Stem

1
Stem Leaf Diagrams

• Representing Data
• Module S1

2
Use a Stem Leaf when

• data set is relatively small
• all exact values known, not grouped advantage
  over Histograms.
• data is in integer or decimal format
• Want to see Shape of data easily

3
Example
LEAVES
KEY
Frequencies
STEMS
4
Constructing a Stem Leaf

• Draw Stems
• Add Leaves
• Re-order leaves for each stem
• Frequencies
• Key
• Median Quartiles

5
Consider Example
1 2 3 4 5 6 7 8 9
5
6
5
2
6
4
1
6
9
2
8
8
6
Reorder Leaves
Key 2 3 means 23
(0) (2) (3) (6) (7) (4) (4) (3) (1)
Frequencies
Modal Group is 50 to 59
7
Calculating Median

• MEDIAN
• Once data is in order
• Calculate as the Middle Value

n number of values
8
Example of Median
25 26 29 35 38 42 49 57
n 8
Lies half way between central two values
9
Lower Quartile is the ¼ x n th value
Q2 36½
25 26 29 35 38 42 49
57
CLEARLY, we want the value between 26 and 29
BUT this gives us ¼ x 8 th value 2nd value
10
Upper Quartile is the ¾ x n th value
Q2 36½
Q1 27½
25 26 29 35 38 42 49
57
CLEARLY, we want the value between 42 and 49
BUT this gives us ¾ x 8 th value 6th value
11
Uneven number of values
25 26 29 35 38 42 49 57 59
n 9
MEDIAN is the 5th value It lies EXACTLY in the
centre of the values
12
Lower Quartile
Q2
Q1
25 26 29 35 38 42 49
57 59
SO we round 2¼ to 3 and take 3rd value
The usual method gives ¼ x 9 th value 2¼th value
2¼ is NOT an integer
13
Upper Quartile
Q2
Q3
Q1
25 26 29 35 38 42 49
57 59
The usual method gives ¾ x 9 th value 6¾th
value
SO we round 6¾ to 7 and take the 7th value
14
In general, for n ordered observations
y1,y2,y3,.,yn

• Median ½th value
• If ½(n1) is an Integer
• Q2 yr
• If ½(n1) is NOT an Integer
• Q2 ½(yr yr1)

15
Quartiles

• If ¼n (or ¾n) is an INTEGER
• Q1 ½(yryr1)
• Q3 ½(yryr1)
• If ¼n (or ¾n) is NOT an INTEGER but lies between
  yr and yr 1
• Q1 yr1
• Q3 yr1

16
Median, Q2
Key 2 3 means 23
(0) (2) (3) (6) (7) (4) (4) (3) (1)
17
Lower Quartile, Q1
Key 2 3 means 23
(0) (2) (3) (6) (7) (4) (4) (3) (1)
Q1 ¼ x nth value
7½th 8th value 42
O
18
Upper Quartile, Q3
Key 2 3 means 23
(0) (2) (3) (6) (7) (4) (4) (3) (1)
Q3 ¾ x nth value
22½th 23rd value 71
O
19
Comparing Results
Compare Medians On average, Boys scores are
LOWER than Girls scores
Compare IQR Girls scores are LESS VARIED than
Boys scores or Girls scores are more
consistent
20
Skew
SYMMETRICAL DISTRIBUTION
Q2 - Q1 Q3 Q2
21
POSITIVELY SKEWED DISTRIBUTION
Q2 - Q1 lt Q3 Q2
22
NEGATIVELY SKEWED DISTRIBUTION
Q2 - Q1 gt Q3 Q2