students in my class spend on homework?
- How many minutes did I spend on homework yesterday?
- How many days are in June?
- What is the sum of 24 and 39?
- What is my house number?
Worked answer
How many minutes do students in my class spend on homework?
A statistical question is one whose answers vary. Every option but the first has exactly one answer, found by looking it up or doing one calculation. Different students spend different amounts of time, so that question produces a spread of answers — a distribution.