349669784-Frequency-Distribution-ppt.ppt

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1. 2- 1
Chapter
Two
McGraw-Hill/Irwin © 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

2. 2- 2
Describing Data: Frequency Distributions and Describing Data: Frequency Distributions and
Graphic PresentationGraphic Presentation
GOALS
When you have completed this chapter, you will be able
to:
ONE
Organize data into a frequency distribution.
TWO
Portray a frequency distribution in a histogram, frequency
polygon, and cumulative frequency polygon.
THREE
Present data using such graphic techniques as line charts,
bar charts, and pie charts.
Goals

3. 2- 3
Frequency Distribution
A Frequency DistributionFrequency Distribution is a
grouping of data into mutually exclusive
categories showing the number of
observations in each class.

4. 2- 4
•Decide the number of classes
•Determine the class interval or class
width
•Set the individual class limit
•Tally the variables into the classes
•Count the number of items in each class
Constructing a Frequency Distribution

5. 2- 5
Class MidpointClass Midpoint:: A point that divides a class
into two equal parts. This is the average of the upper
and lower class limits.
Class FrequencyClass Frequency:
The number of
observations in each
class.
Class intervalClass interval: The
class interval is
obtained by subtracting
the lower limit of a
class from the lower
limit of the next class.
The class intervals
should be equal.
Definitions

6. 2- 6
Example # 1: Mr. Karim of Khulna Automobiles wants to some tables,
charts, and graph. The following tables shows the selling prices of 80
vehicles.
Tk.23197Tk. 23372Tk. 20,454Tk. 23,591Tk. 26,651Tk. 27,453Tk. 17,266
18,02128,683 30,872 19,58723,16935,85119,251
20,04724,285 24,324 24,60928,67015,54619,935
19,87325,251 25,277 28,03424,53327,44319,889
20,00417,357 20,155 19,68823,65726,61320,895
20,20323,765 25,783 25,66132,27720,64221,981
24,05225,799 15,794 18,26335,92517,39917,968
20,35621,442 21,722 19,33122,81719,76620,633
20,96222,845 26,285 27,89629,07632,49218,890
21,74022,374 24,571 25,49928,33720,64223,613
24,22030,655 22,442 17,89120,81726,23720,445
21,55621,639 24,296    

7. 2- 7
Example 1 continued
Step OneStep One:: Decide on the number of classes using
the formula
22
kk
> n > n
where k=number of classes
n=number of observations
oThere are 80 observations so n=80.
oTwo raised to the seven power is 128.
oTherefore, we should have at least 7
classes, i.e., k=7.

8. 2- 8
where H=highest value, L=lowest value
35,925 – 15,546
7
= Tk. 2,911
Step TwoStep Two: Determine the class interval or
width using the formula
H – LH – L
kk
i > =
Round up for an interval of Tk. 30000.
Set the lower limit of the first class at Tk.3000 ,
giving a total of 7 classes.
Example 1 continued

9. 2- 9
Step ThreeStep Three: Set the individual class limits
1.Use the words “ Up to” between two limits of a
class. (10 up to 15, 15 up to 20)
2.Exclusive method- Observation exactly equal to
the upper limit is not included in that class ( 10-
15, 15 -20)

1.Inclusive method – Upper limit of one class is
included in that class ( 10-14, 15-19)

10. 2- 10
EXAMPLE 1 continued
Selling Price
( Tk. Thousand )
Frequency, f
15 up to 18 8
18 up to 21 23
21 up to 24 17
24 up to 27 18
27 up to 30 8
30 up to 33 4
33 up to 36 2
Total 80


Steps Four and FiveSteps Four and Five: Tally and count the number of
items in each class.

11. 2- 11
The Observations are Following
1.The selling price is ranged from Tk. 15,000 up to
about Tk. 36,000
2.The selling prices are concentrated between Tk.
18,000 and Tk. 27,000. A total of 58, or 72.5 percent
of the vehicles sold within this range.
3.The largest concentration, or highest frequency, is in
the Tk. 18,000 up to Tk. 21,000 class. The middle of
the class is Tk. 19,500. So we say the typical selling
price is Tk. 19,500
4.Two of the vehicles sold for Tk. 33,000 or more, and 8
sold for less than Tk. 18,000.

12. 2- 12
Class MidpointClass Midpoint: find the midpoint of each interval, use the following
formula:
Upper limit + lower limit
2

Selling
Price
Midpoint f
15 up to 18 (18+15)/2 =16.5 8
18 up to 21 (21+18)/2=19.5 23
21 up to 24 (24 +21)/2=22.5 17
24 up to 27 (21+24)/2=25.5 18
27 up to 30 (30+27)/2=28.5 8
30 up to 33 (33+30)/2=31.5 4
33 up to 36 (36+33)/2=34.5 2
Total 8-

13. 2- 13
Selling Price ( Tk.
Thousand)
f Relative
Frequency
15 up to 18 8 8/80=0.1000
18 up to 21 23 23/80=0.2875
21 up to 24 17 17/80=0.2125
24 up to 27 18 18/80=0.2250
27 up to 30 8 8/80=0.1000
30 up to 33 4 4/80=0.0500
33 up to 36 2 2/80=0.0250
TOTAL 80 1.000

Example 1 continued
A Relative Frequency DistributionRelative Frequency Distribution shows the
percent of observations in each class.

14. 2- 14
Graphic Presentation of a
Frequency Distribution
A 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.
The three commonly used graphic forms are
Histograms, Frequency PolygonsHistograms, Frequency Polygons, and a
Cumulative FrequencyCumulative Frequency distribution.

15. 2- 15
Graphic Presentation of a Frequency Graphic Presentation of a Frequency
DistributionDistribution
Graphic Presentation of a Frequency Distribution
A Frequency PolygonFrequency Polygon consists of
line segments connecting the points
formed by the class midpoint and the
class frequency.

16. 2- 16
Class MidpointClass Midpoint: The Class mid point with frequency is presented blow


Selling
Price
Midpoint f
15 up to 18 16.5 8
18 up to 21 19.5 23
21 up to 24 22.5 17
24 up to 27 25.5 18
27 up to 30 28.5 8
30 up to 33 31.5 4
33 up to 36 34.5 2
Total 80

17. 2- 17
A Cumulative Cumulative
Frequency Frequency
DistributionDistribution is
used to determine
how many or what
proportion of the
data values are
below or above a
certain value.
Cumulative Frequency DistributionCumulative Frequency Distribution
To create a
cumulative
frequency polygon,
scale the upper limit
of each class along
the X-axis and the
corresponding
cumulative
frequencies along
the Y-axis.
Cumulative Frequency distribution

18. 2- 18
Cumulative Frequency Table for Selling Prices Cumulative Frequency Table for Selling Prices
Selling Price ( Tk.
Thousand)
Upper Limit

f Cumulative
Frequency

15 up to 18 18 8 8
18 up to 21 21 23 31(8+23)
21 up to 24 24 17 48 (31+17)
24 up to 27 27 18 66 (48+18)
27 up to 30 30 8 74 (66+8)
30 up to 33 33 4 78 (74+4)
33 up to 36 36 2 80 (78+2)
Total 80

Cumulative frequency table

19. 2- 19
Line graphs are typically used to show the
change or trend in a variable over time.
Year MalesFemales
1992 30.5 32.9
1993 30.8 33.2
1994 31.1 33.5
1995 31.4 33.8
1996 31.6 34.0
1997 31.9 34.3
1998 32.2 34.6
1999 32.5 34.9
2000 32.8 35.2
2001 33.2 35.5
2002 33.5 35.8
Line Graphs

20. 2- 20
U.S. median age by gender
25
30
35
40
M
e
d
i
a
n
A
g
e
Males
Females
Example 3 continued

21. 2- 21
Construct a bar chart for the number of unemployed per
100,000 population for selected cities during 2001
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


A Bar Chartar Chart can be used to depict any of the
levels of measurement (nominal, ordinal, interval, or
ratio).
Bar Chart

22. 2- 22
Bar Chart for the Unemployment Data
7300
5400
6700
8900
8200
8900
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
1 23 4 56
Cities
#

u
n
e
m
p
l
o
y
e
d
/
1
0
0
,
0
0
0
Atlanta
Boston
Chicago
Los Angeles
New York
Washington

23. 2- 23
•The width of the bars should be uniform
throughout the diagram.
•The gap between one bar and another should be
uniform throughout.
•Bars may be horizontal or vertical
Points to be Kept in Mind in Constructing
Bar Diagram

24. 2- 24
Pie Chart
A sample of 200 runners were asked to indicate their favorite type of
running shoe. Draw a pie chart based on the following information.
Type of shoe # of runners % of total
Nike 92 46.0
Adidas 49 24.5
Reebok 37 18.5
Asics 13 6.5
Other 9 4.5

A Pie ChartPie 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.

25. 2- 25
Pie Chart for Running Shoes
46%
24.50%
18.50%
6.50%
4.50%
Nike
Adidas
Reebok
Asics
Other
Pie Chart for Running Shoes

26. 2- 26
Stem-and-leaf Displays
Note: an
advantage of the
stem-and-leaf
display over a
frequency
distribution is we
do not lose the
identity of each
observation.
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.

27. 2- 27
Stock prices on twelve
consecutive days for a major
publicly traded company
86, 79, 92, 84, 69, 88, 91
83, 96, 78, 82, 85.
Example

28. 2- 28
stem leaf
6 9
7 8 9
8 2 3 4 5 6 8
9 1 2 6


Stem and leaf display of stock prices

29. 2- 29
Exercise # 1
96 93 88 11712795 11396 10894 148153
13914294 107125155155103112127117120
11213513211112510410613913411997 89
118136125143120103113124135
 Organize the data into steam-and-leaf display.
 What is the fewer number of sports purchased by the dealer?
 The largest number of purchased?