Five-number summary and box plots
Find the five-number summary (min, Q1, median, Q3, max) from sorted data; calculate the IQR and range from box plots; identify potential outliers using the 1.5 × IQR rule.
Worked examples
Finding the five-number summary
Straightforward
Problem
Find the five-number summary of the sorted data set: .
1
Identify the minimum and maximum.
Minimum , maximum .
2
Find the median.
There are 8 values, so the median is the mean of the 4th and 5th values.
3
Find Q1 (lower quartile).
The lower half is . Q1 is the mean of the 2nd and 3rd values.
4
Find Q3 (upper quartile).
The upper half is . Q3 is the mean of the 2nd and 3rd values.
Answer
Five-number summary: min , Q1 , median , Q3 , max .
Finding the IQR and range from a box plot
Moderate
Problem
A box plot shows: minimum , Q1 , median , Q3 , maximum . Find: (a) the range, (b) the IQR.
1
Calculate the range.
2
Calculate the IQR.
3
Interpret the IQR.
The IQR of means the middle of data values span a range of units, from to .
Answer
Range ; IQR .
Identifying potential outliers using the IQR rule
Challenging
Problem
A data set has Q1 and Q3 . A value of appears in the data. Determine whether is a potential outlier.
1
Calculate the IQR.
2
Calculate the upper fence.
3
Compare the value to the upper fence.
Since , the value lies beyond the upper fence.
4
State the conclusion.
A value is a potential outlier if it is more than beyond Q1 or Q3. Since , the value is a potential outlier.
Answer
Yes, is a potential outlier because it exceeds the upper fence of .
Practise
Q1·Straightforward
Find the median of the sorted data set: .
Explanation
There are 8 values, so the median lies between the 4th and 5th values. The 4th value is and the 5th is .
Q2·Straightforward
Find the lower quartile (Q1) of the sorted data set: .
Explanation
The median of the 8 values lies between the 4th and 5th values ( and ), so the lower half is . Q1 is the median of this lower half:
Q3·Straightforward
Find the upper quartile (Q3) of the sorted data set: .
Explanation
The median lies between the 4th and 5th values, so the upper half is . Q3 is the median of this upper half:
Q4·Straightforward
Find Q3 for the sorted data set: .
Explanation
There are 11 values, so the median is the 6th value: . The upper half (values above the median) is — five values. Q3 is the median of this upper half, which is the 3rd value:
Q5·Moderate
A box plot shows: minimum , Q1 , median , Q3 , maximum . Calculate the interquartile range (IQR).
Explanation
Q6·Moderate
A box plot shows: minimum , Q1 , median , Q3 , maximum . Calculate the range.
Explanation
Q7·Moderate
A box plot has Q1 and an IQR of . What is Q3?
Explanation
Q8·Moderate
A data set of values is displayed as a box plot with Q1 and Q3 . How many values would you expect to lie between Q1 and Q3?
Explanation
The middle of data lies between Q1 and Q3.
Q9·Challenging
A data set has Q1 and Q3 . Calculate the upper fence value used for potential outlier detection (upper fence Q3 IQR).
Explanation
Step 1: Find the IQR.
Step 2: Calculate the upper fence.
Q10·Challenging
A data set has Q1 and Q3 . A value of appears in the data. Which statement correctly describes this value?
Explanation
Since , the value is beyond the upper fence and is therefore a potential outlier.
Q11·Challenging
Box plot A shows: Q1 , Q3 . Box plot B shows: Q1 , Q3 . Which data set has the greater spread as measured by the IQR?
Explanation
Since , data set B has the greater spread.
Q12·Challenging
A data set has Q1 and Q3 . Calculate the lower fence value used for potential outlier detection (lower fence Q1 IQR).
Explanation
Step 1: Find the IQR.
Step 2: Calculate the lower fence.