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Stem and leaf diagrams
1. Stem and leaf diagrams
A stem-and-leaf diagram provides a quick manual way to create a visual distribution of a list of numbers. It appears much like a horizontal histogram.
2. Stem and leaf diagram
The stem and leaf diagram was invented by
John Tukey (American mathematician and statistician) .
Pct Points: Stem-Leaf diagram for Exam
--- ------- -------------------------
100%+ 70:
90%+ 63: 63 65
80%+ 56: 59 59 60 62
70%+ 49: 50 53 55
60%+ 42: 42 43 43 43 43 44 44 45 45 47 47 47 47 47
50%+ 35: 35 35 37 40
40%+ 28: 28 29 29 30 31 33 34
30%+ 21: 21 23 26 27 27
20%+ 14: 15 20
10%+ 7: 7 9
Also: - -
The method for constructing a stem and leaf diagram is now described by way of an example.
3. The scores
Here is the raw data values that represent the score of each student taking an exam that was worth
70 points.
42 43 47 65 43 7 43 43 59 44
47 20 35 35 55 37 63 40 59 47
44 50 47 28 45 29 9 29 30 62
21 53 45 23 47 15 26 27 27 60
33 34 31 - -
The dashes indicate a student that did not take the exam.
It is not easy to see a pattern to the data when it is presented as an unordered collection of data values.
4. The data values
Here is the raw data as an ordered list of data values.
42
43
47
65
43
7
43
43
59
44
47
20
35
35
55
37
63
40
59
47
44
50
47
28
45
29
9
29
30
62
21
53
45
23
47
15
26
27
27
60
33
34
31
How can we make some sense of this data?
One way is to create a frequency distribution diagram.
First, a stem and leaf diagram will be created. Then a spreadsheet will be used to create a traditional frequency distribution diagram.
5. Steps
The typical steps to manually create a stem and leaf diagram, or to use a spreadsheet to create a traditional frequency distribution diagram from a set of data values, is as follows.
Decide how many intervals to use.
Determine the range of the data.
Split up the range into the desired number of intervals.
Count and record the number of data values in each interval.
Display the results in some form such as a tabular listing, chart, etc.
6. Decide the intervals
In general, you want enough intervals, but not too many. Usually, 5 to 15 intervals are used, unless you have a good reason to use fewer or more intervals.
7. Range of data
In this example, the data ranges from 0 to 70 points. The scores are counts (i.e., nonnegative integer values).
The maximum value in the data is 65 while the maximum possible score an the exam is 70. That is, no student scored 100.0% on the exam.
The minimum value in the data is 9 points. That is, no student scored 0.0% on the exam.
8. Split the range
Once the number of intervals and the range is determined, the buckets into which each data value will fall can be determined.
Suppose that each interval is to be
10.0% of
70 points, or
7 points. Then the buckets into which scores are to be placed could be as follows.
70 and up
63 to 69
56 to 62
48 to 55
42 to 47
35 to 41
28 to 34
21 to 27
14 to 20
7 to 13
0 to 6
These buckets can be written in a more visually informative and concise manner as follows.
Pct Points: Stem-Leaf diagram for Exam
--- ------- -------------------------
100%+ 70:
90%+ 63:
80%+ 56:
70%+ 49:
60%+ 42:
50%+ 35:
40%+ 28:
30%+ 21:
20%+ 14:
10%+ 7:
0%+ 0:
9. Count and record the data values
The data values are then placed into the buckets. Each data value counts as one value in the bucket. The more data values in a bucket, the more that bucket contributes to the frequency (count of data values) in that data range.
10. Add the scores
Next, each score is added to the appropriate bucket.
Pct Points: Stem-Leaf diagram for Exam
--- ------- -------------------------
100%+ 70:
90%+ 63: 63 65
80%+ 56: 59 59 60 62
70%+ 49: 50 53 55
60%+ 42: 42 43 43 43 43 44 44 45 45 47 47 47 47 47
50%+ 35: 35 35 37 40
40%+ 28: 28 29 29 30 31 33 34
30%+ 21: 21 23 26 27 27
20%+ 14: 15 20
10%+ 7: 7 9
Also: - -
In standard form, the data values in each row of a stem-and-leaf diagram are in ascending order, from left to right. When constructing the diagram manually, however, one just adds the scores as they are encountered.
Note that the form used here is that generated by the author's software, so that the "0 to 6" range is not shown as no one scored in that range, and the "Also:" line indicates that two students (indicated by dashes) did not take the exam.
Given a list of scores, the maximum value for any score (70 for the above example), and a point interval (10 points for the above example), you should be able to quickly construct a stem-and-leaf diagram in the form shown above and answer questions about such a diagram.
11. End of page
