Statistics: Centre, Spread & Displays — Challenge
Report a centre, a spread, and the shape. One number on its own has almost never told the whole story.
What these problems 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
What is the mean of 12, 15, 18 and 15?
- 14
- 15
- 16
- 60
- 18
Worked answer
15
Add the values: 12+15+18+15 = 60.
Divide by 4: 604 = 15.
Problem 2
What is the median of 7, 3, 9, 5, 11 and 1?
- 5
- 6
- 7
- 8
- 9
Worked answer
6
Order the values first: 1, 3, 5, 7, 9, 11.
With six values the median is the mean of the middle two, 5 and 7.
5+72 = 6.
Problem 3
On a box plot, what does the line inside the box show?
- The median
- The mean
- The range
- The mode
- The maximum
Worked answer
The median
A box plot is built from five numbers: the minimum, the three quartiles and the maximum.
The box spans the first to the third quartile, and the line inside marks the second quartile, which is the median.
The mean is not shown on a box plot at all.
Problem 4
What is the interquartile range of 2, 4, 6, 8, 10, 12, 14 and 16?
- 6
- 7
- 8
- 9
- 14
Worked answer
8
Split the ordered list in half: the lower half is 2,4,6,8 and the upper half is 10,12,14,16.
The first quartile is the median of the lower half, 4+62 = 5; the third is 12+142 = 13.
The interquartile range is 13-5 = 8. The full range would be 16-2 = 14.
Problem 5
What is the mode of 3, 3, 4, 4, 4 and 5?
- 3
- 4
- 5
- 10
- 23
Worked answer
4
The mode is the value appearing most often.
4 appears three times, 3 twice and 5 once, so the mode is 4.
Problem 6
What is an outlier?
- A value far from the rest
- The value occurring most often
- The middle value
- The average value
- The smallest value
Worked answer
A value far from the rest
An outlier is a value that sits well away from the bulk of the data.
It need not be the smallest or largest by much; what matters is the size of the gap to the other values.
Outliers pull the mean towards themselves while leaving the median largely alone.
Problem 7
Which measure is least affected by an outlier?
- The range
- The median
- The mean
- The standard deviation
- The total
Worked answer
The median
The median depends only on the position of the middle value, so moving an extreme value further out need not change it at all.
The mean, the total and the standard deviation all use every value, so an outlier shifts them.
The range is defined by the extremes, so it is the most affected of all.
Problem 8
Every value in a data set has 5 added to it. What happens to the range?
- It increases by 5
- It decreases by 5
- It stays the same
- It doubles
- It cannot be decided
Worked answer
It stays the same
The range is the largest value minus the smallest.
Adding 5 to both of them adds 5 to each side of that subtraction, so the difference is unchanged.
The mean would rise by 5, but measures of spread are untouched by a shift.