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### Graphing Distribution : Stem and Leaf Displays

#### The Objectives:

• How do we create stem and leaf displays? Its interpretation.

• Back to back stem and leaf display creation and interpretation.

• To examine whether stem and leaf display is appropriate for a dataset. ### What is Stem and Leaf Display?

Stem and Leaf plot or display is basically used for organizing a given statistical dataset. The Stem is the greatest common place value of the statistical data whereas the leaf indicates the next greatest common place value of that data. Such display is good for small or moderate amount of data.

#### Construction and Interpretation of Stem and Leaf Display:

We will consider examples to construct and interpret Stem and Leaf Displays.

Example 1: The following table is the marks obtained by 25 students in economics.

Table 1: Marks obtained by 25 students in economics

 51 43 69 85 36 20 80 50 56 38 78 71 65 60 48 42 25 47 53 45 66 77 39 44 54

To make the construction easier we first arrange the data in ascending order as follows:

 20 25 36 38 39 42 43 44 45 47 48 50 51 53 54 56 60 65 66 69 71 77 78 80 85

We see that the above data lies between 20 to 85. So the stem has a range from 2 to 8. A vertical list of stem is first written for plotting. Assignment is done to the stem list by correctly pairing the unit digit (leaf) with the stem. In this way stem and leaf display is created.

Figure 1: Stem and Leaf Display of the marks obtained by students in economics:

From the above we can see that the marks 20 is plotted by assigning the unit digit 0 to the right of the stem 2. In these the other marks are also plotted.

Thus, in figure 1 there are two leaves 0 and 5 to the right, for the stem 2. If we combine the stem with the leaves, we will obtain the marks of the students for stem 2, viz. 20 and 25. Now the next row comprises of 4 leaves 6, 8, 9 with the stem 3. Combining the stem and leaves we get 3 marks viz 36, 38, 39. In these way if we observe the remaining rows.

The shape of the distribution can be understood more easily through the stem and leaf display. We can easily figure out the distribution from figure 1 than from table 1. From figure 1 we can see that majority of the students obtained marks between 42 to 56. So from stem and Leaf display we can easily understand the distribution of the data set.

A further experimentation can be done by splitting each of the stem into two portions. The division is done in such a way that one part consists of the leaves 0 to 4 and the other part consists of the leaves 5 to 9.This has been shown in fig 2.

Fig 2: Stem and Leaf display with splitting stems:

#### Leaf

2

0

2

5

3

6 8 9

4

2 3 4

4

5 7 8

5

0 1 2 3 4

5

6

6

0

6

5 6 9

7

1

7

7 8

8

0

8

5

The first row represents the marks from 20 to 24. The second row represents the marks from 25 to 29 and so on. Figure 2 is more revealing as compared to Figure 1. The former that is figure 1 contained too many values in a row. In Figure 2 we have splitted the stem into two halves and the display becomes better. So it is advisable to split the stems into two or more parts if a row gets too long with a particular stem.

#### Back to back stem and Leaf Display:

Stem and Leaf Display can also be used to compare distributions.

Example 2:The following table consists of the scores in two years 1999 and 2001

Table 2: The scores in two years 1999 and 2001

#### 2001

34,36,40,45,46,49,50,51,51,52,

33,35,37,41,44,46,48,49,51,52

58,59,61,64,67,69,72,72,72,76,

55,56,57,62,62,63,64,65,72,73,

78,81,82,84,87

78,79,80,83,89

In a back to back stem leaf display the plotting is done back to back by using the common stem. In this way the comparison can be done between the two distributions of scores in 1999 and 2001. There are three columns in such display, the middle column represents the common stem, the column left of the middle are the leaves of the scores of 1999, and the column right of the middle comprises leaves of the scores of 2001. Figure 3: Back to Back Stem and Leaf display of Table 2:

 4 3 3 6 3 57 0 4 14 569 4 689 0112 5 12 89 5 567 14 6 223 79 6 4 222 7 23 68 7 89 124 8 03 7 8 9

The above shows the difference of distributions between scores in 1999 and 2001.Hence with the help of it , the distributions can be compared. We can see that the differences between distributions of scores in 1999 and 2001 are more or less similar.

Stem and Leaf display is not only limited to whole number and positive numbers. It can also be performed when the data contain numbers that are decimal or have negative value. In addition to that the stem and leaf display can also be done when the numbers have three or more digits. We can see from table 1 and table 2 that all the numbers are whole numbers and positive for which one digit is used as stem and the other one is used as a leaf.And the numbers have two digits. Three digit numbers or decimals are rounded and transformed to numbers with two digits.Non-positive numbers can also be handled. These can be shown by an example.

Example 3: The deviations of the scores of a test before and after training are given in the following table. The difference is carried out by subtracting score before training from score after training. The deviations can be negative or positive, if the score is negative then the score decreased after training, otherwise if it is positive the score has improved after the training is given.

Table 3: Scores and Deviations of scores before and after training:

#### Deviation

41.3

49.6

8.3

56.3

43.5

-12.8

58.2

62.3

4.1

59.1

51.8

-7.3

69.3

65.7

-3.6

52.3

60.0

7.7

49.1

47.4

-1.7

75.6

71.4

-4.2

57.4

63.8

6.4

67.3

60.5

-6.8

The differences range from -12.8 to 8.3. The decimal values are rounded and for the negative values we use negative stem. For example the first value 8.3 is considered as 8 and the value -12.8 is considered as – 13, the stem is -1. So the numbers after rounding becomes 8,-13, 4,-7,-4, 8,-2,-4, 6,-7.

Figure 4: Stem and Leaf Display of negative number and decimals

#### Leaf

0

846

-0

4 7 4 2 4 7

-1

3

The stem with 0 contains leaves with numbers from 0 to 9, the stem -0 has leaves with numbers 0 to -9. If there was a value 0 then it is written twice, one with stem 0 and other with stem -0. Stem and Leaf Display are appropriate for small and moderate dataset. It is not so good for large dataset.

Three digit number can also be handled and stem and leaf display can be used to plot the numbers.

Table 4: Population of a country for 15 years is given in the following table:

 23650 24560 25030 29630 36998 37836 39664 401225 41963 51236 55994 56914 63218 649321 75261

The numbers are rounded to two digits. For example the first number is approximated to 23000, the next 24000 and so on.

Figure5: Stem and Leaf of three digit number

#### Leaf

2

3 4

2

5 9

3

6 7 9

4

0 1

5

1

5

5 6

6

3 4

7

5

To find whether the data is appropriate for a Stem and Leaf Display we should know that rounding off doesn’t cause any loss of information. Judgment should be made to find what graph is appropriate for a given data.

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