MTH410 S14 Lecture 01 May 12 -Mo-Wed

1MTH410

Probability and Statistics

Spring 2014

Nursel S. Ruzgar

Mathematics Department

[email protected]

416-979 5000/ext. 3173

MTH410 S14- Lecture 1

MTH410 S14- Lecture 1 2/153

Discussion of SyllabusRequired Text: Solved problems in Statistics, Part I- P. Ghargbouri, B. Todorow

Exercises in Statistics, Part I- P. Gharghbouri, B. Todorow

Meets

Mondays: 2:00-5:00pm-KHE221,

Wednesdays: 2:00-5:00pm- KHE221

Office Hours: Tuesdays: 5pm-5:45pm-VIC707

Labs: Section 1: Fridays: 10:00-12:00pm-ENGLG12

Section 2: Fridays: 13:00-15:00pm-ENGLG12

Section 3: Fridays: 16:00-18:00pm-ENGLG12

Section 4: Wednesdays: 11:00-13:00pm-ENG102


Discussion of Syllabus (contd)

Course Web Site: Blackboard

Labs and Quizzes

Labs will start in the first week, May 12.

There will be a quiz each week, except the

first week.


Discussion of Syllabus (contd)

Academic Dishonesty (Strongly

discouraged)

Refer to the senate policy

Tentative Course Outline


Course Objectives

Identify and formulate problems where statistics can have an impact.

See the relevance of statistics. Apply what has been learned to other engineering courses and to career practice.

Understand the basics of Statistics and Probability Theory

Interpret the statistical results and retrieve necessary information to help decision making

Develop the bases for the other courses.


Evaluation

30% MidtermTest (100 minutes) 10:00am,

Saturday, June 14, 2014

60% Final exam (180 minutes), room: TBA

10% Lab quizzes


OUTLINE Lecture 1Statistics-Descriptive and Inferential Statistics

Populations, Parameters, and Samples, Statistic, Variable

Data & Types of Data Cross-Sectional vs Time Series Data

Interval, Nominal Data, Ordinal

Graphical descriptive techniques for each type of data Histograms, Pie and Bar Charts

Scatter Diagrams, Contingency Table

Line Chart


In todays world

we are constantly being surrounded by statistics and statistical information. For example:

Political Polls, Customer Surveys

Interest rates, Economic Predictions

Course Marks, Job Market Information

How can we make sense out of all these data?

How can we differentiate valid from flawed claims?

What is Statistics?!


What is Statistics? (Contd)

Statistics is a way to get information from data

Statistics

Data

Data: Facts, especially

numerical facts, collected

together for reference or

information.

Information

Information: Knowledge

communicated concerning

some particular fact.

Definitions: Oxford English Dictionary

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

MTH410 S14- Lecture 1 10/153

Example

A student is somewhat apprehensive about the statistics

course because the student believes the myth that the

course is difficult. The professor provides last terms marks to the student. What information can the student obtain from this

list? Statistics

Data

List of last terms marks.

95

89

70

65

78

57

:

Information

New information about

the statistics class.

E.g. Median of all marks,

Typical mark, i.e. average,

Mark distribution, etc.

MTH410 S14- Lecture 1 11/153

Population, Parameter, &

Sample, Statistics, VariablePopulation: group of all items of interest to the statistics practitioner. All the members of the Ryerson University.

Parameter: A descriptive measure of a population. Mean number of soft drinks sold at Ryerson every week.

Sample: A set of items drawn from the population. 500 students surveyed.

Statistic: A descriptive measure of a sample. Average number of soft drinks these students buy per week.

Variable: A characteristic of population or sample that is of interest for us. Number of soft drinks a student buys every week.

MTH410 S14- Lecture 1 12/153

Key Statistical Concepts

Sample

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 polls on election day.

Population

a population is the entire set of all items under study.

frequently very large, sometimes infinite.

E.g. All 5 million Florida voters

MTH410 S14- Lecture 1 13/153

Key Statistical Concepts (Contd)

Statistic

A descriptive measure of a sample.

E.g. The proportion of the sample of 765 Floridians who voted

for Obama.

Parameter

A descriptive measure of a population.

In most applications of inferential statistics, the parameter

represents the information we need.

E.g. The proportion of the 5 million Florida voters who voted

for Obama.

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Key Statistical Concepts (Contd)

Parameter

Populations have

Parameters

Population

Sample

Subset

Statistic

Samples have

Statistics

Inference

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Types of Statistics

Descriptive statistics: involves the

arrangement, summary, and presentation of

data, to enable meaningful interpretation, and

to support decision making.

Inferential Statistics: a set of methods used

to draw conclusions about characteristics of a

population based on sample data.

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Descriptive Statistics

The actual method used depends on what information we

would like to extract. Are we interested in:

measure(s) of central location? and/or

measure(s) of variability (dispersion)?

Descriptive Statistics is a set of methods of organizing, summarizing, and presenting data in a convenient and informative way. These methods include:

Graphical Techniques

Numerical Techniques

MTH410 S14- Lecture 1 17/15317

Inferential Statistics

Descriptive Statistics describe the data set thats being

analyzed, but doesnt allow us to draw any conclusions

or make any inferences about the data. Hence we need

another branch of statistics: inferential statistics.

Inferential statistics is also a set of methods, but it is used

to draw conclusions or inferences about characteristics of

populations based on data from a sample.

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Statistical Inference

Statistical inference is the process of making an estimate,

prediction, or decision about a population based on a sample.

Parameter

Population

Sample

Statistic

Inference

What can we infer about a Populations Parameters

based on a Samples Statistics?

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Statistical Inference(Contd)

We use statistics to make inferences about parameters.

Therefore, we can make an estimate, prediction, or

decision about a population based on sample data.

Then, we can apply what we know about a sample to the

larger population from which the sample was drawn!

What is the purpose or/and which kind of benefits

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Statistical Inference (Contd)

Rationale:

Large populations make investigating each member

impractical, extremely expensive and time-consuming.

Easier and cheaper to take a sample and make estimates

about the population from the sample.

However:

Such conclusions and estimates are not always going to be

correct.

Hence, we have to build into the statistical inference

measures of reliability, namely confidence level and

significance level.

MTH410 S14- Lecture 1 21/153

Confidence & Significance

LevelsThe confidence level is the proportion of times that an

estimating procedure will be correct, if the sampling

procedure were repeated a very large number of times.

E.g. a confidence level of 95% means that, estimates based on

this form of statistical inference will be correct 95% of the time.

When the purpose of the statistical inference is to draw a

conclusion about a population, the significance level

measures how frequently the conclusion will be wrong in

the long run.

E.g. a 5% significance level means that, in repeated samples,

this type of conclusion will be wrong 5% of the time.

MTH410 S14- Lecture 1 22/153

Confidence & Significance Levels

(Contd)

So if we use (Greek letter alpha) to represent significance level (how frequently

the conclusion will be wrong) , then our

confidence level is 1 .

Confidence Level

+ Significance Level

= 1

This relationship can also be stated as:

MTH410 S14- Lecture 1 23/153

Confidence & Significance Levels

(Contd)Consider a statement from polling data you may hear about in the news these days:

This poll is considered accurate within 3.4 percentage points, 19 times out of 20.

In this case, the confidence level is 95% (19/20 = 0.95), and the significance level is 5%.

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Graphical Descriptive

Techniques

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Agenda

Types of Data and Information

Graphical and Tabular Techniques for Nominal

Data

Graphical Techniques for Interval Data

Describing Time-Series Data

Describing the Relationship Between Two

Variables

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Definitions

A variable is some characteristic of a population or sample.

Typically denoted with a capital letter: X, Y, Z

E.g. student marks. No all students achieve the same mark. The

marks vary from student to student, so the name variable.

Values of a variable are all possible observations of the variable.

E.g. student marks: all integers between 0 and 100.

Data are the observed values of a variable.

E.g. marks of 6 students in an exam: {67, 74, 71, 83, 93, 48}

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Types of data analysis

Knowing the type of data is necessary to properly

select the technique to be used when analyzing data.

Type of analysis allowed for each type of data

Interval data arithmetic calculations

Nominal data counting the number of observation in each category

Ordinal data - computations based on an ordering process

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Types of data and information

Variable - a characteristic of population or

sample that is of interest for us.

Number of soft drinks a student buys every week

The waiting time for medical services

The score of a student in the Stats Exam.

Data - the actual values of variables

Interval data are numerical observations

Nominal data are categorical observations

Ordinal data are ordered categorical observations

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Types of Data & Information

Data (at least for purposes of Statistics)

fall into three main groups:

Interval Data

Nominal Data

Ordinal Data

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Interval Data

Real numbers, i.e. weights, prices,

distance, etc.

Also called as quantitative or numerical.

Arithmetic operations can be performed on

Interval Data, so its meaningful to talk about 2*Weight, or Price + $1.5, and so on.

MTH410 S14- Lecture 1 31/153

Nominal Data

The values of nominal data are categories.

E.g. responses to questions about marital status, coded as:

Single = 1, Married = 2, Divorced = 3, Widowed = 4

Because the numbers are arbitrary, arithmetic operations dont make any sense (e.g. does Widowed 2 = Married?!)

Any other numbering system is also valid provided that each category has a different number assigned to it.

E.g. Another coding system as valid as the previous one:


Nominal data are also called qualitative or categorical.

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Ordinal Data

Ordinal Data appear to be nominal, but their values have an

order, a ranking to them.

E.g. The most active stocks traded on the NASDAQ in

descending order

MSFT = 1, CSCO = 2, Dell = 3, SunW = 4

Any other numbering system is valid provided the order is

maintained.

E.g. Another coding system as valid as the previous one:


It is still not meaningful to do arithmetic operations on this kind of data (e.g.

does 2*MSFT = CSCO?!).

We can say something like the number of stocks traded from:

Microsoft > Cisco or Sun Microsystems < Dell

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Nominal data vs. Ordinal data

The critical difference between nominal data and ordinal

data is that the values of the latter are in order.

E.g. It is valid for nominal data to have:


However, it wont be valid for ordinal data:MSFT = 7, CSCO = 4, Dell = 13, SunW = 1

(The order changed to be: SunW, CSCO, MSFT,

Dell.

We must keep the order of MSFT, CSCO, Dell, SunW)

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Interval data vs. Ordinal data

The critical difference between interval data and ordinal data is that the intervals or differences between values of interval data are consistent and meaningful.

E.g. The difference between marks of 85 and 80 is the same five-mark difference as that between 75 and 70.

However for coding system like:


We can see CSCO MSFT = 1, and SunW Dell = 1

But we cant conclude that the difference between the number of stocks traded in Microsoft and Cisco Systems is the same as the difference in the number of stocks traded between Dell Computer and Sun Microsystems.

MTH410 S14- Lecture 1 35/153

Types of Data & Information(Contd)

Categorical?DataInterval

DataN

Ranked?

Y

Ordinal

DataY

Nominal

Data

N

Categorical

Data

Knowing the type of data is necessary to properly select

the technique to be used when analyzing data.

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Calculations for Types of Data

All calculations are permitted on interval data.

Only calculations involving a ranking process are allowed for ordinal data.

No calculations are allowed for nominal data, only allowed to count the number of observations in each

category.

This leads to the following hierarchy of data

MTH410 S14- Lecture 1 37/153

Hierarchy of Data

Higher

level

may be

treated

as lower

level(s)

Interval

Values are real numbers.

All calculations are valid.

Data may be treated as ordinal or nominal.

Ordinal

Values must represent the ranked order of the data.

Calculations based on an ordering process are valid.

Data may be treated as nominal but not as interval.

Nominal

Values are the arbitrary numbers that represent categories.

Only calculations based on the frequencies of occurrence are valid.

Data may not be treated as ordinal or interval.

MTH410 S14- Lecture 1 38/153

E.g. Representing Student Grades

Categorical?DataInterval Data

Nominal Data

Ordinal Data

N

Ranked?

Y

Y

NCategorical

DataRanked order to data

NO ranked order to data

e.g. integers in {0..100}

e.g. {F, D, C, B, A}

e.g. {Pass | Fail}

MTH410 S14- Lecture 1 39/153

Agenda


Graphical and Tabular Techniques for

Nominal Data

MTH410 S14- Lecture 1 40/153

Graphical & Tabular Techniques for

Nominal DataThe only allowable calculation on nominal data is to

count the frequency of each value of the variable.

We can summarize the data in a table that presents the

categories and their counts called a frequency

distribution.

A relative frequency distribution lists the categories

and the proportion with which each occurs.

MTH410 S14- Lecture 1 41/153

It is often preferable to show the relative frequency

(proportion) of observations falling into each class, rather

than the frequency itself.

Relative frequencies should be used when

the population relative frequencies are studied

comparing two or more histograms

the number of observations of the samples studied are different

Class relative frequency = Class frequency

Total number of observations

Relative frequency

MTH410 S14- Lecture 1 42/153

It is generally best to use equal class width, but sometimes unequal class width are called for.

Unequal class width is used when the frequency

associated with some classes is too low. Then,

several classes are combined together to form a

wider and more populated class.

It is possible to form an open ended class at the

higher end or lower end of the histogram.

Class width

MTH410 S14- Lecture 1 43/153

Example: Light Beer Preference

SurveyIn 2006 total light beer sales in the United States was

approximately 3 million gallons

With this large a market breweries often need to know more

about who is buying their product.

The marketing manager of a major brewery wanted to

analyze the light beer sales among college and university

students who do drink light beer.

A random sample of 285 graduating students was asked to

report which of the following is their favorite light beer.

MTH410 S14- Lecture 1 44/153

Example

1. Budweiser Light

2. Busch Light

3. Coors Light

4. Michelob Light

5. Miller Lite

6. Natural Light

7. Other brand

The responses were recorded using the codes. Construct a

frequency and relative frequency distribution for these data

and graphically summarize the data by producing a bar

chart and a pie chart.

MTH410 S14- Lecture 1 45/153

Example 1 1 1 1 2 4 3 5 1 3 1 3 7 5 1

1 5 2 1 5 1 3 3 3 1 1 5 3 1 5

5 1 1 3 3 5 5 6 3 5 3 5 5 5 1

1 2 1 1 5 5 3 2 1 6 1 1 4 5 1

3 3 5 4 7 6 6 4 4 6 5 2 1 1 5

3 3 1 3 5 3 3 7 3 7 2 1 5 7

3 6 2 6 3 6 6 6 5 6 1 1 6 3

7 1 1 1 5 1 3 1 3 7 7 2 1 1

2 5 3 1 1 3 1 1 7 5 3 2 1 1

6 5 7 1 3 2 1 3 1 1 7 5 5 6

1 4 6 1 3 1 1 5 5 5 5 1 5 5

6 1 3 3 1 3 7 1 1 1 2 4 1 1

3 3 7 5 5 1 1 3 5 1 5 4 5 3

4 1 4 5 3 1 5 3 3 3 1 1 5 3

5 6 4 3 5 6 4 6 5 5 5 5 3 1

2 3 2 7 5 1 6 6 2 3 3 3 1 1

5 1 4 6 3 5 1 1 2 1 5 6 1 1

5 1 3 5 1 1 1 3 7 3 1 6 3 1

2 2 5 1 3 5 5 2 3 1 1 3 6 1

1 1 1 7 3 1 5 3 3 3 5 3 1 7

MTH410 S14- Lecture 1 46/1532014/5/8

Frequency and Relative Frequency

Distributions Light Beer Brand Frequency Relative Frequency

Budweiser Light 90 31.6%

Busch Light 19 6.7

Coors Light 62 21.8

Michelob Light 13 4.6

Miller Lite 59 20.7

Natural Light 25 8.8

Other brands 17 6.0

Total 285 100

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Nominal Data (Frequency)

Bar Charts are often used to display frequencies

90

19

62

13

59

25

17

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7

MTH410 S14- Lecture 1 48/153

Nominal Data (Relative

Frequency)

Pie Charts show relative frequencies

131%

27%

322%

44%

521%

69%

76%

MTH410 S14- Lecture 1 49/153

Nominal Data

It all the same information,

(based on the same data).

Just different presentation.

Light Beer Brand Frequency Relative Frequency

Budweiser Light 90 31.6%

Busch Light 19 6.7

Coors Light 62 21.8

Michelob Light 13 4.6

Miller Lite 59 20.7

Natural Light 25 8.8

Other brands 17 6.0

90

19

62

13

59

25

17

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7

131%

27%

322%

44%

521%

69%

76%

MTH410 S14- Lecture 1 50/153

Agenda



Nominal Data


MTH410 S14- Lecture 1 51/153


There are several graphical methods that are used when the data are interval.

The most important of these graphical methods is the histogram, which is created by drawing rectangles

whose bases are the intervals and whose heights are the

frequencies.

The histogram is not only a powerful graphical technique used to summarize interval data, but also used

to help explain probabilities.

MTH410 S14- Lecture 1 52/153

Building a Histogram

Example The marketing manager of a long-distance telephone

company conducted a survey of 200 new costumers wherein the

first months bills are recorded. What information can be extracted

from those data?

This manager was only able to find that the smallest bill is $0, and

the largest bill is $119.63, and most of bills are less than $100

However, there is a lot of information may be more interesting.

Bill distribution,

Are there many small bills and few large bills?

What is the typical bill?

Are the bills somewhat similar or different?

MTH410 S14- Lecture 1 53/1532014/5/8

Building a Histogram(Contd)

Alternatively, we could use Sturges formula:

Number of class intervals = 1 + 3.3 log (n), n

is the number of observations, then we get 9.

1) Collect the Data

2) Create a frequency distribution for the data a) Determine the number of classes to use

Refer to Table.

With 200 observations, we should have between 7 & 10 classes 9 seems the best. For our purpose, let us pick 8.

MTH410 S14- Lecture 1 54/153


1) Collect the Data 2) Create a frequency distribution for the data

a) Determine the number of classes to use. [8]

b) Determine how large to make each class

Look at the range of the data, that is,

Range = Largest Observation Smallest Observation

Range = $119.63 $0 = $119.63

Then each class width becomes:

Range (# of classes) = 119.63 8 15

FYI: if pick 9, the width should be 13.3

MTH410 S14- Lecture 1 55/153


1) Collect the Data 2) Create a frequency distribution for the data

a) Determine the number of classes to use. [8] b) Determine how large to make each class. [15]

c) Place the data into each class

each item can only belong to one class;

each class contains observations greater than its

lower limit and less than or equal to its upper limit.

That means, there is not overlapping between any

two classes.

MTH410 S14- Lecture 1 56/153


3) Draw the Histogram

1) Collect the Data

2) Create a frequency

distribution for the data.

MTH410 S14- Lecture 1 57/153


1) Collect the Data 2) Create a frequency distribution for the data. 3) Draw the Histogram

MTH410 S14- Lecture 1 58/153

Example : Interpret

0

20

40

60

80

15 30

45

60

75

90

10

5

12

0Bills

Fre

qu

en

cy

About half of all

the bills are smallA few bills are in

the middle range

Relatively, large

number of bills

are large

18+28+14=60

13+9+10=3271+37=108

MTH410 S14- Lecture 1 59/153

FYI The difference with Bar chart

All data are nominal,

Each bin is one category,

There is a gap between

two neighbor bins.

All data are interval,

Each bin is an interval of values,

There is no gap between two

neighbor bins.

MTH410 S14- Lecture 1 60/153

There are four typical shape characteristics

Shapes of histograms

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Shapes of Histograms

Symmetry

A histogram is said to be symmetric if, when we

draw a vertical line down the center of the histogram,

the two sides are identical in shape and size:

Fre

quency

Variable

Fre

quency

Variable

Fre

quency

Variable

MTH410 S14- Lecture 1 62/153

Shapes of Histograms(Contd)

Skewness

A skewed histogram is one with a long tail extending to

either the right or the left:

Negatively skewedPositively skewed

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Shapes of Histograms(Contd)

Modality

A unimodal histogram is one with a single peak,

while a bimodal histogram is one with two peaks:

Fre

qu

en

cy

Variable

Unimodal

Variable

Bimodal

Fre

qu

en

cy

A modal class is the class with the largest number of observations

MTH410 S14- Lecture 1 64/153

A modal class is the one with the largest number of observations.

A unimodal histogram

The modal class

Modal classes

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Modal classes

A bimodal histogram

A modal class A modal class

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Bell Shaped Histograms

A special type of symmetric unimodal histogram

is Bell Shaped:

Bell Shaped

Fre

qu

en

cy

VariableMany statistical techniques

require that the population

be bell shaped.

MTH410 S14- Lecture 1 67/153

Many statistical techniques require that the population be bell shaped.

Drawing the histogram helps verify the shape of the population in question

Bell shaped histograms

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Stem & Leaf Display

Retains information about individual observations that would normally be lost in the creation of a histogram.

Split each observation into two parts, a stem and a leaf:

e.g. Observation value: 42.19

There are several ways to split it up

We could split it at the decimal point:

Or split it at the tens position (while rounding to the nearest integer in the ones position)

Stem Leaf

42 19

4 2

MTH410 S14- Lecture 1 69/1532014/5/8

Stem & Leaf Display

Continue this process for all the observations.

Then, use the stems for the classes and each leaf becomes part of the histogram as

follows

Stem Leaf0 00000000001111122222233333455555566666667788889999991 0000011112333333344555556678899992 00001111123446667789993 0013355894 1244455895 335666 34587 0222245567898 3344578899999 0011222223334455599910 00134444669911 124557889

Thus, we still have access to our

original data points value!

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Histogram and Stem & Leaf

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Ogive

(pronounced Oh-jive) is a graph of

a cumulative relative frequency distribution.

We create an ogive in three steps

First, from the frequency distribution created earlier, calculate relative frequencies

MTH410 S14- Lecture 1 72/1532014/5/8

Relative Frequencies

For example, we had 71 observations in the first class (telephone

bills from $0.00 to $15.00). Hence, the relative frequency for this

class is 71 200 (the total # of phone bills) = 0.355 (or 35.5%)

MTH410 S14- Lecture 1 73/1532014/5/8

Ogive(Contd)

is a graph of a cumulative frequency distribution.

We create an ogive in three steps

1) Calculate relative frequencies.

2) Calculate cumulative relative frequencies by adding

the current class relative frequency to the previous

class cumulative relative frequency.(For the first class, its cumulative relative frequency is just its relative

frequency)

MTH410 S14- Lecture 1 74/1532014/5/8

Cumulative Relative

Frequencies

first class, just itself

next class: .355+.185=.540

last class: .930+.070=1.00

Always or by chance?

MTH410 S14- Lecture 1 75/1532014/5/8

Ogive(Contd)

is a graph of a cumulative frequency distribution. 1) Calculate relative frequencies. 2) Calculate cumulative relative frequencies. 3) Graph the cumulative relative frequencies.

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Ogive(Contd)

The ogive can be

used to answer

questions like:

What telephone bill

value is at the 50th

percentile?

around $35

MTH410 S14- Lecture 1 77/1532014/5/8

Agenda



Nominal Data

Graphical Techniques for Interval

Data


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Describing Time Series Data

Observations measured at the same point in time

are called cross-sectional data.

Observations measured at successive points in

time are called time-series data.

Time-series data graphed on a line chart, which

plots the value of the variable on the vertical axis

against the time periods on the horizontal axis.

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Example

We recorded the monthly average retail

price of gasoline since 1978.

Draw a line chart to describe these data

and briefly describe the results.

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Example

0

0.5

1

1.5

2

2.5

3

3.5

1 25 49 73 97 121 145 169 193 217 241 265 289 313 337

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Agenda



Nominal Data




Variables

Two Nominal Variables

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Relationship between Two

Nominal VariablesSo far weve looked at tabular and graphical techniques for one variable (either nominal or

interval data).

A cross-classification table (or cross-tabulation

table) is used to describe the relationship between

two nominal variables.

A cross-classification table lists the frequency of

each combination of the values of the two

variables

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Example

In a major North American city there are four competing

newspapers: the Post, Globe and Mail, Sun, and Star.

To help design advertising campaigns, the advertising

managers of the newspapers need to know which segments of

the newspaper market are reading their papers.

A survey was conducted to analyze the relationship between

newspapers read and occupation.

A sample of newspaper readers was asked to report which

newspaper they read: Globe and Mail (1) Post (2), Star (3),

Sun (4), and to indicate whether they were blue-collar worker

(1), white-collar worker (2), or professional (3).

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Example

By counting the number of times each of the 12 combinations occurs,

we produced the Table

Occupation

Newspaper Blue Collar White Collar Professional Total

G&M 27 29 33 89

Post 18 43 51 112

Star 38 21 22 81

Sun 37 15 20 72

Total 120 108 126 354

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Example

If occupation and newspaper are related, then there will be differences in

the newspapers read among the occupations. An easy way to see this is

to covert the frequencies in each column to relative frequencies in each

column. That is, compute the column totals and divide each frequency by

its column total.

Occupation

Newspaper Blue Collar White Collar Professional

G&M 27/120 =.23 29/108 = .27 33/126 = .26

Post 18/120 = .15 43/108 = .40 51/126 = .40

Star 38/120 = .32 21/108 = .19 22/126 = .17

Sun 37/120 = .31 15/108 = .14 20/126 = .16

MTH410 S14- Lecture 1 86/1532014/5/8

Example

Interpretation: The relative frequencies in the columns 2 & 3 are similar,

but there are large differences between columns 1 and 2 and between

columns 1 and 3.

This tells us that blue collar workers tend to read different newspapers

from both white collar workers and professionals and that white collar and

professionals are quite similar in their newspaper choice.

dissimilar

similar

MTH410 S14- Lecture 1 87/1532014/5/8

Graphing the Relationship Between Two Nominal

Variables

Use the data from the cross-classification table to create bar charts

Professionals tend

to read the Globe &

Mail more than

twice as often as the

Star or Sun

MTH410 S14- Lecture 1 88/1532014/5/8

Agenda


Graphical and Tabular Techniques for Nominal

Data




Variables

Two Nominal Variables

Two Interval Variables

MTH410 S14- Lecture 1 89/1532014/5/8

Graphing the Relationship Between Two

Interval Variables

Moving from nominal data to interval data, we are

frequently interested in how two interval variables are

related.

To explore this relationship, we employ a scatter

diagram, which plots two variables against one another.

The independent variable is labeled X and is usually

placed on the horizontal axis, while the other, dependent

variable, Y, is mapped to the vertical axis.

MTH410 S14- Lecture 1 90/1532014/5/8

Example

A real estate agent wanted to know to what extent the selling

price of a home is related to its size. To acquire this

information he took a sample of 12 homes that had recently

sold, recording the price in thousands of dollars and the size

in hundreds of square feet. These data are listed in the

accompanying table. Use a graphical technique to describe

the relationship between size and price.

Size 23 18 26 20 22 14 33 28 23 20 27 18

Price 315 229 355 261 234 216 308 306 289 204 265 195

MTH410 S14- Lecture 1 91/1532014/5/8

Example

It appears that in fact there is a relationship,

that is, the greater the house size the greater

the selling price

MTH410 S14- Lecture 1 92/1532014/5/8

Patterns of Scatter Diagrams

Linearity and Direction are two concepts we are interested in.

Positive Linear RelationshipNegative Linear Relationship

Non-Linear RelationshipNo Relationship

MTH410 S14- Lecture 1 93/1532014/5/8

Summary

Histogram, Ogive

Frequency and

Relative Frequency

Tables, Bar and Pie

Charts

Scatter Diagram Cross-classification

Table, Bar Charts

IntervalData

NominalData

Single Set of Data

Relationship Between

Two Variables

MTH410 S14- Lecture 1 94/153

Agenda

Introduction

Measures of Central Location

Measures of Variability

Measures of Relative Standing

MTH410 S14- Lecture 1 95/153

Numerical Descriptive Techniques


Mean, Median, Mode


Range, Standard Deviation, Variance,

Coefficient of Variation


Percentiles, Quartiles

MTH410 S14- Lecture 1 96/153

Agenda

Introduction


MTH410 S14- Lecture 1 97/153


Usually, we focus our attention on two

types of measures when describing

population characteristics:

Central location (e.g. average)

Variability or spread

The measure of central location

reflects the locations of all the actual

data points.

MTH410 S14- Lecture 1 98/153

With one data point

clearly the central

location is at the point

itself.

The measure of central location reflects the locations

of all the actual data points.

How?

But if the third data point

appears on the left hand-side

of the midrange, it should pullthe central location to the left.

With two data points,

the central location

should fall in the middle

between them (in order

to reflect the location of

both of them).


MTH410 S14- Lecture 1 99/153

Arithmetic Mean

The arithmetic mean, or average, simply

as mean, is the most popular & useful

measure of central location.

It is computed by simply adding up all the

observations and dividing by the total

number of observations:

Sum of the observations

Number of observationsMean =

The arithmetic mean for a sample is denoted with an

x-bar:

MTH410 S14- Lecture 1 100/153

Notation

When referring to the number of

observations in a population, we use

uppercase letter N

When referring to the number of

observations in a sample, we use lower

case letter n

The arithmetic mean for a population is

denoted with Greek letter mu:

MTH410 S14- Lecture 1 101/153

Statistics is a pattern language

Population Sample

Size N n

Mean

MTH410 S14- Lecture 1 102/153

Mean(Contd)

Population Mean Sample Mean

MTH410 S14- Lecture 1 103/153


Population Sample

Size N n

Mean

MTH410 S14- Lecture 1 104/153

Mean(Contd)

is appropriate for describing interval data, e.g. heights of people, marks of student papers, etc.

is seriously affected by extreme values called outliers.

E.g. If Bill Gates moved into any neighborhood, the average household income for that neighborhood would increase dramatically beyond what it was previously!

MTH410 S14- Lecture 1 105/153

10

...

10

102110

1 xxxxx ii

Example

The reported time on the Internet of 10 adults are 0, 7, 12, 5, 33,

14, 8, 0, 9, 22 hours. Find the mean time on the Internet.

0 7 2211.0

Example

Suppose the telephone bills of Example 2.1 represent

the population of measurements. The population mean is

200

x...xx

200

x 20021i200

1i 42.19 38.45 45.77 43.59

The Arithmetic Mean

MTH410 S14- Lecture 1 106/153

Properties of Mean

Calculated by using every data point.

Every interval data has a unique mean.

Sum of deviations from mean is 0.

Effected from extreme (very large or small)

values

Not meaningful for nominal or ordinal data.

Useful comparing 2 or more data sets.

MTH410 S14- Lecture 1 107/153

MedianThe median is calculated by placing all the observations in

order; the observation that falls in the middle is the median.

Data: {0, 7, 12, 5, 14, 8, 0, 9, 22} N=9 (odd)

Sort them bottom to top, find the middle:

0 0 5 7 8 9 12 14 22

Data: {0, 7, 12, 5, 14, 8, 0, 9, 22, 33} N=10 (even)

Sort them bottom to top, there are two elements in

the middle:

0 0 5 7 8 9 12 14 22 33

median = (8 + 9) 2 = 8.5

Sample and population medians are computed the same way.

MTH410 S14- Lecture 1 108/153

Properties of Median

Calculated by using only 1 or at most 2

values.

Every interval data has a unique median.

Not affected from extreme values.

Can be calculated for ordinal data as well,

but cant be interpreted as the centre of location.

MTH410 S14- Lecture 1 109/153

Mode

The mode of a set of observations is the value that occurs most frequently. Sometimes we say

MODE = PEAK of a curve.

A set of data may have one mode (or modal class), or two modes, or more modes.

Mode can be used for all data types, although mainly used for nominal data.

For populations and large samples the modal classis more preferable.

Sample and population modes are computed the same way.

MTH410 S14- Lecture 1 110/153

Mode(Contd)

E.g. Data: {0, 7, 12, 5, 14, 8, 0, 9, 22, 33}

N=10

Which observation appears most often?

The mode for this data set is 0. How about

this as a measure of central location?

In a small sample, it may not be a good measure.

MTH410 S14- Lecture 1 111/153

Mode(Contd)

The mode may be not unique, i.e. 2 modes for

bimodal data.

Note: if you are using Excel for your data

analysis and your data is multi-modal (i.e.

there is more than one mode), Excel only

calculates the smallest one.

You will have to use other techniques (i.e.

histogram) to determine if your data is bimodal,

trimodal, etc.

MTH410 S14- Lecture 1 112/153

Properties of Mode

Not affected from extreme values.

Multiple modes possible, hence not a good

measure of central location.

No mode exists sometimes, all observations

have the same value.

Can be calculated for nominal data as well,

but cant be interpreted as the centre of location

MTH410 S14- Lecture 1 113/153

Mean, Median, Mode

If a distribution is symmetrical, the mean,

median and mode may coincide

mode

mean

median

MTH410 S14- Lecture 1 114/153

Mean, Median, Mode(Contd)If a distribution is asymmetrical, say skewed to the

left or to the right, the three measures may differ.

E.g.:

MeanMedian

Mode

A negatively skewed distribution

(skewed to the left)

A positively skewed distribution

(skewed to the right)

MeanModeMedian

Note: Median not as sensitive

as Mean for the skewness.

modemedian

mean

MTH410 S14- Lecture 1 115/153

Mean, Median, Mode(Contd)

If data are symmetric, the mean, median,

and mode will be approximately the same.

If data are multimodal, report the mean,

median and/or mode for each subgroup.

If data are skewed, report the median.

MTH410 S14- Lecture 1 116/153

About Ordinal & Nominal Data

For ordinal and nominal data the calculation

of the mean is not valid.

Median is appropriate for ordinal data.

For nominal data, a mode calculation is

useful for determining highest frequency but

not central location.

MTH410 S14- Lecture 1 117/153

Example: Assume you got 35 marks for one exam,

and the average was 45 marks, which kind of result

would you expect? Failed?

No sure. Dependent on the actual marks for all

students.

If all marks like:

15, 20, 25, 25, 25, 30, 35, 75, 100, 100

Congratulation! Good job, youre the fourth highest!

25 30

2

How about telling you the median was 27.5 ( ),

would you worry again?

Course Marks: Mean & Median

MTH410 S14- Lecture 1 118/153

The mean is generally the first choice.

When the following scenarios, the median is

the best

there are extreme observations

determine the rank of a particular value

relative to the data set

The mode is rare the best measure.

Mean, Median, Mode: Which is Best

MTH410 S14- Lecture 1 119/153

Geometric Mean

The geometric mean is used when the variable is a growth rate or rate of change, such as the value of an investment over periods of time.

If the rate of return was Rg in every

period, the nth period return would

be calculated by:n

g )R1( )R1)...(R1)(R1( n21

For the given series of rate of

returns the nth period return is

calculated by:

The geometric mean Rg is selected such that

1)R1)...(R1)(R1(R n n21g

MTH410 S14- Lecture 1 120/153

Finance Example

Suppose a 2-year investment of $1,000 grows by 100% to

$2,000 in the first year, but loses 50% from $2,000 back to

the original $1,000 in the second year. What is the average

return?

The upper case Greek Letter Pi represents a product of terms

Solving for the geometric mean yields a rate of 0%.

This would indicate having more than $1,000 at the end of the second

year, however in fact we only have $1,000.

Using the arithmetic mean, misleading

more precise

MTH410 S14- Lecture 1 121/153

Measures of Central Location SummaryCompute mean to

Describe the central location of a single set of

interval data

Compute median to

Describe the central location of a single set of

ordinal or interval data (with extreme observations)

Compute mode to

Describe a single set of nominal, ordinal or interval

data

Compute Geometric mean to

Describe a single set of interval data based on

growth rates

MTH410 S14- Lecture 1 122/153

Agenda

Introduction



MTH410 S14- Lecture 1 123/153

Measures of variability

Measures of central location fail to tell the whole story about the distribution.

A question of interest still remains unanswered:

How much are the observations spread out

around the mean value?

MTH410 S14- Lecture 1 124/153

The average value provides

a good representation of the

observations in the data set.

Small variability

This data set is now

changing to...

Why not use mean

Observe two data sets:

MTH410 S14- Lecture 1 125/153

Why not use mean(Contd)

Observe two data sets:

The average value provides

a good representation of the

observations in the data set.

Small variability

Larger variabilityThe same average value does not

provide as good representation of the

observations in the data set as before.

MTH410 S14- Lecture 1 126/153

Range

The range is the simplest measure of variability, and

calculated as:

Range = Largest observation Smallest observation

E.g. Data set: {4, 4, 4, 4, 4, 50} Range = 46

Data set: {4, 8, 15, 24, 39, 50} Range = 46

The range is the same in both cases, but the data

sets have very different distributions

MTH410 S14- Lecture 1 127/153

Range(Contd)

? ? ?

But, how do all the observations spread out?

Smallest

observation

Largest

observation

The range cannot assist in answering this question

Range

MTH410 S14- Lecture 1 128/153

Variance

Variance and its related measure,

standard deviation, are arguably the most

important statistics. Used to measure

variability, they also play a vital role in

almost all statistical inference procedures.

Population variance is denoted by

(Lower case Greek letter sigma squared)

Sample variance is denoted by

(Lower case s squared)

MTH410 S14- Lecture 1 129/153


Population Sample

Size N n

Mean

Variance

MTH410 S14- Lecture 1 130/153

Variance(Contd)

The variance of a population is:

population mean

sample mean

Sample size minus one !

The reason we will discuss later

population size

The variance of a sample is:

MTH410 S14- Lecture 1 131/153

Variance(Contd)

Alternatively, there is a short-cut formulation

to calculate sample variance directly from the

data without the intermediate step of

calculating the mean. Its given by:

As you can see, you have to calculate the sample mean

in order to calculate the sample variance.

MTH410 S14- Lecture 1 132/153

Why not use the sum of deviations...

Consider two small populations:

1098

74 10

11 12

13 16

8-10 = -2

9-10 = -1

11-10 = +1

12-10 = +2

4-10 = - 6

7-10 = -3

13-10 = +3

16-10 = +6

Sum = 0

Sum = 0

The mean of both

populations is 10...

but measurements in B

are more dispersed

than those in A.

Any good measure of

dispersion should agree

with this observation.

Can the sum of deviations be a

good measure of variability?

A

B

The sum of deviations is zero for all populations,

therefore, is not a good measure of variability.

10-10 = 0

10-10 = 0

MTH410 S14- Lecture 1 133/153

Let us calculate the variances

185

)1016()1013()1010()107()104( 222222B

25

)1012()1011()1010()109()108( 222222A

Why is the variance defined as

the average squared deviation

rather than the sum of squared

deviations?

MTH410 S14- Lecture 1 134/153

Let us calculate the sum of squared deviations for both

data sets in this example

Which data set has a larger dispersion?

1 3 1 32 5

A B

Data set B is

more dispersed

around the mean

Why not use the sum of squared deviations...

Date set A:

{1, 1, 1, 1, 1

3, 3, 3, 3, 3}

Date set B:{1, 5}

MTH410 S14- Lecture 1 135/153

1 3 1 32 5

A B

Sum of squared deviation for A = 5(1-2)2 + 5(3-2)2= 10

Sum of squared deviation for B = (1-3)2 + (5-3)2 = 8

SumA > SumB. This is inconsistent

with the observation that set B is

more dispersed.

Why not use the sum of squared deviations...

Date set A:

{1, 1, 1, 1, 1

3, 3, 3, 3, 3}

Date set B:{1, 5}

MTH410 S14- Lecture 1 136/153

1 3 1 32 5

A B

When calculated on per observation basis (variance),

the data set dispersions are properly ranked.

A2 = SumA/N = 10/10 = 1

B2 = SumB/N = 8/2 = 4

How about averaged squared deviations...

MTH410 S14- Lecture 1 137/153

Application

Example

The following sample consists of the

number of jobs six students applied for:

17, 15, 23, 7, 9, 13.

Finds its mean and variance.

What are we looking to calculate?

MTH410 S14- Lecture 1 138/153

Sample Mean & Variance

Sample Mean

Sample Variance

Sample Variance (shortcut method)

MTH410 S14- Lecture 1 139/153

Standard Deviation

The standard deviation is the square root of

the variance.

Population standard deviation:

Sample standard deviation:

MTH410 S14- Lecture 1 140/153


Population Sample

Size N n

Mean

Variance

Standard

Deviation

MTH410 S14- Lecture 1 141/153

Mean Absolute Deviation

There is another deviation: Mean Absolute Deviation

(MAD), which is calculated by averaging the absolute

value of the deviation. However, this statistic is rarely

used.

E.g. Given data set {17, 15, 23, 7, 9, 13}

|17 14| |15 14| |23 14| |7 14| |9 14| |13 14| 1MAD 4

6 3

n

xxMAD

n

i i 1)(

MTH410 S14- Lecture 1 142/153

Measures of Variability Summary

If data are symmetric, with no serious outliers, use range and standard deviation.

If comparing variation across two data sets, use coefficient of variation.

The measures of variability introduced in this section can be used only for interval data.

The next section will discuss a measure that can be used to describe the variability of ordinal data.

There are no measures of variability for nominaldata.

MTH410 S14- Lecture 1 143/153

Agenda

Introduction




MTH410 S14- Lecture 1 144/153


Measures of relative standing are designed to provide

information about the position of particular values relative

to the entire data set.

Percentile: the Pth percentile is the smallest point in a

distribution at or below which p percentage of cases is

found.

Your score

60% of all the scores lie here 40%

Example: Suppose your score is the 60th percentile of a

GMAT test. That is

Note: The 60th percentile doesnt mean you scored 60% on the

exam. It means that 60% of your peers scored lower than you on

the exam..

MTH410 S14- Lecture 1 145/153

Quartiles

We have special names for the 25th, 50th, and 75th

percentiles, namely quartiles.

The first or lower quartile is labeled Q1 = 25th percentile.

The second quartile, Q2 = 50th percentile (also the

median).

The third or upper quartile, Q3 = 75th percentile.

We can also convert percentiles into quintiles (fifths) and

deciles (tenths).

MTH410 S14- Lecture 1 146/153

Quartiles vs. VariabilityQuartiles can provide an idea about the shape of a histogram

Q1 Q2 Q3

Positively skewed

histogram

Q1 Q2 Q3

Negatively skewed

histogram

< >

MTH410 S14- Lecture 1 147/153

Commonly Used Percentiles

First (lower) decile = 10th percentile

First (lower) quartile, Q1, = 25th percentile

Second (middle) quartile, Q2, = 50th percentile

Third quartile, Q3, = 75th percentile

Ninth (upper) decile, = 90th percentile

MTH410 S14- Lecture 1 148/153

Location of Percentiles

The following formula allows us to

approximate the location of any percentile:

percentilePtheoflocationtheisLwhere

100

P)1n(L

thP

P

MTH410 S14- Lecture 1 149/153

Location of Percentiles(Contd)

Given the data :

0 0 5 7 8 9 12 14 22 33

Where is the location of the 25th percentile?

0 0 5 7 8 9 12 14 22 33

The 25th percentile is three-quarters of the distance between the second (which is 0) and the third observations (which is 5). Three-quarters of the distance is: (.75)(5 0) = 3.75; because the second observation is 0, the 25th percentile is

0 + 3.75 = 3.75

L25 = (10+1)(25/100) = 2.75

MTH410 S14- Lecture 1 150/153


What about the upper quartile?

L75 = (10+1)(75/100) = 8.25

0 0 5 7 8 9 12 14 22 33

It is located one-quarter of the distance between the eighth and the ninth observations, which are 14 and 22, respectively. One-quarter of the distance is: (.25)(22 - 14) = 2, which means the 75th percentile is at: 14 + 2 = 16

MTH410 S14- Lecture 1 151/153


0 0 5 7 8 9 12 14 22 33

16

Lp determines the position in the data set where the percentile value

lies, not the percentile itself.

We have already shown how to find the Median, which is the 50th

percentile. It is the 5.5th observation, (8+9)/2=8.5 The 50th percentile

is halfway between the fifth and sixth observations (in the middle

between 8 and 9), that is 8.5.

3.75position

8.25

position2.75

5.5100

50)110(L50

MTH410 S14- Lecture 1 152/153

Interquartile Range

The quartiles can be used to create another

measure of variability, the interquartile range,

which is defined as follows:

The interquartile range measures the spread of the

middle 50% of the observations.

Large values of this statistic mean that the 1st and 3rd

quartiles are far apart indicating a high level of

variability.

Interquartile range = Q3 Q1

MTH410 S14- Lecture 1 153/153

1. It is a summary.

2. It is also a guideline for

selecting techniques.

MTH410 S14 Lecture 01 May 12 -Mo-Wed

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