The document discusses statistical techniques for summarizing data through stem-and-leaf displays, quartiles, and frequency distributions. It provides examples of calculating percentiles and constructing frequency distributions, including organizing and presenting data graphically with histograms and pie charts. Essential principles include determining class limits and frequencies to effectively analyze and present data.

1. Chapter 2
Summarizing Data: Listing and
Grouping

2. Stem-and-Leaf Displays
• A statistical technique for displaying a set of data.
• Each numerical value is divided into two parts: the
leading digits become the stem and the trailing
digits become the leaf.
• Ex: Stock prices on twelve consecutive days for a
major publicly traded company

3. Example
• The first row of a stem-and-leaf chart
appears as follows: 62 1 3 3 7 9. Assume
whole number values.
a. What is the “possible range” of the values
in this row?
b. How many data values are in this row?
c. List the actual values in this row of data.

4. Quartiles
- Divide a set of observations into 4 equal parts
Lp
= (n+1)

5. Example
• 20, 15, 12, 11, 18, 11, 19, 15, 18
Find the Q1, Q2 and Q3!

6. Example of Quartiles
Using the twelve stock prices, we can find the
median, 25th
, and 75th
percentiles as follows:
Quartile 1
Quartile 3
Median

7. 96
92
91
88
86
85
84
83
82
79
78
69
12
11
10
9
8
7
6
5
4
3
2
1
25th
percentile
Price at 3.25 observation = 79 + .25(82-79)
= 79.75
50th
percentile: Median
Price at 6.50 observation = 84 + .5(85-84)
= 84.50
75th
percentile
Price at 9.75 observation = 88 + .75(91-88)
= 90.25
Q1
Q2
Q3
Q4
Percentile

8. Example
• Find 10th
and 76th
percentile from the data
below:
20 35 42 35 26 37 47 38 31 40
33 41 35 24 36 46 37 30 39 50
41 21 22 43 44 27 26 49 48 31

9. Frequency Distributions
• A grouping of data into mutually exclusive
categories showing the number of observations in
each class.
• Constructing a frequency distribution involves:
- Determining the question to be addressed
- Collecting raw data
- Organizing data (Frequency Distributions)
- Presenting data (Graph)
- Drawing Conclusion

10. Frequency and Categorical
Distributions
Frequency Distribution
Categorical Distribution

11. Some Rules in Choosing the
Classes
• Hardly ever use fewer than 6 or more than
15 classes
• Make sure each item (measurement or
observation) goes into only one class
• Use classes covering equal ranges (or
intervals) of values

12. Organizing Data (Frequency
Distributions)
• Arrange data from the smallest to the largest
• Determine “Range” from data
Range = Largest data – Smallest data
• Determine how many class (k);
A. Trial and Error
B. [Sturgess Formula] k = 1 + 3.3 log n ; k need to be rounded, k =
class(s), n = data
C. 2k
> n; n = data
• Determine the length of class interval (i)
i = R/k
• Determine the first lower class limit
• Write down frequency in table with tally according with the data

13. Parts of Frequency Distributions
• Class
• Class limits; lower class limits and upper
class limits
• Class boundary or real class limits
• Class midpoint or class marks
• Class interval
• Class Frequency

14. Two Ways to Modify Frequency
Distributions to Suit Particular Needs
• Percentage
Distribution
By dividing each class
frequency by the total
number of items grouped
and then multiplying by
100%
• Cumulative
Distribution
(Frequency and
Percentage)
To convert it into a “less than”,
or “more than”

15. Example
• Below is the data of 50 employee's salary in
a year (in million Rupiah)
80 18 69 51 71 92 35 28 60 45
63 59 64 98 47 49 48 64 58 74
85 56 72 38 89 55 28 67 84 78
37 73 65 66 86 96 57 76 57 19
54 76 49 53 83 55 83 47 64 39
Construct a frequency distribution!

16. Graphical Presentation
• Histogram is a graph in which the class midpoints
or limits are marked on the horizontal axis and the
class frequencies on the vertical axis. The class
frequencies are represented by the heights
of the bars and the bars are drawn adjacent
to each other.

17. Graphical Presentation
• Frequency Polygon consists of line segments
connecting the points formed by the class
midpoint and the class frequency.

18. Graphical Presentation
• Cumulative Frequency Distribution is used to
determine how many or what proportion of the
data values are below or above a certain value.

19. Other Useful Charts
• Bar Chart can be used to depict any of the levels
of measurement
• Pie Chart is useful for displaying a relative
frequency distribution. A circle is divided
proportionally to the relative frequency and
portions of the circle are allocated for the different
groups.

20. Bar Chart and Pie Chart
Bar Chart Pie Chart

21. End of Lecture