Statistics: Centre, Spread & Displays — Practice
Report a centre, a spread, and the shape. One number on its own has almost never told the whole story.
What these questions cover
These questions use 10 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.
- Median — Two kinds of centre: the balance point, and the middle value.
- Range — The set of outputs that actually occur.
- Scatter plot — A graph of paired data, one dot per case.
- Outlier — A dot far away from the rest of the pattern.
- Consistent — Has a solution / has none.
- Standard deviation — A measure of typical distance from the mean.
- Skew — A long tail on one side of the distribution.
- Five-number summary — Minimum, Q1, median, Q3, maximum.
- Dot plot — A chart with one dot per value, stacked above a number line.
Problem 1
Find the mean of 4, 8, 10, 14.
Worked answer
9
36 ÷ 4 = 9
Problem 2
Find the median of 3, 9, 4, 12, 7.
Worked answer
7
3, 4, 7, 9, 12
The middle value is 7
Problem 3
Find the median of 2, 5, 8, 11.
Worked answer
6.5
Middle two are 5 and 8
5 + 82 = 6.5
Problem 4
Find the range of 12, 5, 20, 9.
Worked answer
15
20 - 5 = 15
Problem 5
Find Q1 of 2, 4, 6, 8, 10, 12, 14, 16.
Worked answer
5
Lower half: 2, 4, 6, 8
4 + 62 = 5
Problem 6
Find the IQR of 1, 3, 5, 7, 9, 11, 13, 15.
Worked answer
8
Q1 = 3+52 = 4
Q3 = 11+132 = 12
12 - 4 = 8
Problem 7
Data: 20, 22, 24, 25, 400. Should you report the mean or the median?
Worked answer
median
400 is far from the rest
It drags the mean up to 98.2
The median, 24, is representative
Problem 8
The mean is 62 and the median is 70. Describe the skew.
Worked answer
skewed left
Mean below median
So the tail points left
Skewed left
Problem 9
The mean is 85 and the median is 74. Describe the skew.
Worked answer
skewed right
Mean above median
The tail points right
Skewed right
Problem 10
Which pair of statistics is resistant to outliers?
- Median and IQR
- Mean and standard deviation
- Mean and range
- Median and range
Worked answer
Median and IQR
Median ignores the extremes
IQR uses only the middle half
Mean, range and standard deviation all move a lot
Problem 11
Find the five-number summary of 5, 7, 9, 12, 15, 18, 21.
Worked answer
5, 7, 12, 18, 21
Min 5, max 21
Median 12
Q1 = 7, Q3 = 18
Problem 12
Which display best compares three groups at once?
- Side-by-side box plots
- A single dot plot
- A pie chart
- A scatter plot
Worked answer
Side-by-side box plots
Box plots show the five-number summary
Placing them side by side compares groups directly
Problem 13
Four numbers have a mean of 15. Three of them are 10, 14 and 20. Find the fourth.
Worked answer
16
Total = 60
10 + 14 + 20 = 44
60 - 44 = 16
Problem 14
Using the 1.5 × IQR rule with Q1 = 20 and Q3 = 32, is 55 an outlier? Answer yes or no.
Worked answer
yes
IQR = 12
1.5 × 12 = 18
Upper fence = 32 + 18 = 50
55 > 50, so yes
Problem 15
Group A: median 60, IQR 5. Group B: median 60, IQR 22. Which is more consistent?
Worked answer
A
A has the smaller IQR
So A is more consistent
The centres are identical
Problem 16
A data set has mean 50. Every value is increased by 8. Find the new mean and the new standard deviation, given the old one was 6.
Worked answer
mean 58, sd 6
Every value up 8, so the mean is 58
Distances between values are unchanged
So the standard deviation is still 6