Measures of spread
Calculate the range and interquartile range of datasets, and identify the effect of outliers on measures of spread.
Worked examples
Finding the range
Straightforward
Problem
Find the range of the dataset:
1
Identify the highest and lowest values.
Highest value , lowest value
2
Apply the formula: Range = highest − lowest.
Range
Answer
The range is .
Finding the interquartile range
Moderate
Problem
Find the interquartile range (IQR) of the ordered dataset:
1
Find the median.
There are 7 values. The median is the 4th value: .
2
Identify the lower half (values below the median) and find .
Lower half: . The median of this half is .
3
Identify the upper half (values above the median) and find .
Upper half: . The median of this half is .
4
Calculate the IQR.
Answer
The IQR is .
Comparing range and IQR when an outlier is present
Challenging
Problem
For the dataset , calculate the range and the IQR, and explain which better describes the spread.
1
Calculate the range.
Range
2
Find the median to split the data.
There are 7 values. The median is the 4th value: .
3
Find from the lower half.
Lower half: . .
4
Find from the upper half.
Upper half: . .
5
Calculate the IQR and compare with the range.
. The range is because the outlier pulls the maximum far from the rest. The IQR is only , reflecting the true spread of the central data. The IQR is the better measure here.
Answer
Range ; IQR . The IQR better describes the spread because it is not distorted by the outlier .
Practise
Q1·Straightforward
Find the range of this dataset:
Explanation
The highest value is and the lowest is , so the range .
Q2·Straightforward
Find the range of this dataset:
Explanation
The highest value is and the lowest is , so the range .
Q3·Straightforward
Find the range of this dataset:
Explanation
The highest value is and the lowest is , so the range .
Q4·Straightforward
Find the range of this dataset:
Explanation
The highest value is and the lowest is , so the range .
Q5·Moderate
Find the interquartile range (IQR) of this ordered dataset:
Explanation
There are 7 values, so the median is the 4th value: . The lower half is , giving . The upper half is , giving . .
Q6·Moderate
Find the interquartile range (IQR) of this ordered dataset:
Explanation
There are 6 values. The lower half is , giving . The upper half is , giving . .
Q7·Moderate
Find the interquartile range (IQR) of this ordered dataset:
Explanation
There are 8 values. The lower half is , so . The upper half is , so . .
Q8·Moderate
Find the interquartile range (IQR) of this ordered dataset:
Explanation
There are 9 values, so the median is the 5th value: . The lower half is , so . The upper half is , so . .
Q9·Challenging
A dataset is: . Find the range.
Explanation
The maximum is and the minimum is , so the range . The outlier greatly inflates the range.
Q10·Challenging
The value is an outlier and is removed from the dataset . What is the range of the remaining data?
Explanation
Removing leaves . The range . The range dropped from to — a large change caused by one outlier.
Q11·Challenging
Find the IQR of the dataset:
Explanation
Lower half: , so . Upper half: , so . . Compare this with the range of — the IQR is much less affected by the outlier .
Q12·Challenging
A dataset contains an extreme outlier. Which measure of spread is the better choice to describe the spread of the data, and why?
Explanation
The IQR measures the spread of the middle 50% of the data, so an outlier in the top or bottom quarter has little or no effect on it. The range, by contrast, uses only the minimum and maximum, so a single extreme value can make the range very large and misleading.