Business Statistics **** Management Information Systems Business Statistics Third level First mid-term: 1436-1437 Instructor: Dr. ZRELLI Houyem Majmaah

Business Statistics **** Management Information Systems

Business Statistics

Third levelFirst mid-term: 1436-1437

Instructor: Dr. ZRELLI Houyem

Majmaah University ***** Faculty of Science and Humanities in Ghat

الرحيم الرحمان الله بسم

Third level Majmaah University Dr: ZRELLI Houyem 1

Chapter 2;

Describing Data:Frequency Tables, Frequency

Distributions, and Graphic Presentation


Data Presentation

Data Presentation

QualitativeData

QuantitativeData

Continues Variables



Discrete Variables

• Qualitative Data are nonnumerical– Major Discipline– Political Party– Gender– Eye color



Describing Qualitative Data

• Summarized in two ways:– Class Frequency– Class Relative Frequency




• Class Frequency– A class is one of the categories into which

qualitative data can be classified– Class frequency is the number of observations in

the data set that fall into a particular class




Student Evaluation Student Evaluation

1 Very Good 12 Very Good

2 Good 13 Good

3 Good 14 Very Good

4 Excellent 15 Good

5 Very Good 16 Good

6 Excellent 17 Good

7 Excellent 18 Excellent

8 Good 19 Very Good

9 Excellent 20 Good

10 Good 21 Good

11 Excellent 22 Excellent



Describing Qualitative Data Example: Student

evaluations

Evaluation Frequency

Good 10

Very Good 5

Excellent 7

Total 22




evaluations

• Class Relative Frequency– Class frequency divided by the total number of

observations in the data set

class frequencyclass relative frequency =

n




• Class Percentage– Class relative frequency multiplied by 100

class percentage = (class relative frequency) 100




Evaluation Relative Frequency

Class Percentage

Good 10/22 = .455 45.5%

Very Good 5/22 = .227 22.7%

Excellent 7/22 = .318 31.8%

Total 22/22 = 1.00 100%




evaluations

0

2

4

6

8

10

Good Very Good Excellent

Bar Graph: The categories (classes) of the qualitative variable are represented by bars, where the height of each bar is either the class frequency, class relative frequency or class percentage.




evaluations

Good163.8

Very Good81.72

Excellent114.48

Pie Chart: The categories (classes) of the qualitative variable are represented by slices of a pie. The size of each slice is proportional to the class relative frequency.




evaluations

Pareto Diagram: A bar graph with the categories (classes) of the qualitative variable (i.e., the bars) arranged in height in descending order from left to right.

Good Very Good Excellent




evaluations

Quantitative data: Discrete Variable


A discrete variable can have only countable number of

values

Examples:

Family size (x = 0, 1, 2, 3, … )

Number of patients (x = 0, 1, 2, 3, … )


Example:

The following data represent the number of children of 16 Saudi women:

3, 5, 2, 4, 0, 1, 3, 5, 2, 3, 2, 3, 3, 2, 4, 1

- Variable = X = no. of children (discrete, quantitative)

- Sample size = n = 16

- The possible values of the variable are: 0, 1, 2, 3, 4, 5




no. of children(variable)

Frequency(no. of women)

Relative Freq. (R.F.) (=Freq /n)

Percentage Freq.(= R.F. * 100%)

0 1 0.0625 6.25%

1 2 0.125 12.5%

2 4 0.25 25%

3 5 0.3125 31.25%

4 2 0.125 12.5%

5 2 0.125 12.5%

Total n=16 1.00 100%

NoteTotal of the frequencies = n = e sample size·Relative frequency = frequency/nPercentage frequency = Relative frequency *100%

Simple frequency distribution of the no. of children




The most common form of graphs for discrete variables is the bar chart.



Graphical representation of discrete variables

· Frequency bar chart



Graphical representation of discrete variables



Cumulative representation of discrete variables

No. Of children

Frequency Relative frequency

Cumulative relative frequency

0 1 0.0625 0

1 2 0.125 0.0625

2 4 0.25 0.1875

3 5 0.3125 0.4375

4 2 0.125 0.75

5 2 0.125 0.875

Total N=10 1.00 1



Cumulative representation of discrete variables

modalities

Fi

1 2 3 4 5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Cumulative distribution

0.9

1

0

. ..

. .

.

.

A continuous variable can have any

value within a certain interval of values.

Quantitative data: Continuous variablesBusiness Statistics **** Management Information Systems


Examples: - height (140 < x < 190)

- blood sugar level (10 < x < 15)

A continuous frequency distribution CANNOT be represented by a bar chart. It is most appropriately represented by a histogram



Graphical representation of continuous variables

Example: Hudson Auto Repair

The manager of Hudson Autowould like to have a betterunderstanding of the costof parts used in the enginetune-ups performed in theshop. She examines 50customer invoices for tune-ups. The costs of

parts,rounded to the nearest dollar, are listed on the

nextslide.



Example: Hudson Auto Repair Sample of Parts Cost for 50 Tune-ups

91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73

Including a line in the table for every possible cost is not a good idea.

Need to categorize.



Frequency Distribution• Guidelines for Selecting Number of Classes

• Use between 5 and 20 classes.

• Data sets with a larger number of elements usually require a larger number of classes.

• Smaller data sets usually require fewer classes



Frequency Distribution• Guidelines for Selecting Width of Classes

Largest Data Value Smallest Data ValueNumber of Classes

•Use classes of equal width.

•Approximate Class Width =



Frequency Distribution

For Hudson Auto Repair, if we choose six classes:

[50-60[ [60-70[ [70-80[ [80-90[ [90-100[ [100-110[

2 13 16 7 7 5Total 50

Parts Cost ($)Frequency

Approximate Class Width = (109 - 52)/6 = 9.5 10



Relative Frequency and Percent Frequency Distributions

[50-60[ [60-70[ [70-80[ [80-90[ [90-100[ [100-110[

PartsCost ($)

.04 .26 .32 .14 .14 .10Total 1.00

RelativeFrequency

4 26 32 14 14 10 100

Percent Frequency

2/50 .04(100)

Prev

iew

cum

ulati

ve fr

eque

ncie

s he

re.



• Only 4% of the parts costs are in the $50-60 class.

• The greatest percentage (32% or almost one-third) of the parts costs are in the $70-80 class.

• 30% of the parts costs are under $70.

• 10% of the parts costs are $100 or more.

Insights Gained from the Percent Frequency Distribution

Relative Frequency and Percent Frequency Distributions



Histogram Another common graphical presentation of quantitative data is a histogram.

The variable of interest is placed on the horizontal axis. A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency.

Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes.

In informal discussions bar graphs and histograms are often equated. In this class you should be careful to keep them straight.



Histogram

22

44

66

88

1010

1212

1414

1616

1818

PartsCost ($) PartsCost ($)

Fre

qu

en

cy

Fre

qu

en

cy

50 60 70 80 90 100 11050 60 70 80 90 100 110

Tune-up Parts Cost



• Symmetric– Left tail is the mirror image of the right tail– Examples: heights and weights of people

Histogram (Common categories)

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00



Cumulative frequency distribution - shows the number of items with values less than or equal to the upper limit of each class..

Cumulative relative frequency distribution – shows the proportion of items with values less than or equal to the upper limit of each class.

Cumulative Distributions

Cumulative percent frequency distribution – shows the percentage of items with values less than or equal to the upper limit of each class.



Cumulative Distributions• Hudson Auto Repair

[50-60[ [60-70[ [70-80[ [80-90[ [90-100[ [100-110[

Cost ($) CumulativeFrequency

CumulativeRelativeFrequency

CumulativePercent Frequency

2 15 31 38 45 50

0 .04

.30 .62 .76 .90 1.00

4 30 62 76 90 100

2 + 13

15/50 .30(100)



Ogive

An ogive is a graph of a cumulative distribution. The data values are shown on the horizontal axis. Shown on the vertical axis are the:• cumulative frequencies, or• cumulative relative frequencies, or• cumulative percent frequencies

The frequency (one of the above) of each class is plotted as a point.

The plotted points are connected by straight lines.



PartsCost ($) PartsCost ($)

2020

4040

6060

8080

100100

Cu

mu

lati

ve P

erc

en

t Fr

eq

uen

cyC

um

ula

tive P

erc

en

t Fr

eq

uen

cy

50 60 70 80 90 100 11050 60 70 80 90 100 110

(89.5, 76)

Ogive with Cumulative Percent Frequencies

Tune-up Parts Cost



End of Chapter 2

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Business Statistics **** Management Information Systems Business Statistics Third level First mid-term: 1436-1437 Instructor: Dr. ZRELLI Houyem Majmaah