Centre & Spread — Challenge
One number for the middle is never enough, because two very different groups can share an average.
What these problems cover
These questions use 6 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.
- MAD — Mean absolute deviation — the average distance from the mean; a measure of spread.
- Mode — The value that appears most often.
- Range — The largest value minus the smallest.
- Mean absolute deviation — The average distance of the values from the mean.
Problem 1
What is the mean of 3, 5, 7 and 9?
- 5
- 6
- 7
- 24
- 4
Worked answer
6
Add the values: 3+5+7+9 = 24.
Divide by how many there are: 244 = 6.
Problem 2
What is the median of 2, 4, 6, 8 and 10?
- 4
- 5
- 6
- 8
- 30
Worked answer
6
The values are already in order, and there are five of them.
With an odd count the median is the middle value, which is the third one: 6.
Here the mean happens to be 6 as well, but that is a coincidence of this evenly spaced list.
Problem 3
What is the range of 3, 15, 8 and 2?
- 12
- 13
- 15
- 17
- 2
Worked answer
13
The range is the largest value minus the smallest.
The largest is 15 and the smallest is 2, so the range is 15-2 = 13.
Problem 4
What is the mean absolute deviation of 2, 4, 6 and 8?
- 1
- 1.5
- 2
- 2.5
- 5
Worked answer
2
First find the mean: 2+4+6+84 = 5.
Then find how far each value is from 5, ignoring direction: 3, 1, 1 and 3.
The mean of those distances is 3+1+1+34 = 2.
Problem 5
What is the mode of 4, 5, 5, 6 and 7?
- 4
- 5
- 6
- 7
- 27
Worked answer
5
The mode is the value that occurs most often.
5 appears twice and every other value appears once, so the mode is 5.
Problem 6
Data set A has a range of 5 and data set B has a range of 12. Which set is more spread out?
- A
- B
- They are equally spread
- It cannot be decided
- Neither is spread out
Worked answer
B
The range measures how far apart the extreme values are, so a larger range means the values are more spread out.
B has the larger range at 12, so B is more spread out.
The range says nothing about where the centre is, only about the width.
Problem 7
What is the mean of 10, 20 and 30?
- 15
- 20
- 25
- 60
- 30
Worked answer
20
Add the values: 10+20+30 = 60.
Divide by 3: 603 = 20.
Problem 8
A new value is added to a data set, and it is exactly equal to the old mean. What happens to the mean?
- It increases
- It decreases
- It stays the same
- It doubles
- It cannot be decided
Worked answer
It stays the same
Suppose the set has n values with mean m, so the total is nm.
Adding another value equal to m makes the total nm+m = (n+1)m across n+1 values.
The new mean is (n+1)mn+1 = m, unchanged. A value at the mean carries no pull in either direction.