Stem-and-Leaf Stem-and-Leaf display is another graphical device that is useful for representing quantitative data sets. A stem-and-leaf display bears a strong resemblance to a histogram and serves the same purpose. To construct a stem-and-leaf display we partition each measurement into two parts. The first part is called the stem, and the second part is called the leaf. The stem consists of one or more of the initial digits of the measurement, and the leaf is composed of one or more of the remaining digits. When leaves consist of more than one digit, all digits after the first may be rounded off. Decimals when present in the original data are omitted in the stem-and-leaf display. The stems are separated from their leaves by a vertical line. Thus, we see that a stem-and-leaf display is also an ordered array of the data.

1. Lecture No 06
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Muhammad Alfahad
Farwa Butt

2. Stem-and-Leaf
 Stem-and-Leaf display is another graphical device that is useful for representing quantitative
data sets. A stem-and-leaf display bears a strong resemblance to a histogram and serves the
same purpose.
 To construct a stem-and-leaf display we partition each measurement into two parts. The first
part is called the stem, and the second part is called the leaf.
 The stem consists of one or more of the initial digits of the measurement, and the leaf is
composed of one or more of the remaining digits.
 When leaves consist of more than one digit, all digits after the first may be rounded off.
Decimals when present in the original data are omitted in the stem-and-leaf display.
 The stems are separated from their leaves by a vertical line. Thus, we see that a stem-and-
leaf display is also an ordered array of the data.
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3. Steps for stem and leaf display
 The choice of stem unit is important when constructing stem and leaf display.
 Note the minimum and maximum value.
 Take a stem unit and also mention the width of class also.
 Draw a vertical line and display the stem to the left of the vertical bar.
 Attached the leaves to the right hand side of vertical bar.
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4. Example
The following data is the final plant height (cm) of thirty plants of wheat. Construct a stem and
leaf plot by taking 10 as a stem unit.
87 91 89 88 89 91 87 92 90 98 95
97 96 100 101 96 98 99 98 100 102 99
101 105 103 107 105 106 107 112
Solution:
Minimum value = 87, Maximum value = 112
Stem unit = 10, Width of class = 10
Stem Leaf
8
9
10
11
79897
1120857668989
01021537567
2
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5. Stem Leaf
8*
8.
9*
9.
10*
10.
11*
11.
-
79897
1120
857668989
010213
57567
2
-
Alternate solution: Taking width of class width =
5
*Indicates 0-4 , .Indicates 5-9
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6. Stem Leaf
6
7
10
13
6
6
5
2
Using Other stem units
Use the 100’s digit as the stem.
Round off the 10’s digit to form the leaves. For example consider the following values 665,756,
1053,1322.
 665 would become
 756 would become
 1053 would become
 1322 would become
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7. Steam and Leaf Plots for Data containing decimal values
Example: Construct a stem-and-leaf plot for the following list of values:
23.25, 24.13, 24.76, 24.81, 24.98, 25.31, 25.57, 25.89, 26.28, 26.34, 27.09
Solution
Before construction of steam and leaf plot, we round off the numbers to one decimal place
23.2, 24.1, 24.8, 24.8, 25.0, 25.3, 25.6, 25.9, 26.3, 26.3, 27.1
Stem Leaf
23
24
25
26
27
2
188
0369
33
27
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8. Question: Construct steam and leaf plot for the following data using 100 digit as stem.
613,632,658,717,722,750,776,827,841,859,863,891,894,906,928,933,955,982,1034,1056,11401169,1224.
Question: The following are the ages of 30 patients seen in the emergency room of a hospital on
a Friday night. Construct a stem-and-leaf display from these data.
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9. Measures of Central Tendency
 In statistics the condensation of huge data to a meaningful representative form is
absolutely necessary.
 In order to reduce the complexity of data obtained form different sources, it is
necessary that data should be reduced to a single figure.
 It is, therefore, the single figure which is used to represent a whole series should be
neither the lowest value nor a highest value but value somewhere between the two
values, possibly in the center where most of the item of a series are clustered.
 Such figures are called measures of central tendency or averages. Average is a
single value which is representative of a series and contain major characteristics of
series.
Averages can be broadly classified into two types
i) Mathematical averages
ii) Positional average
 Mathematical average is further divided into Arithmetic mean, Geometric mean and
Harmonic Mean
 Positional averages are further divided into Mode and Median
Types of Averages
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10. The Greek letter Σ (sigma) is used as a symbol for summation, and for the value for
observation i.
The sum of numbers can be expressed:
The sum of squares of n numbers is:
The sum of products of two sets of n numbers and
Example: Consider a set of three numbers: 1, 3 and 6. The numbers are symbolized by: y1 =
1, y2 = 3 and y3 = 6.
The sum and sum of squares of those numbers are: and
1 2
, ,..., n
y y y
1 2
1
...
n
n
i
yi y y y

   

2 2 2 2
1 2
1
...
n
i n
i
y y y y

   

1 1 2 2
1
...
n
i i n n
i
x y x y x y x y

   

1 2 3
, , ,..., n
x x x x 1 2
, ,..., n
y y y
1 2
, ,..., n
y y y
i
y
3
1
1 3 6 10
i
i
y

   

3
2 2 2 2
1
1 3 6 46
i
i
y

   

n
Symbolic Notation
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11. 1. The sum of addition of two sets of numbers is equal to the addition of the sums:
2. The sum of products of a constant k and a variable y is equal to the product of the
constant and the sum of the values of the variable:
3. The sum of n constants with value k is equal to the product nk:
1 1 1
( )
n n n
i i
i i i
Xi Yi X Y
  
  
  
1 1
( )
n n
i i
i i
KY K Y
 

 
1
( )
n
i
K nK



Three main rules of
addition:
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