mata kuliah statistika untuk mahasiswa Perguruan Tinggi

1. Frequency Distributions
and Graphic Presentation

2. Frequency Distribution
Frequency distribution: A grouping of data into categories
showing the number of observations in each mutually exclusive
category.
Types of Frequency Distribution
 FD Numerical
 FD Categorical
Construction of a Frequency Distribution
q u e s tio n to
b e a d re s s e d
c o lle c t d a ta
(ra w d a ta )
fre q u e n c y d is trib u tio n
o rg a n iz e d a ta p re s e n t d a ta
(g ra p h )
d ra w
c o n c lu s io n

3. The steps for Frequency Distribution
Decide on the number of classes (k)
 2k
> N
 H.A. Sturgess : k = 1 + 3,322 log N
Determine the class interval or width (i)
 i ≥ (highest value-lowest value)/number of classes
 The class intervals used in the FD should be equal.
Set the individual class limit
Tally data into the classes
Count the number of items in each class (classes frequency)

4. EXAMPLE :
The salaries of XYZ corp. employees ($)
55 48 20 49 78 59 27 41 68 54
34 80 68 42 73 51 76 45 32 53
66 32 64 47 76 58 75 60 35 57
73 38 30 44 54 57 72 67 51 89
25 37 69 71 52 25 47 63 59 64

5. Organizing data into a Frequency Distribution
k = 1 + 3,322 log 50
= 6,64 (7)
 i = (89-20)/7
= 9,86 (10)
20 25 25 27 30 32 32 34 35 37
38 41 42 44 45 47 47 48 49 51
51 52 53 54 54 55 57 57 58 59
59 60 63 64 64 66 67 68 68 69
71 72 73 75 75 76 76 78 80 89

6. Frequency Distribution
Class
Number
Number of
Employess (f)
1 4
2 7
3 8
4 12
5 9
6 8
7 2
50
Salaries
20-< 30
30-< 40
40-< 50
50-< 60
60-< 70
70-< 80
80-< 90

7. Terms in Frequency Distribution
Class interval
Class limit
Class boundary
Class mark (midpoint)
Relative Frequency Distribution
Cumulative Frequency Distribution
Relative Cumulative Frequency Distribution

8. Data cumulative
lower upper midpoint frequency percent frequency percent
20 < 30 25 4 8.0 4 8.0
30 < 40 35 7 14.0 11 22.0
40 < 50 45 8 16.0 19 38.0
50 < 60 55 12 24.0 31 62.0
60 < 70 65 9 18.0 40 80.0
70 < 80 75 8 16.0 48 96.0
80 ≤ 90 85 2 4.0 50 100.0
50 100.0
Printout of Megastat (software)

9. Advantages and Disadvantages of FD
ADVANTAGE : We get a quick visual picture of the shape of the
distribution without doing any further calculation
DISADVANTAGES :
 We lose the exact identity of each value
 We are not sure how the values within each class are distributed

10. Stem-and-Leaf Displays
Stem-and-Leaf Display: 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 the
leaf.
Note : An advantage of the stem-and-leaf display over a
frequency distribution is we do not lose the identity of each
observation.

11. EXAMPLE
Colin achieved the following scores on his twelve
accounting quizzes this semester: 86, 79, 92, 84, 69, 88,
91, 83, 96, 78, 82, 85. Construct a stem-and-leaf chart for
the data.
stem leaf
6 9
7 8 9
8 2 3 4 5 6 8
9 1 2 6

12. Stem-and-Leaf Display: Salaries
Stem-and-leaf of Salaries N = 50
Leaf Unit = 1.0
4 2 0557
11 3 0224578
19 4 12457789
(12) 5 112344577899
19 6 034467889
10 7 12335668
2 8 09
Printout of Minitab (software)

13. Graphic Presentation of a Frequency
Distribution
The three commonly used graphic forms are histograms,
frequency polygons, and a cumulative frequency
distribution (ogive).
Histogram: A graph in which the classes 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.

14. Histogram
Histogram
0
5
10
15
20
25
30
20
30
40
50
60
70
80
90
Data
Percent

15. Cont..
A frequency polygon consists of line segments connecting
the points formed by the class midpoint and the class
frequency.
A cumulative frequency distribution (ogive) is used to
determine how many or what proportion of the data values
are below or above a certain value.

16. Frequency Polygon
0
2
4
6
8
10
12
14
15 25 35 45 55 65 75 85 95
Data
Frequency

17. Cumulative FD “less than” (Ogive)
0
10
20
30
40
50
60
20 30 40 50 60 70 80 90
Data
CumulativeFrequency

18. Cumulative FD “or greater” (Ogive)
0
10
20
30
40
50
60
20 30 40 50 60 70 80 90
Data
CumulativeFrequency

19. Bar Chart
A bar chart can be used to depict any of the levels of measurement
(nominal, ordinal, interval, or ratio).
 EXAMPLE : Construct a bar chart for the number of unemployed
people per 100,000 population for selected cities of 1995.

20. EXAMPLE
City Number of unemployed
per 100,000 population
Atlanta, GA 7300
Boston, MA 5400
Chicago, IL 6700
Los Angeles, CA 8900
New York, NY 8200
Washington, D.C. 8900

21. Bar Chart for the Unemployment Data
7300
5400
6700
8900
8200
8900
0
2000
4000
6000
8000
10000
1 2 3 4 5 6
Cities
#unemployed/100,000
Atlanta
Boston
Chicago
Los Angeles
New York
Washington

22. Pie Chart
A pie chart is especially useful in 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.
 EXAMPLE : A sample of 200 runners were asked to indicate their
favorite type of running shoe.

23. EXAMPLE
Draw a pie chart based on the following information.
Type of shoe # of runners
Nike 92
Adidas 49
Reebok 37
Asics 13
Other 9

24. Pie Chart for Running Shoes
Nike
Adidas
Reebok
Asics
Other
Nike
Adidas
Reebok
Asics
Other