1 introduction to statistics_handouts

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Introduction to Statistics

By

Dr. Vishal Singh Patyal

Learning Objectives

In this chapter, you will learn:

What is StatisticsWhy StatisticsBasic vocabulary used in StatisticsHow statistics is used in Business

The sources of data and its types used in BusinessTypes of VariablesLevel of ManagementTabular and Graphical Presentation of Data

2Introduction to Statistics


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What is Statistics?

The science of collecting, describing, and interpreting data.

“Statistics is a way to get information from data”

Data

Statistics

Information

Data: Facts, especiallynumerical facts, collectedtogether for reference orinformation.

Information: Knowledgecommunicatedconcerning someparticular fact.

Statistics is a tool for creating new understanding from a set of numbers. 4Introduction to Statistics


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Why Study Statistics?

Decision Makers Use Statistics To:

Present and describe business data and information properlyDraw conclusions about large populations, using informationcollected from samplesMake reliable forecasts about a business activityImprove business processes


Types of Statistics

StatisticsThe branch of mathematics that transforms data into

useful information for decision makers.

Descriptive Statistics

Collecting, summarizing,and describing data

Inferential Statistics

Drawing conclusionsand/or making decisionsconcerning a population

based only on sample data



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Descriptive Statistics

Collect dataex. Survey

Present dataex. Tables and graphs

Characterize dataex. Sample mean = i X

nCollect

OrganizeSummarizeDisplayAnalyze 7Introduction to Statistics

Inferential Statistics

Estimationex. Estimate the populationmean weight using the samplemean weight

Hypothesis testingex. Test the claim that thepopulation mean weight is 120pounds

Drawing conclusions and/or making decisions concerning a populationbased on sample results.

Predict and forecast valuesof population parametersTest hypotheses aboutvalues of populationparametersMake decisions



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Example

A recent study examined the QUANTand VERBALCAT scores of students across the country. Which of the followingstatements are descriptive in nature and which are inferential.

• The mean QUANTCAT score was 492. D• The mean VERBALSAT score was 475. D• Students in the Northeast scored higher in QUANTbut lower

in VERBAL. I• 80% of all students taking the exam were headed for IIMs. I• 32% of the students scored above 610 on the VERBALCAT. D• The QUANTCAT scores are higher than they were 10 years

ago. I


PopulationA population consists of all the items or individuals about

which you want to draw a conclusion.A population is the group of all items of interest to a

statistics practitioner.frequently very large; sometimes infinite.

E.g. All 1.252 Billion Indian population i.e. census data.Sample

A subset of the population.A sample is a set of data drawn from the population.Potentially very large, but less than the population.

E.g. a sample of 765 voters exit polled on election day

Basic Vocabulary of Statistics



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Parameter

PopulationSample

Statistic

Subset

Measures used to describethe population are calledparameters

Measures computedfrom sample data arecalled statistics



VariableA variable is some characteristic of a population or sample.E.g. student grades. Typically denoted with a capital letter: A,B, C…

The values of the variable are the range of possible values fora variable.

E.g. student marks (0..100)Data

Data are the observed values of a variable.Data are the different values associated with a variable.E.g. student marks: {67, 74, 71, 83, 93, 55, 48}




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ExampleNICMAR Institute dean is interested in learning about the averageage of faculty. Identify the basic terms in this situation.

The population is the age of all faculty members at the Institute.A sample is any subset of that population. For example, we might

select 10 faculty members and determine their age.The variable is the “age” of each faculty member.The data would be the set of values in the sample.The parameter of interest is the “average” age of all faculty at the

Institute.The statistic is the “average” age for all faculty in the sample.


Why Collect Data?

A marketing research analyst needs to assess theeffectiveness of a new television advertisement.A pharmaceutical manufacturer needs to determine whethera new drug is more effective than those currently in use.An operations manager wants to monitor a manufacturingprocess to find out whether the quality of product being

manufactured is conforming to company standards.An auditor wants to review the financial transactions of acompany in order to determine whether the company is incompliance with generally accepted accounting principles.



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Sources of Data

Primary Sources:The data collector is the one using the data for analysis

Data from a political surveyData collected from an experimentObserved data

Secondary SourcesThe person performing data analysis is not the data collector

Analyzing census data

Examining data from print journals or data published onthe internet.


Types of Variables

Data

Categorical Numerical

Discrete Continuous

Examples:

Marital StatusPolitical PartyEye Color

(Defined categories)Examples:

Number of ChildrenDefects per hour

(Counted items)

Examples:

WeightVoltage(Measured characteristics) 16Introduction to Statistics


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Types of Variables

CategoricalQualitative variables have values that can only be placed intocategories, such as “yes” and “no.”

A variable that categorizes or describes an element of apopulation.

Note : Arithmetic operations, such as addition and averaging, are not meaningful for data resulting from a qualitative variable

NumericalQuantitative variables have values that represent quantities.A variable that quantifies an element of a population.Note : Arithmetic operations such as addition and averaging, are

meaningful for data resulting from a quantitative variable.17Introduction to Statistics

ExampleIdentify each of the following examples as attribute (qualitative) ornumerical (quantitative) variables.

The amount of CNG pumped by the next 10 customers at the localhp PUMP . (Numerical)The amount of radon in the basement of each of 25 homes in anew development. (Numerical)

The color of the baseball cap worn by each of 20 students.(Attribute)The length of time to complete a mathematics homeworkassignment. (Numerical)The state in which each truck is registered when stopped andinspected at a weigh station. (Attribute)



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Question?

Identify each of the following as examples of qualitative orquantitative variables:

The temperature in Barrow, Alaska at 12:00 pm on anygiven day.

The make of automobile driven by each faculty member.Whether or not a 6 volt lantern battery is defective.The weight of a lead pencil.The length of time billed for a long distance telephone call.The brand of cereal children eat for breakfast.

The type of book taken out of the library by an adult.


Level of Measurement

Introduction to Statistics 20

Nominal

Ordinal

Interval

Ratio

NOIR


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Nominal scale

A nominal scale classifies data into distinct categories inwhich no ranking is implied.

Categorical Variables Categories

Personal ComputerOwnership

Type of StocksOwned

Internet Provider

Yes / No

Microsoft Network /AOL

Growth, Value, Other


Ordinal scale

An ordinal scale classifies data into distinctcategories in which ranking is implied

Student class designation Freshman, Junior, Senior

Product satisfaction Satisfied, Neutral, Unsatisfied

Faculty rank Professor, Associate Professor, Assistant Professor, Instructor

Standard & Poor’s bond ratings AAA, AA, A, BBB, BB, B, CCC, CC,C, DDD, DD, D

Student Grades A, B, C, D, F

Categorical Variable Ordered Categories

Chap 1-22Introduction to Statistics


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Interval scale

An interval scale is an ordered scale in which thedifference between measurements is a meaningfulquantity but the measurements do not have a true zeropoint.

Example:the difference between 1 and 2 years of age is thesame amount as the difference between 21 and 22years of age, or 50 and 51, or 65 and 66.the difference between a height of 60 inches and aheight of 55 inches is the same amount of difference asa height of 72 inches and a height of 67 inches.


NOTE : For interval level variables, it is mathematically legitimate to do arithmetic (add, subtract,and divide) as well as count the values, and sort or rank the values .



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A ratio scale is an ordered scale in which the differencebetween the measurements is a meaningful quantity.

Ratio level variables have the additional property of having a true zero value so that ratios between values aremeaningful, but practically speaking, ratio level data istreated the same as interval level.Example

number of clients in past six months

It is meaningful to say that “...we had twice as manyclients in this period as we did in the previous six months.

Levels of Measurement

Example



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The Hierarchy of Levels

Nominal27Introduction to Statistics


Nominal Attributes are only named; weakest28Introduction to Statistics


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Nominal Attributes are only named; weakest

Ordinal



Nominal Attributes are only named; weakest

Attributes can be orderedOrdinal



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Nominal

Interval

Attributes are only named; weakest

Attributes can be orderedOrdinal



Nominal

Interval


Attributes can be ordered

Distance is meaningful

Ordinal



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Nominal

Interval

Ratio




Ordinal



Nominal

Interval

Ratio




Absolute zero

Ordinal



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Level of Measurement :Decision Tree


Level of Measurement :Characteristics


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Level of Measurement:Statistical Tests


Example

Identify each of the following as examples of (1) nominal, (2)ordinal, (3) discrete, or (4) continuous variables:

The length of time until a pain reliever begins to work.The number of chocolate chips in a cookie.The number of colors used in a statistics textbook.The brand of refrigerator in a home.The overall satisfaction rating of a new car.The number of files on a computer’s hard disk.The pH level of the water in a swimming pool.The number of staples in a stapler.



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Class Exercise

Q 1: Determine whether the variable is categoricalor numerical If numerical, determine whether thevariable is discrete or continuous .Determine thelevel of measurement

Amount of money spent on clothing in pastmonth?

Favorite department store?

Most likely time period during which shopping forclothing takes place?

Number of pairs of shoes owned?

Q 2: A manufacturer of dog food was planning tosurvey household in India to determine purchasinghabit of dog owners. Among the variables to becollected are

The primary place of purchase of dog food?Whether dry or moist food can be purchased ?Number of dogs living in the household?Whether the dog is pedigreed?

Class Exercise


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Q3 : Suppose the following information collected fromMr X on his application for a home loan at the HDFCbank Loan departmenta. Monthly payment : Rs 25100b. Annual Family income:c. Marital status: Marriedd. No of job changed in past 10 years: 2

Classify each of the response by type of data and level ofmeasurement.

Class Exercise

Organizing and VisualizingCategorical and Numerical Data



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Categorical Data Are Organized ByUtilizing Tables

Categorical Data

Tallying Data

Summary Table

One CategoricalVariable

Two CategoricalVariables

Contingency Table


Organizing Categorical Data:Summary Table

A summary table indicates the frequency, amount, orpercentage of items in a set of categories so that you cansee differences between categories.

How do you spend the holidays? Percent

At home with family 45%

Travel to visit family 38%

Vacation 5%

Catching up on work 5%

Other 7%



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Contingency Table

Used to study patterns that may existbetween the responses of two or morecategorical variables

Cross tabulates or tallies jointly the responsesof the categorical variables

For two variables the tallies for one variableare located in the rows and the tallies for thesecond variable are located in the columns


Contingency Table - Example

A random sample of 400invoices is drawn.Each invoice is categorized as asmall, medium, or largeamount.Each invoice is also examined toidentify if there are any errors.This data are then organized inthe contingency table to theright.

NoErrors Errors Total

Small

Amount

170 20 190

MediumAmount

100 40 140

LargeAmount

65 5 70

Total335 65 400

Contingency Table ShowingFrequency of Invoices CategorizedBy Size and The Presence Of Errors



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Contingency Table Based on% of Overall Total


SmallAmount

170 20 190

MediumAmount

100 40 140

LargeAmount

65 5 70

Total335 65 400


SmallAmount

42.50% 5.00% 47.50%

MediumAmount

25.00% 10.00% 35.00%

LargeAmount

16.25% 1.25% 17.50%

Total83.75% 16.25% 100.0%

42.50% = 170 / 40025.00% = 100 / 40016.25% = 65 / 400

83.75% of sampled invoices have no

errors and 47.50% of sampled invoicesare for small amounts.


Contingency Table Based on% of Row TotalsNo

Errors Errors Total

SmallAmount

170 20 190

MediumAmount

100 40 140

LargeAmount

65 5 70

Total335 65 400


Small

Amount

89.47% 10.53% 100.0%

MediumAmount

71.43% 28.57% 100.0%

LargeAmount

92.86% 7.14% 100.0%

Total83.75% 16.25% 100.0%

89.47% = 170 / 19071.43% = 100 / 14092.86% = 65 / 70

Medium invoices have a larger chance(28.57%) of having errors than small(10.53%) or large (7.14%) invoices.



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Contingency Table Based onPercentage of Column Total


SmallAmount

170 20 190

MediumAmount

100 40 140

LargeAmount

65 5 70

Total335 65 400


SmallAmount

50.75% 30.77% 47.50%

MediumAmount

29.85% 61.54% 35.00%

LargeAmount

19.40% 7.69% 17.50%

Total100.0% 100.0% 100.0%

50.75% = 170 / 33530.77% = 20 / 65

There is a 61.54% chance that invoiceswith errors are of medium size.


Tables Used For OrganizingNumerical Data

Numerical Data

Ordered ArrayCumulative

DistributionsFrequency

Distributions



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Organizing Numerical Data:Ordered Array

An ordered array is a sequence of data, in rank order,from the smallest value to the largest value.

Age ofSurveyedCollege

Students

Day Students

16 17 17 18 18 18

19 19 20 20 21 22

22 25 27 32 38 42Night Students

18 18 19 19 20 21

23 28 32 33 41 45


Organizing Numerical Data:Frequency Distribution

The frequency distribution is a summary table in which thedata are arranged into numerically ordered class groupings.You must give attention to selecting the appropriate numberof class groupings for the table, determining a suitable widthof a class grouping, and establishing the boundaries of eachclass grouping to avoid overlapping.

The number of classes depends on the number of values inthe data. With a larger number of values, typically there aremore classes. In general, a frequency distribution shouldhave at least 5 but no more than 15 classes.To determine the width of a class interval, you divide therange (Highest value –Lowest value) of the data by thenumber of class groupings desired.



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Organizing Numerical Data:Frequency Distribution Example

Example: A manufacturer of insulation randomlyselects 20 winter days and records the dailyhigh temperature

24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41,

43, 44, 27, 53, 27


STEPS

1. Sort raw data in ascending order:12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

2. Find range: 58 - 12 = 463. Select number of classes: 5 (usually between 5 and 15)4. Compute class interval (width): 10 (46/5 then round up)5. Determine class boundaries (limits):

1. Class 1: 10 to less than 202. Class 2: 20 to less than 303. Class 3: 30 to less than 404. Class 4: 40 to less than 505. Class 5: 50 to less than 60

6. Compute class midpoints: 15, 25, 35, 45, 557. Count observations & assign to classes 54Introduction to Statistics


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Organizing Numerical Data:Frequency Distribution Example

Class Midpoints Frequency

10 but less than 20 15 320 but less than 30 25 630 but less than 40 35 540 but less than 50 45 450 but less than 60 55 2

Total 20

Data in ordered array:

12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58


Organizing Numerical Data:Relative & Percent FrequencyDistribution

Class Frequency

10 but less than 20 3 .15 15

20 but less than 30 6 .30 3030 but less than 40 5 .25 2540 but less than 50 4 .20 2050 but less than 60 2 .10 10

Total 20 1.00 100

RelativeFrequency Percentage


12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58



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Organizing Numerical Data:Cumulative FrequencyDistribution

Class

10 but less than 20 3 15% 3 15%

20 but less than 30 6 30% 9 45%

30 but less than 40 5 25% 14 70%

40 but less than 50 4 20% 18 90%50 but less than 60 2 10% 20 100%

Total 20 100 20 100%

Percentage CumulativePercentage


12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

FrequencyCumulativeFrequency


Why Use a Frequency Distribution?

It condenses the raw data into a more useful form

It allows for a quick visual interpretation of the data

It enables the determination of the majorcharacteristics of the data set including where the dataare concentrated / clustered



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Frequency Distributions:Some Tips

Different class boundaries may provide differentpictures for the same data (especially for smallerdata sets)Shifts in data concentration may show up whendifferent class boundaries are chosenAs the size of the data set increases, the impact ofalterations in the selection of class boundaries isgreatly reduced

When comparing two or more groups with differentsample sizes, you must use either a relativefrequency or a percentage distribution


Visualizing Categorical DataThrough Graphical DisplaysCategorical Data

Visualizing Data

BarChart

Summary TableFor One Variable

ContingencyTable For Two

Variables

Side By Side BarChart

Pie Chart

ParetoChart



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In a bar chart, a bar shows each category, the lengthof which represents the amount, frequency orpercentage of values falling into a category.

45%38%

5%

5%

7%

0% 10% 20% 30% 40% 50%

At home with …Travel to visit …

Vacation

Catching up …

Other

How Do You Spend the Holidays?

Organizing Categorical Data:Summary Table


Organizing Categorical Data:Pie Chart

The pie chart is a circle broken up into slices thatrepresent categories. The size of each slice of the pievaries according to the percentage in each category.

45%

38%

5% 5%7%

How Do You Spend the Holiday's

At home with family

Travel to visit family

Vacation

Catching up on work

OtherChap 1-62

Introduction to Statistics


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Organizing Categorical Data:Pareto Diagram

Used to portray categorical data

A bar chart, where categories are shown indescending order of frequency

A cumulative polygon is shown in the same graph

Used to separate the “vital few” from the “trivial

many ”


Organizing Categorical Data:Pareto Diagram

c u m u l a t i v e % i n v e s t e d

( l i n e g r a p h )

% i

n v e s t e

d i n e a c h

c a t e g o r y

( b a r

g r a p h

)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

Stocks Bonds Savings CD

Current Investment Portfolio



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Visualizing Categorical Data:Side By Side Bar Charts

The side by side bar chart represents the data from a contingency table.

0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0%

No Errors

Errors

Invoice Size Split Out By Errors & NoErrors

Large Medium Small

Invoices with errors are much more likely to be of medium size (61.54% vs 30.77% and 7.69%)


SmallAmount

50.75% 30.77% 47.50%

MediumAmount

29.85% 61.54% 35.00%

LargeAmount

19.40% 7.69% 17.50%

Total

100.0% 100.0% 100.0%


Visualizing Numerical Data ByUsing Graphical Displays

Numerical Data

Ordered Array

Stem-and-Leaf Display Histogram Polygon Ogive

Frequency Distributions andCumulative Distributions



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Organizing Numerical Data:Stem and Leaf Display

A stem-and-leaf display organizes data into groups(called stems) so that the values within each group(the leaves) branch out to the right on each row.

Stem Leaf

1 67788899

2 0012257

3 28

4 2

Age of College Students

Day Students Night Students

Stem Leaf

1 8899

2 0138

3 23

4 15

Age ofSurveyedCollegeStudents

Day Students16 17 17 18 18 18

19 19 20 20 21 22

22 25 27 32 38 42

Night Students18 18 19 19 20 2123 28 32 33 41 45

2-67Introduction to Statistics

Organizing Numerical Data:Stem and Leaf Display

A stem-and-leaf display organizes data into groups(called stems) so that the values within each group(the leaves) branch out to the right on each row.

Stem Leaf 1 67788899

2 0012257

3 28

4 2

Age of College Students

Day Students Night Students

Stem Leaf 1 8899

2 0138

3 23

4 15



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Visualizing Numerical Data:The Histogram

A graph of the data in a frequency distribution iscalled a histogram.In a histogram there are no gaps between adjacentbars.The class boundaries (or class midpoints ) are shownon the horizontal axis.The vertical axis is either frequency, relativefrequency, or percentage .

Bars of the appropriate heights are used to representthe number of observations within each class.


Class Frequency

10 but less than 20 3 .15 1520 but less than 30 6 .30 3030 but less than 40 5 .25 25

40 but less than 50 4 .20 2050 but less than 60 2 .10 10

Total 20 1.00 100

RelativeFrequency Percentage

0

5

10

5 15 25 35 45 55 More

F r e q u e n c y

Histogram: Daily HighTemperature

Visualizing Numerical Data:The Histogram



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A percentage polygon is formed by having themidpoint of each class represent the data in that classand then connecting the sequence of midpoints attheir respective class percentages.The cumulative percentage polygon, orogive, displays the variable of interest along the X axis, and the cumulative percentages along the Y axis.

Useful when there are two or more groups tocompare

Visualizing Numerical Data:The Polygon


Visualizing Numerical Data:The Frequency Polygon

0

5

10

5 15 25 35 45 55 More

F r e q u e n c y

Frequency Polygon: Daily HighTemperature

Class Frequency

10 but less than 20 3 .15 1520 but less than 30 6 .30 3030 but less than 40 5 .25 2540 but less than 50 4 .20 2050 but less than 60 2 .10 10

Total 20 1.00 100

RelativeFrequency

Percentage

(In a percentage polygonthe vertical axis would bedefined to show thepercentage of observationsper class)



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Organizing Numerical Data:The Cumulative Percentage Polygon

0

50

100

10 20 30 40 50 60

C u m u

l a t i v e P e r c e n t a g e

Ogive: Daily High Temperature

Class LowerBoundary

% Less ThanLowerBoundary

10


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Scatter Plot Example

Volumeper day

Cost perday

23 125

26 140

29 146

33 160

38 167

42 170

50 18855 195

60 200

0

50

100

150

200

250

20 30 40 50 60 70

C o s t p e r D a y

Volume per Day

Cost per Day vs. ProductionVolume


Time SeriesA Time Series Plot is used to study patterns in thevalues of a numeric variable over timeThe Time Series Plot:Numeric variable is measured on the vertical axis and

the time period is measured on the horizontal axis

Attendance (in millions) at USA amusement/theme parks from 2000-2005

Year Year Number Attendance2000 0 317

2001 1 319

2002 2 324

2003 3 322

2004 4 328

2005 5 33576Introduction to Statistics


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Time Series Example

316

320

324

328

332

336

0 1 2 3 4 5 6

A t t e n

d a n c e

Year (Since 2000 )

Attendance (in millions) at US ThemeParks


Principles of Excellent Graphs

The graph should not distort the data.The graph should not contain unnecessaryadornments (sometimes referred to as chart junk ).The scale on the vertical axis should begin at zero.

All axes should be properly labeled.The graph should contain a title.The simplest possible graph should be used for agiven set of data.



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Graphical Errors: Chart Junk

1960: $1.00

1970: $1.60

1980: $3.10

1990: $3.80

Minimum Wage

Bad Presentation

Minimum Wage

0

2

4

1960 1970 1980 1990

$

Good Presentation


Graphical Errors:No Relative Basis

A’s received bystudents.

A’s received bystudents.

Bad Presentation

0

200

300

FR SO JR SR

Freq.

10%

30%

FR SO JR SR

FR = Freshmen, SO = Sophomore, JR = Junior, SR = Senior

100

20%

0%

%

Good Presentation



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Graphical Errors:Compressing the Vertical Axis

Good Presentation

Quarterly Sales Quarterly Sales

Bad Presentation

0

25

50

Q1 Q2 Q3 Q4

$

0

100

200

Q1 Q2 Q3 Q4

$


Class Exercise 1The owner of the restaurant wanted to study the demand fordessert. He decided that in addition to studying whether the desertwas ordered, he would also study the gender of individual. Datawere collected from 600 customers and organized in the followingcontingency tables.

a.Construct a contingency tables for row, column and total percentage?b.Which type of percentage (row, column and total ), do you think more

informative for each gender?c.What conclusions concerning the pattern of dessert ordering can the

restaurant owner reach?

GenderDessert Ordered Male Female Total

Yes 40 96 136No 240 224 464

Total 280 320 600


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Class Exercise 2

The Following Table represents estimated green power salesby renewable energy source 2008Source Percentage

Geothermal 2.8

hydro 11.3

Landfill mass and biomass 28.1

Solar 0.2

Unreported 2.5

Wind 55.1

a. Construct a bar chart, pie chart and Pareto chartb. What conclusion can you reach about the sources of green

powerSource: National renewable energy laboratory,2008

Class Exercise 3


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Calculate the following ?

a. Divide the data into classesb. Absolute frequency

c. Relative frequency

d. Percentages

e. Cumulative frequencyf. Cumulative percentage

g. Midpoints

h. Draw Histogram and relative frequencypolygon

THANKS

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