“Teach A Level Maths”
Statistics 1
Stem and Leaf
Diagrams
© Christine Crisp
Stem and Leaf Diagrams
Statistics 1
AQA
EDEXCEL
MEI/OCR
OCR
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Stem and Leaf Diagrams
You met some statistical diagrams when you did GCSE.
The next three presentations and this one remind you of
them and point out some details that you may not have
met before.
We will start with stem and leaf diagrams
( including back-to-back ).
Stem and leaf diagrams are sometimes called stem plots.
Stem and Leaf Diagrams
e.g. The table below gives the number of hours
worked in a particular week by a sample of 30 men
35
41
33
31
30
45
35
36
51
32
32
30
28
35
34
33
35
36
32
42
33
31
41
21
34
35
34
46
32
35
I’ll use intervals of 5 hours to draw the diagram i.e.
20-25, 26-30 etc.
5 1 Weekly hours of 30 men
The stem shows the
4 5 6
tens . . . and the
4 1 1 2
leaves the units
3 5 5 5 5 5 5 6 6
e.g. 46 is 4 tens and 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
units
2 8
2 1
Stem and Leaf Diagrams
e.g. The table below gives the number of hours
worked in a particular week by a sample of 30 men
35
41
33
31
30
45
35
36
51
32
32
30
28
35
34
33
35
36
32
42
33
31
41
21
34
35
34
46
32
35
I’ll use intervals of 5 hours to draw the diagram i.e.
20-25, 26-30 etc.
5 1 Weekly hours of 30 men
The stem shows the
4 5 6
tens . . . and the
4 1 1 2
leaves the units
3 5 5 5 5 5 5 6 6
e.g. 46 is 4 tens and 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
units
2 8
2 1
Stem and Leaf Diagrams
e.g. The table below gives the number of hours
worked in a particular week by a sample of 30 men
35
41
33
31
30
45
35
36
51
32
32
30
28
35
34
33
35
36
32
42
33
31
41
21
34
35
34
46
32
35
I’ll use intervals of 5 hours to draw the diagram i.e.
20-25, 26-30 etc.
5 1 Weekly hours of 30 men
The stem shows the
4 5 6
tens . . . and the
4 1 1 2
leaves the units
3 5 5 5 5 5 5 6 6
e.g. 46 is 4 tens and 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
units
2 8
N.B. 35 goes here . . .
2 1
not in the line below.
Stem and Leaf Diagrams
e.g. The table below gives the number of hours
worked in a particular week by a sample of 30 men
35
41
33
31
30
45
35
36
51
32
32
30
28
35
34
33
35
36
32
42
33
31
41
21
34
35
34
46
32
35
I’ll use intervals of 5 hours to draw the diagram i.e.
20-25, 26-30 etc.
5 1 Weekly hours of 30 men
The stem shows the
4 5 6
tens . . . and the
4 1 1 2
leaves the units
3 5 5 5 5 5 5 6 6
e.g. 46 is 4 tens and 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
units
2 8
We must show a key.
2 1
Key: 3 5 means 35 hours
Stem and Leaf Diagrams
Weekly hours of 30 men
5 1
4 5 6
4 1 1 2
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
Key: 3 5 means 35 hours
2 1
If you tip your head to the right and look at the diagram
you can see it is just a bar chart with more detail.
Points to notice:
•
•
The leaves are in numerical order
The diagram uses raw ( not grouped ) data
Stem and Leaf Diagrams
Finding the median and quartiles
Finding these is easy because the data are in order.
5 1
Weekly hours of 30 men
4 5 6
4 1 1 2
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
Key: 3 5 means 35 hours
2 1
Median: The median is the middle item, . . .
so with 30 observations we need the 15  5n th
 1item, the
Tip:
To16.
find the middle use
where n
average of 15
and
2
is the number of items of data.
Stem and Leaf Diagrams
Finding the median and quartiles
Finding these is easy because the data are in order.
5 1
Weekly hours of 30 men
4 5 6
4 1 1 2
15th
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
Key: 3 5 means 35 hours
2 1
Median: The median is the middle item, . . .
so with 30 observations we need the 15  5 th item, the
average of 15 and 16.
Stem and Leaf Diagrams
Finding the median and quartiles
Finding these is easy because the data are in order.
5 1
Weekly hours of 30 men
4 5 6
4 1 1 2
3 5 5 5 5 5 5 6 6
15th
16th
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
Key: 3 5 means 35 hours
2 1
Median: The median is the middle item, . . .
so with 30 observations we need the 15  5 th item, the
average of 15 and 16.
Stem and Leaf Diagrams
Finding the median and quartiles
Finding these is easy because the data are in order.
5 1
Weekly hours of 30 men
4 5 6
4 1 1 2
3 5 5 5 5 5 5 6 6
15th
16th
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
Key: 3 5 means 35 hours
2 1
Median: The median is the middle item, . . .
so with 30 observations we need the 15  5 th item, the
average of 15 and 16.
Since the 15th and 16th items are both 34, the median is 34.
( If the values are not the same we average them. )
Stem and Leaf Diagrams
Finding the median and quartiles
Finding these is easy because the data are in order.
Weekly hours of 30 men
7th
4 5 6
5 1
4 1 1 2
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
2 1
Key: 3 5 means 35 hours
For the lower quartile (LQ) we first need
n1
4
(  7  75 )
Stem and Leaf Diagrams
Finding the median and quartiles
Finding these is easy because the data are in order.
Weekly hours of 30 men
7th
4 5 6
5 1
4 1 1 2
8th
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
2 1
Key: 3 5 means 35 hours
For the lower quartile (LQ) we first need
n1
4
(  7  75 )
Stem and Leaf Diagrams
Finding the median and quartiles
Finding these is easy because the data are in order.
Weekly hours of 30 men
7th
4 5 6
5 1
8th
4 1 1 2
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
2 1
Key: 3 5 means 35 hours
For the lower quartile (LQ) we first need
The lower quartile is 32.
n1
4
(  7  75 )
Stem and Leaf Diagrams
Finding the median and quartiles
If the values of the 7th and 8th observation are not the
same, we interpolate to find the LQ.
e.g. If we had
7th value:
8th value:
32
36
and we want the 7·75th value, we need to add 0·75 of the
gap between the 7th and 8th to the 7th value.
So,
The gap is 36 – 32 = 4.
0.75 of 4 is 3.
LQ = 32 + 3 = 35
Stem and Leaf Diagrams
Finding the median and quartiles
5 1
Weekly hours of 30 men
4 5 6
4 1 1 2
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
2 1
Key: 3 5 means 35 hours
For the upper quartile (UQ), we first need
( or 7  75 from the top end )
3( n  1)
4
(  23  25 )
Stem and Leaf Diagrams
Finding the median and quartiles
5 1
Weekly hours of 30 men
4 5 6
23rd
4 1 1 2
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
2 1
Key: 3 5 means 35 hours
For the upper quartile (UQ), we first need
( or 7  75 from the top end )
3( n  1)
4
(  23  25 )
Stem and Leaf Diagrams
Finding the median and quartiles
5 1
Weekly hours of 30 men
4 5 6
4 1 1 2
23rd
24th
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
2 1
Key: 3 5 means 35 hours
For the upper quartile (UQ), we first need
( or 7  75 from the top end )
3( n  1)
4
(  23  25 )
Stem and Leaf Diagrams
Finding the median and quartiles
5 1
Weekly hours of 30 men
4 5 6
4 1 1 2
23rd
24th
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
2 1
Key: 3 5 means 35 hours
For the upper quartile (UQ), we first need
( or 7  75 from the top end )
The upper quartile is 36.
The interquartile range (IQR) = UQ - LQ
= 36 – 32 = 4
3( n  1)
4
(  23  25 )
SUMMARY
Stem and Leaf Diagrams
 Stem and Leaf diagrams are used for small, raw data
sets (not grouped data).
e.g.
Weekly hours of 30 men
5 1
4 5 6
4 1 1 2
leaves
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
2 1
stem
Key: 3 5 means 35 hours
 The diagram must have a title and a key.
 The leaves are in numerical order ( away from
the stem )
continued
Stem and Leaf Diagrams
 The median is the
 n  1  th


 2 
piece of data.
( If necessary, average 2 data items )
 The quartiles are found at the  n  1 th position
and the
 3( n  1)  th


4



4

position.
 The interquartile range (IQR) = UQ - LQ
( upper quartile – lower quartile )
 If necessary, we can interpolate to find the LQ
and UQ.
Back-to-Back Stem and Leaf Diagrams
Back-to-back stem and leaf diagrams can be used to
compare 2 sets of data relating to the same subject.
In our example we could add the data for 30 women.
Weekly hours
Women
5 1
Men
4 3 4 5 6
0 4 1 1 2
8 8 6 6 5 5 5 5 3 5 5 5 5 5 5 6 6
4 4 4 3 3 3 2 2 2 3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
8 2 8
3 3 1 0 2 1
5 5 1
4 4 2 1
Key: 5 3 means 35 hrs
Notice how easily we can
compare the variability
of the 2 data sets
Key: 3 5 means 35 hrs
Back-to-Back Stem and Leaf Diagrams
Exercise
January Max Temperatures 1985 to 2005
Braemar
Sheffield
9 0
2 8 1 2 4 5 9
7 1 2 2 4 8
8 5 2 1 6 4 5 9
8 6 1 5 0 2 3 4 5
9 8 4 3 0 0 4
9 7 5 1 3 4 7
5 2
6 1 1
Key: 3 4 means 4·3C
Key: 3 4 means 3·4C
Find the medians and quartiles. What can you say about
the temperatures of the 2 places?
Back-to-Back Stem and Leaf Diagrams
Answer:
January Max Temperatures 1985 to 2005
Braemar
Sheffield
2
8 5
8
9 84 3
9 7
Key: 3 4 means 4·3C
Median
Braemar
4·4
Sheffield
7·1
LQ
UQ
2
6
0
5
1
1
0
1
5
6 1
9
8
7
6
5
4
3
2
1
0
1
1
4
0
2
2
5
2
4 5 9
2 4 8
9
3 4 5
4 7
Key: 3 4 means 3·4C
The places have similar
variability in temperature but
3·6 5·95
Sheffield is about 2 ·7C
5·35 8·15 warmer than Braemar.
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Stem and Leaf Diagrams
SUMMARY
 This is used for small, raw data sets (not grouped data).
e.g.
stem
5 1
Weekly hours of 30 men
4 5 6
leaves
4 1 1 2
3 5 5 5 5 5 5 6 6
3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
2 8
2 1
Key: 3 5 means 35 hours
 The diagram must have a title and a key.
 The leaves are in numerical order ( away from the
stem )
Stem and Leaf Diagrams
 n  1  th

 2 
 The median is the 
piece of data.
( If necessary, average 2 data items )
 The quartiles are found at the  n  1 th position
and the  3( n  1)  th position.

4

4


 The interquartile range (IQR) = UQ - LQ
( upper quartile – lower quartile )
 If necessary, we can interpolate to find the LQ
and UQ.
Stem and Leaf Diagrams
Back-to-back stem and leaf diagrams can be used to
compare 2 sets of data relating to the same subject.
In our example we could add the data for 30 women.
Women
Weekly hours
5 1
Men
4 3 4 5 6
0 4 1 1 2
8 8 6 6 5 5 5 5 3 5 5 5 5 5 5 6 6
4 4 4 3 3 3 2 2 2 3 0 0 1 1 2 2 2 2 3 3 3 4 4 4
8 2 8
3 3 1 0 2 1
5 5 1
4 4 2 1
Key: 5 3 means 35 hours
Notice how easily we can
compare the variability
of the 2 data sets
Key: 3 5 means 35 hours