Back-to-back stem plots
Read and interpret back-to-back stem-and-leaf plots; find median, mode and range for each group; and compare two distributions using summary statistics.
Worked examples
Finding the median from a back-to-back stem plot
Straightforward
Problem
A back-to-back stem-and-leaf plot shows the number of hours per week two groups of students spend on hobbies.
Group P | Stem | Group Q
6 | 1 | 2 5 8
9 5 2 | 2 | 0 4 7
7 4 | 3 | 1 6 9
3 | 4 | 5
Leaves for Group P are read from the stem outward (right to left).
Find the median for Group P.
Group P | Stem | Group Q
6 | 1 | 2 5 8
9 5 2 | 2 | 0 4 7
7 4 | 3 | 1 6 9
3 | 4 | 5
Leaves for Group P are read from the stem outward (right to left).
Find the median for Group P.
1
Read the Group P leaves for each stem and write out the values.
Stem 1, leaf 6 → 16
Stem 2, leaves 2, 5, 9 (outward from stem) → 22, 25, 29
Stem 3, leaves 4, 7 (outward) → 34, 37
Stem 4, leaf 3 → 43
Stem 2, leaves 2, 5, 9 (outward from stem) → 22, 25, 29
Stem 3, leaves 4, 7 (outward) → 34, 37
Stem 4, leaf 3 → 43
2
Write the values in ascending order and count them.
16, 22, 25, 29, 34, 37, 43
There are 7 values.
There are 7 values.
3
Locate the middle value.
For values, the median is the th value.
4th value = 29
4th value = 29
Answer
The median for Group P is 29 hours.
Finding median, mode and range from a back-to-back stem plot
Moderate
Problem
A back-to-back stem-and-leaf plot shows the number of points scored by players in two sports teams.
Team X | Stem | Team Y
8 6 4 | 2 | 1 3 5
9 5 3 | 3 | 0 4 8
6 6 1 | 4 | 2 6 9
5 | 5 | 1 4
Leaves for Team X are read from the stem outward (right to left).
For Team X, find: (a) the median, (b) the mode, and (c) the range.
Team X | Stem | Team Y
8 6 4 | 2 | 1 3 5
9 5 3 | 3 | 0 4 8
6 6 1 | 4 | 2 6 9
5 | 5 | 1 4
Leaves for Team X are read from the stem outward (right to left).
For Team X, find: (a) the median, (b) the mode, and (c) the range.
1
List the Team X values in ascending order.
Stem 2, leaves 4, 6, 8 → 24, 26, 28
Stem 3, leaves 3, 5, 9 → 33, 35, 39
Stem 4, leaves 1, 6, 6 → 41, 46, 46
Stem 5, leaf 5 → 55
Ordered: 24, 26, 28, 33, 35, 39, 41, 46, 46, 55 (10 values)
Stem 3, leaves 3, 5, 9 → 33, 35, 39
Stem 4, leaves 1, 6, 6 → 41, 46, 46
Stem 5, leaf 5 → 55
Ordered: 24, 26, 28, 33, 35, 39, 41, 46, 46, 55 (10 values)
2
Find the median.
10 values → median is the average of the 5th and 6th values.
5th value = 35, 6th value = 39
Median
5th value = 35, 6th value = 39
Median
3
Find the mode.
Scan for repeated values: 46 appears twice; all other values appear once.
Mode = 46
Mode = 46
4
Find the range.
Range = maximum − minimum = 55 − 24 = 31
Answer
(a) Median = 37. (b) Mode = 46. (c) Range = 31.
Comparing two distributions using a back-to-back stem plot
Challenging
Problem
A back-to-back stem-and-leaf plot shows the marks (out of 80) achieved by two exam classes.
Team A | Stem | Team B
8 | 3 | 1 4 6
9 7 5 | 4 | 0 3 8
8 6 4 | 5 | 2 5 9
9 7 3 | 6 | 1 7
5 | 7 | 4 8
Leaves for Team A are read from the stem outward (right to left).
Use the median and range to compare the performance of the two classes.
Team A | Stem | Team B
8 | 3 | 1 4 6
9 7 5 | 4 | 0 3 8
8 6 4 | 5 | 2 5 9
9 7 3 | 6 | 1 7
5 | 7 | 4 8
Leaves for Team A are read from the stem outward (right to left).
Use the median and range to compare the performance of the two classes.
1
List the Team A values in ascending order and find the median and range.
Stem 3: 38
Stem 4: 45, 47, 49
Stem 5: 54, 56, 58
Stem 6: 63, 67, 69
Stem 7: 75
Ordered: 38, 45, 47, 49, 54, 56, 58, 63, 67, 69, 75 (11 values)
Median = 6th value = 56
Range = 75 − 38 = 37
Stem 4: 45, 47, 49
Stem 5: 54, 56, 58
Stem 6: 63, 67, 69
Stem 7: 75
Ordered: 38, 45, 47, 49, 54, 56, 58, 63, 67, 69, 75 (11 values)
Median = 6th value = 56
Range = 75 − 38 = 37
2
List the Team B values in ascending order and find the median and range.
Stem 3: 31, 34, 36
Stem 4: 40, 43, 48
Stem 5: 52, 55, 59
Stem 6: 61, 67
Stem 7: 74, 78
Ordered: 31, 34, 36, 40, 43, 48, 52, 55, 59, 61, 67, 74, 78 (13 values)
Median = 7th value = 52
Range = 78 − 31 = 47
Stem 4: 40, 43, 48
Stem 5: 52, 55, 59
Stem 6: 61, 67
Stem 7: 74, 78
Ordered: 31, 34, 36, 40, 43, 48, 52, 55, 59, 61, 67, 74, 78 (13 values)
Median = 7th value = 52
Range = 78 − 31 = 47
3
Compare the medians.
Team A median = 56, Team B median = 52.
Team A has a higher median, suggesting Team A generally achieved higher marks.
Team A has a higher median, suggesting Team A generally achieved higher marks.
4
Compare the ranges.
Team A range = 37, Team B range = 47.
Team B has a greater range, indicating more variation in Team B's results.
Team B has a greater range, indicating more variation in Team B's results.
Answer
Team A has a higher median mark (56 compared to 52), suggesting Team A generally performed better. Team B has a greater range (47 compared to 37), indicating more spread in Team B's results.
Practise
Q1·Straightforward
A back-to-back stem-and-leaf plot shows the number of books read in a year by students in two groups.
Group A | Stem | Group B
6 | 1 | 2 4 9
8 5 2 | 2 | 1 5 7
7 4 0 | 3 | 3 6
Leaves for Group A are read from the stem outward (right to left).
Find the median for Group A.
Group A | Stem | Group B
6 | 1 | 2 4 9
8 5 2 | 2 | 1 5 7
7 4 0 | 3 | 3 6
Leaves for Group A are read from the stem outward (right to left).
Find the median for Group A.
Explanation
Group A values in ascending order: 16, 22, 25, 28, 30, 34, 37.
There are 7 values, so the median is the 4th value = 28.
There are 7 values, so the median is the 4th value = 28.
Q2·Straightforward
A back-to-back stem-and-leaf plot shows the distances (in metres) thrown in a shot put event.
Class A | Stem | Class B
5 | 1 | 3 6 8
9 6 3 | 2 | 0 4 7
8 4 1 | 3 | 2 5 9
7 | 4 | 1 6
Leaves for Class A are read from the stem outward (right to left).
How many students in Class A achieved a distance of at least 30 metres?
Class A | Stem | Class B
5 | 1 | 3 6 8
9 6 3 | 2 | 0 4 7
8 4 1 | 3 | 2 5 9
7 | 4 | 1 6
Leaves for Class A are read from the stem outward (right to left).
How many students in Class A achieved a distance of at least 30 metres?
Explanation
Class A values: 15, 23, 26, 29, 31, 34, 38, 47.
Values of at least 30 m: 31, 34, 38, 47.
That is 4 students.
Values of at least 30 m: 31, 34, 38, 47.
That is 4 students.
Q3·Straightforward
A back-to-back stem-and-leaf plot shows the ages (in months) of dogs at two animal shelters.
Shelter A | Stem | Shelter B
8 | 1 | 2 5 7
9 5 1 | 2 | 0 4 6
7 4 2 | 3 | 1 8
Leaves for Shelter A are read from the stem outward (right to left).
Find the median age (in months) of dogs at Shelter A.
Shelter A | Stem | Shelter B
8 | 1 | 2 5 7
9 5 1 | 2 | 0 4 6
7 4 2 | 3 | 1 8
Leaves for Shelter A are read from the stem outward (right to left).
Find the median age (in months) of dogs at Shelter A.
Explanation
Shelter A values in ascending order: 18, 21, 25, 29, 32, 34, 37.
There are 7 values, so the median is the 4th value = 29 months.
There are 7 values, so the median is the 4th value = 29 months.
Q4·Straightforward
A back-to-back stem-and-leaf plot shows the ages of members of two running clubs.
Club A | Stem | Club B
7 | 2 | 4 8
9 6 3 | 3 | 1 5 7
8 4 2 | 4 | 0 3 6
5 | 5 | 2 9
Leaves for Club A are read from the stem outward (right to left).
What is the minimum age of Club A members?
Club A | Stem | Club B
7 | 2 | 4 8
9 6 3 | 3 | 1 5 7
8 4 2 | 4 | 0 3 6
5 | 5 | 2 9
Leaves for Club A are read from the stem outward (right to left).
What is the minimum age of Club A members?
Explanation
The smallest stem row is stem 2. The Club A leaf furthest from the stem in that row is 7, giving the value 27.
Minimum age = 27 years.
Minimum age = 27 years.
Q5·Moderate
A back-to-back stem-and-leaf plot shows the scores in a spelling competition for two school teams.
Team A | Stem | Team B
9 5 5 | 1 | 2 4 8
8 6 3 | 2 | 1 3 7
7 4 2 | 3 | 5 5 8
6 | 4 | 1
Leaves for Team A are read from the stem outward (right to left).
Find the median score for Team A.
Team A | Stem | Team B
9 5 5 | 1 | 2 4 8
8 6 3 | 2 | 1 3 7
7 4 2 | 3 | 5 5 8
6 | 4 | 1
Leaves for Team A are read from the stem outward (right to left).
Find the median score for Team A.
Explanation
Team A values in order: 15, 15, 19, 23, 26, 28, 32, 34, 37, 46.
There are 10 values, so the median is the average of the 5th and 6th values.
Median
There are 10 values, so the median is the average of the 5th and 6th values.
Median
Q6·Moderate
A back-to-back stem-and-leaf plot shows the scores in a spelling competition for two school teams.
Team A | Stem | Team B
9 5 5 | 1 | 2 4 8
8 6 3 | 2 | 1 3 7
7 4 2 | 3 | 5 5 8
6 | 4 | 1
Leaves for Team A are read from the stem outward (right to left).
Find the range of scores for Team B.
Team A | Stem | Team B
9 5 5 | 1 | 2 4 8
8 6 3 | 2 | 1 3 7
7 4 2 | 3 | 5 5 8
6 | 4 | 1
Leaves for Team A are read from the stem outward (right to left).
Find the range of scores for Team B.
Explanation
Team B values: 12, 14, 18, 21, 23, 27, 35, 35, 38, 41.
Maximum = 41, minimum = 12.
Range = 41 − 12 = 29.
Maximum = 41, minimum = 12.
Range = 41 − 12 = 29.
Q7·Moderate
A back-to-back stem-and-leaf plot shows the heights (in cm) of plants grown in two different gardens.
Garden A | Stem | Garden B
9 6 3 | 2 | 1 4 7
8 8 5 | 3 | 0 3 6
7 4 1 | 4 | 2 5 8
6 | 5 | 1 1
Leaves for Garden A are read from the stem outward (right to left).
Find the mode height for Garden A.
Garden A | Stem | Garden B
9 6 3 | 2 | 1 4 7
8 8 5 | 3 | 0 3 6
7 4 1 | 4 | 2 5 8
6 | 5 | 1 1
Leaves for Garden A are read from the stem outward (right to left).
Find the mode height for Garden A.
Explanation
Garden A values: 23, 26, 29, 35, 38, 38, 41, 44, 47, 56.
The value 38 appears twice; all other values appear once.
Mode = 38 cm.
The value 38 appears twice; all other values appear once.
Mode = 38 cm.
Q8·Moderate
A back-to-back stem-and-leaf plot shows the heights (in cm) of plants grown in two different gardens.
Garden A | Stem | Garden B
9 6 3 | 2 | 1 4 7
8 8 5 | 3 | 0 3 6
7 4 1 | 4 | 2 5 8
6 | 5 | 1 1
Leaves for Garden A are read from the stem outward (right to left).
Find the median height for Garden B.
Garden A | Stem | Garden B
9 6 3 | 2 | 1 4 7
8 8 5 | 3 | 0 3 6
7 4 1 | 4 | 2 5 8
6 | 5 | 1 1
Leaves for Garden A are read from the stem outward (right to left).
Find the median height for Garden B.
Explanation
Garden B values in ascending order: 21, 24, 27, 30, 33, 36, 42, 45, 48, 51, 51.
There are 11 values, so the median is the 6th value = 36 cm.
There are 11 values, so the median is the 6th value = 36 cm.
Q9·Challenging
A back-to-back stem-and-leaf plot shows the ages of participants at two sporting events.
Event A | Stem | Event B
8 3 | 1 | 4 6 9
9 7 5 | 2 | 2 5 8
8 6 4 | 3 | 1 7
7 3 1 | 4 | 0 5 8
2 | 5 | 3 6
Leaves for Event A are read from the stem outward (right to left).
Find the median age for Event A.
Event A | Stem | Event B
8 3 | 1 | 4 6 9
9 7 5 | 2 | 2 5 8
8 6 4 | 3 | 1 7
7 3 1 | 4 | 0 5 8
2 | 5 | 3 6
Leaves for Event A are read from the stem outward (right to left).
Find the median age for Event A.
Explanation
Event A values in ascending order: 13, 18, 25, 27, 29, 34, 36, 38, 41, 43, 47, 52.
There are 12 values, so the median is the average of the 6th and 7th values.
Median years.
There are 12 values, so the median is the average of the 6th and 7th values.
Median years.
Q10·Challenging
A back-to-back stem-and-leaf plot shows the ages of participants at two sporting events.
Event A | Stem | Event B
8 3 | 1 | 4 6 9
9 7 5 | 2 | 2 5 8
8 6 4 | 3 | 1 7
7 3 1 | 4 | 0 5 8
2 | 5 | 3 6
Leaves for Event A are read from the stem outward (right to left).
Find the median age for Event B.
Event A | Stem | Event B
8 3 | 1 | 4 6 9
9 7 5 | 2 | 2 5 8
8 6 4 | 3 | 1 7
7 3 1 | 4 | 0 5 8
2 | 5 | 3 6
Leaves for Event A are read from the stem outward (right to left).
Find the median age for Event B.
Explanation
Event B values in ascending order: 14, 16, 19, 22, 25, 28, 31, 37, 40, 45, 48, 53, 56.
There are 13 values, so the median is the 7th value = 31 years.
There are 13 values, so the median is the 7th value = 31 years.
Q11·Challenging
A back-to-back stem-and-leaf plot shows the number of customers served per hour at two coffee shops over an eleven-day period.
Shop A | Stem | Shop B
9 6 | 2 | 1 4 7
8 6 3 | 3 | 0 2 8
9 7 4 | 4 | 1 5 9
8 5 2 | 5 | 3 6
Leaves for Shop A are read from the stem outward (right to left).
Find the range of customers per hour for Shop A.
Shop A | Stem | Shop B
9 6 | 2 | 1 4 7
8 6 3 | 3 | 0 2 8
9 7 4 | 4 | 1 5 9
8 5 2 | 5 | 3 6
Leaves for Shop A are read from the stem outward (right to left).
Find the range of customers per hour for Shop A.
Explanation
Shop A values: 26, 29, 33, 36, 38, 44, 47, 49, 52, 55, 58.
Maximum = 58, minimum = 26.
Range = 58 − 26 = 32.
Maximum = 58, minimum = 26.
Range = 58 − 26 = 32.
Q12·Challenging
A back-to-back stem-and-leaf plot shows the number of customers served per hour at two coffee shops over an eleven-day period.
Shop A | Stem | Shop B
9 6 | 2 | 1 4 7
8 6 3 | 3 | 0 2 8
9 7 4 | 4 | 1 5 9
8 5 2 | 5 | 3 6
Leaves for Shop A are read from the stem outward (right to left).
Which statement correctly compares the two coffee shops?
Shop A | Stem | Shop B
9 6 | 2 | 1 4 7
8 6 3 | 3 | 0 2 8
9 7 4 | 4 | 1 5 9
8 5 2 | 5 | 3 6
Leaves for Shop A are read from the stem outward (right to left).
Which statement correctly compares the two coffee shops?
Explanation
Shop A values: 26, 29, 33, 36, 38, 44, 47, 49, 52, 55, 58 (11 values). Median = 6th value = 44. Range = 58 − 26 = 32.
Shop B values: 21, 24, 27, 30, 32, 38, 41, 45, 49, 53, 56 (11 values). Median = 6th value = 38. Range = 56 − 21 = 35.
Shop A has a higher median (44 > 38) and a smaller range (32 < 35), so Shop A has a higher median and smaller spread.
Shop B values: 21, 24, 27, 30, 32, 38, 41, 45, 49, 53, 56 (11 values). Median = 6th value = 38. Range = 56 − 21 = 35.
Shop A has a higher median (44 > 38) and a smaller range (32 < 35), so Shop A has a higher median and smaller spread.