Measures of spread
Find and interpret the range and interquartile range (IQR) as measures of spread for a data set.
Worked examples
Finding the range
Straightforward
Problem
Find the range of the data set .
1
Identify the highest value in the data set.
The highest value is .
2
Identify the lowest value in the data set.
The lowest value is .
3
Subtract the lowest value from the highest value.
Answer
The range is .
Finding the interquartile range
Moderate
Problem
Find the interquartile range (IQR) of the data set .
1
Confirm the data is sorted, then split it into a lower half and an upper half.
Lower half: . Upper half: .
2
Find Q1, the median of the lower half.
The median of is , so Q1 .
3
Find Q3, the median of the upper half.
The median of is , so Q3 .
4
Calculate the IQR.
Answer
The IQR is .
Comparing the spread of two data sets
Challenging
Problem
Compare the spread of Set A: and Set B: using both range and IQR.
1
Find the range of each set.
Set A: range . Set B: range .
2
Find Q1 and Q3 for Set A.
Lower half: — Q1 . Upper half: — Q3 . IQR .
3
Find Q1 and Q3 for Set B.
Lower half: — Q1 . Upper half: — Q3 . IQR .
4
Compare the measures and draw a conclusion.
Set B has a greater range () and a greater IQR (), so Set B is more spread out by both measures.
Answer
Set B is more spread out, with a greater range ( vs ) and a greater IQR ( vs ).
Practise
Q1·Straightforward
Find the range of the data set .
Explanation
The highest value is and the lowest is . Range .
Q2·Straightforward
Find the range of the data set .
Explanation
The highest value is and the lowest is . Range .
Q3·Straightforward
Find the range of the data set .
Explanation
The highest value is and the lowest is . Range .
Q4·Straightforward
The scores in a quiz are . Find the range of the scores.
Explanation
The highest score is and the lowest is . Range .
Q5·Moderate
Find the interquartile range (IQR) of the data set .
Explanation
Lower half: — Q1 . Upper half: — Q3 . IQR .
Q6·Moderate
Find the interquartile range (IQR) of the data set .
Explanation
Lower half: — Q1 . Upper half: — Q3 . IQR .
Q7·Moderate
Find the interquartile range (IQR) of the data set .
Explanation
Lower half: — Q1 . Upper half: — Q3 . IQR .
Q8·Moderate
Find the interquartile range (IQR) of the data set .
Explanation
Lower half: — Q1 . Upper half: — Q3 . IQR .
Q9·Challenging
Which data set has the greater range?
Set A:
Set B:
Set A:
Set B:
Explanation
Set A: range . Set B: range . Set B has the greater range.
Q10·Challenging
Which data set is more consistent?
Set A:
Set B:
Set A:
Set B:
Explanation
Set A: range . Set B: range . Set A has the smaller range, so it is more consistent.
Q11·Challenging
Which data set has the greater IQR?
Set A:
Set B:
Set A:
Set B:
Explanation
Set A: Q1 , Q3 , IQR . Set B: Q1 , Q3 , IQR . Set B has the greater IQR.
Q12·Challenging
Two data sets are summarised below.
Set A: range , IQR
Set B: range , IQR
Which statement best describes the comparison?
Set A: range , IQR
Set B: range , IQR
Which statement best describes the comparison?
Explanation
Set A has a greater range (), so it has greater overall spread. Set B has a greater IQR (), so it has greater spread in the middle 50% of values.