Frequency Distributions and Graphic Presentation

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 t i o n t ob e a d r e s s e d

c o l l e c t d a t a( r a w d a t a )

f r e q u e n c y d i s t r i b u t i o n

o r g a n i z e d a t a p r e s e n t d a t a( g r a p h )

d r a wc o n c l u s i o n

The steps for Frequency DistributionDecide 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 limitTally data into the classesCount the number of items in each class (classes frequency)

EXAMPLE : The salaries of XYZ corp. employees ($)

55 48 20 49 78 59 27 41 68 5434 80 68 42 73 51 76 45 32 5366 32 64 47 76 58 75 60 35 5773 38 30 44 54 57 72 67 51 8925 37 69 71 52 25 47 63 59 64

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 3738 41 42 44 45 47 47 48 49 5151 52 53 54 54 55 57 57 58 5959 60 63 64 64 66 67 68 68 6971 72 73 75 75 76 76 78 80 89

Frequency Distribution

Class Number

Number of Employess ( f)

1 42 73 84 125 96 87 2

50

Salar ies

20-< 3030-< 4040-< 5050-< 6060-< 7070-< 8080-< 90

Terms in Frequency Distribution

Class intervalClass limitClass boundaryClass mark (midpoint)Relative Frequency DistributionCumulative Frequency DistributionRelative Cumulative Frequency Distribution

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)

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

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.

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

Stem-and-Leaf Display: Salaries

Stem-and-leaf of Salaries N = 50Leaf 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)

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.

Histogram

Histogram

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Data

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cent

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.

Frequency Polygon

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15 25 35 45 55 65 75 85 95

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Cumulative FD “less than” (Ogive)

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Cumulative FD “or greater” (Ogive)

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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.

EXAMPLE

City Number of unemployedper 100,000 population

Atlanta, GA 7300Boston, MA 5400Chicago, IL 6700

Los Angeles, CA 8900New York, NY 8200

Washington, D.C. 8900

Bar Chart for the Unemployment Data

7300

5400

6700

89008200

8900

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# u

nem

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/100,0

00

AtlantaBostonChicagoLos AngelesNew YorkWashington

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.

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

Pie Chart for Running Shoes

Nike

Adidas

ReebokAsics

Other

Nike

Adidas

ReebokAsics

Other