Data Variables & Unit of Observation

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DataVariables & Unit of Observation

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Statistics(The field of) Statistics is the systematic study of data.

The word “data” is plural…“The data are the price gains of 200 stocks on the

NYSE.”

Singular? “Datum.” (Uncommon.)Shares of Exxon-Mobil gained 2.3%. The datum is 2.3%.

What characterizes data is variability.

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Variables / Units of Observation

Units of observation: Set of entities (things / objects) being studied

Variable: An attribute of units

Suppose X describes a variable and U describes the units.

“X varies among the (statistical) units.”

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Units of ObsMath 158-800 students.

Variable:Gender.

Gender is a Categorical Variable

Gender varies among Math 158-800 students.

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Units of ObsMath 158-800 students.

Variable:Number of FB friends.

Number of FB friends is a Quantitative Variable

Number of FB friends varies among Math 158-800 students.

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1. An experiment was conducted to test the performance of four brands of batteries in three different environments (room temperature; hot and humid; cold). For each combination of brand and environment, batteries were put into a flashlight. The flashlight was then turned on and allowed to run until the light went out. The amount of time until the flashlight stopped shining (in minutes) was recorded. Do brand and environment play a role in the lifetime of these batteries?Minutes are measurement units. Most quantitative variables have a measurement unit. If I want the measurement unit, I’ll say exactly that. By “unit” I mean “unit of observation” = thing / object that is studied.

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2. 55 year old men are recruited into a study about heart attacks. The heart rate of each man is recorded. Each is tracked for a one-year period, and whether or not he has a heart attack is determined.

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3. A student runs an experiment to study the effect of tire pressure on gas mileage. He devises a system so that his car uses gasoline from a one-liter container. Each time the container is filled, he randomly selects a tire pressure between 20 and 35 psi, then drives the car at 60 mph on a divided highway. When he runs out of gas, he records the distance driven on that fill. Does tire pressure impact the distance driven?

Something like “drives” (the noun, not the verb) would also suffice for the units.

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Types of variablesQuantitative VariableNaturally measured as numbers for which ordering and at least some of the usual operations (addition, multiplication, subtraction, etc.) make sense.

DiscreteAll the possible values are easily listedFrequent “ties”Often count or related to counts

ContinuousTechnically: “ties” are impossibleIn practice ties are uncommon

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Types of variablesCategorical (Qualitative) VariableNot quantitative (usually verbal, but sometimes expressed as numbers having little or no number meaning).

Categorical variables are discrete. So, the term discrete is rarely used in speaking about categorical variables – it is redundant.

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DistributionA variable’s distribution is a description of what values it takes and how often it takes them.Categorical Variables

Distributions are always summarized in terms of percents (falling into each category).Graphs include the pie chart, bar chart, and Pareto chart.

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DistributionA variable’s distribution is a description of what values it takes and how often it takes them.Quantitative Variables

There are many ways to summarize quantitative variables.

Graphs: Relative frequency charts and histograms (very similar); boxplots; dotplots; stemplots; etc.Quantitative summaries: Mode; Range; Mean + Standard Deviation; Median + Interquartile Range

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In a data table each unit takes a row; each variable occupies a column.Column headers identify variable names.

There are other ways to organize data, and some are preferable when the idea is to display the data efficiently. However, in most cases, a data table is how data are organized in a spreadsheet.

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Here are the monthly fees (in $) paid by a random sample of 50 users of internet service providers in 2008:

42 31 33 34 65 47 37 38 32 4032 36 31 42 32 32 72 42 45 3741 46 39 38 34 31 41 51 42 3732 42 31 43 40 32 37 34 44 4136 34 45 45 42 35 39 83 30 39

VARIABLE: ____________

UNITS: ____________

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Here are the monthly fees (in $) paid by a random sample of 50 users of internet service providers in 2008:

42 31 33 34 65 47 37 38 32 4032 36 31 42 32 32 72 42 45 3741 46 39 38 34 31 41 51 42 3732 42 31 43 40 32 37 34 44 4136 34 45 45 42 35 39 83 30 39

VARIABLE: Monthly fee (for use of internet)

UNITS: Users of internet service

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User Monthly Fee ($)

User 1* 42

User 2* 31

User 3* 33

:  

*Perhaps identified by name? (Names aren’t given here.)Often, unit identifiers will not be given or displayed.

You can start almost any problem in this course by first asking:

What are the units?What is the variable?

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Variables / Statistical UnitsThe units of observation are the companies listed on the New York Stock Exchange.Describe a variable.Write the sentencevariable_________ varies from company to company.

Is the variable quantitative or categorical?If quantitative, is it discrete or continuous?

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Distribution

The distribution of a variable tells us what values it takes and the likelihood of those values.

User Monthly Fee ($)

User 1* 42

User 2* 31

User 3* 33

:  

What the fees are.

How often those fees occur.

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Car model Vehicle typeTransmission

typeNumber of cylinders

City MPG

Highway MPG

:          

BMW 3030CI Subcompact Automatic 6 19 27

BMW 3030CI Subcompact Manual 6 21 30

Buick Century Midsize Automatic 6 20 29

Chevrolet Blazer 4-wheel drive Automatic 6 15 20

:          

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Car model Vehicle typeTransmission

typeNumber of cylinders

City MPG

Highway MPG

:          

BMW 3030CI Subcompact Automatic 6 19 27

BMW 3030CI Subcompact Manual 6 21 30

Buick Century Midsize Automatic 6 20 29

Chevrolet Blazer 4-wheel drive Automatic 6 15 20

:          

UNITSVARIABLES (there are 5)

varies from unit to unit.X (Variable)

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Car model Vehicle typeTransmission

typeNumber of cylinders

City MPG

Highway MPG

:          

BMW 3030CI Subcompact Automatic 6 19 27

BMW 3030CI Subcompact Manual 6 21 30

Buick Century Midsize Automatic 6 20 29

Chevrolet Blazer 4-wheel drive Automatic 6 15 20

:          

UNITSVARIABLES (there are 5)

varies from car model to car model.City MPG

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Car model Vehicle typeTransmission

typeNumber of cylinders

City MPG

Highway MPG

:          

BMW 3030CI Subcompact Automatic 6 19 27

BMW 3030CI Subcompact Manual 6 21 30

Buick Century Midsize Automatic 6 20 29

Chevrolet Blazer 4-wheel drive Automatic 6 15 20

:          

UNITSVARIABLES (there are 5)

varies from car model to car model.Number of cylinders

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Car model Vehicle typeTransmission

typeNumber of cylinders

City MPG

Highway MPG

:          

BMW 3030CI Subcompact Automatic 6 19 27

BMW 3030CI Subcompact Manual 6 21 30

Buick Century Midsize Automatic 6 20 29

Chevrolet Blazer 4-wheel drive Automatic 6 15 20

:          

UNITSVARIABLES (there are 5)

varies from car model to car model.Transmission type

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Car model Vehicle typeTransmission

typeNumber of cylinders

City MPG

Highway MPG

:          

BMW 3030CI Subcompact Automatic 6 19 27

BMW 3030CI Subcompact Manual 6 21 30

Buick Century Midsize Automatic 6 20 29

Chevrolet Blazer 4-wheel drive Automatic 6 15 20

:          

UNITSVARIABLES (there are 5)

CATEGORICAL VARIABLETransmission type

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Car model Vehicle typeTransmission

typeNumber of cylinders

City MPG

Highway MPG

:          

BMW 3030CI Subcompact Automatic 6 19 27

BMW 3030CI Subcompact Manual 6 21 30

Buick Century Midsize Automatic 6 20 29

Chevrolet Blazer 4-wheel drive Automatic 6 15 20

:          

UNITSVARIABLES (there are 5)

QUANTITATIVE VARIABLECity MPG

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Mutual Fund CategoryNet assets ($ millions)

2008 return Expense Ratio

:    

Fidelity Low-Priced Stock

Small cap value 19,378 -36.2% 0.98%

Price International Stock International 3,828 -48.0% 0.87%

Vanguard 500 Index Large cap blend 74,886 -37.0% 0.15%

:    

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Mutual Fund CategoryNet assets ($ millions)

2008 return Expense Ratio

:    

Fidelity Low-Priced Stock

Small cap value 19,378 -36.2% 0.98%

Price International Stock International 3,828 -48.0% 0.87%

Vanguard 500 Index Large cap blend 74,886 -37.0% 0.15%

:    

VARIABLES (there are 4)UNITS

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The Science of StatisticsData vary

A population is a collection of all the units of interest. If we have information on all the units of a population we have a complete description of the variation in the data. Such a description of a population is a census. Characteristics of populations are parameters.A sample is an incomplete collection of units from the population. A sample necessarily provides incomplete information. Characteristics of samples are called (the word) statistics.

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Variables / Statistical Units

The units of observation are the countries (of the world).Describe a variable.Write the sentence

_________ varies among __________.

Is the variable quantitative or categorical?

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Variables / Statistical Units

The units of observation are the countries (of the world).Describe a variable.Write the sentence

_________ varies among countries.

Is the variable quantitative or categorical?

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GDP per capita and LongevityCountry GDP / person Longevity (years)

Qatar $86006 75.6

U. S. $47440 78.2

Spain $30589 80.9

[world average] $10433 67.2

Haiti $1317 60.9

: :   :

32NOT a unit

GDP per capita and LongevityCountry GDP / person Longevity (years)

Qatar $86006 75.6

U. S. $47440 78.2

Spain $30589 80.9

[world average] $10433 67.2

Haiti $1317 60.9

: :   :

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The Science of StatisticsData vary

A population is a collection of all the units of interest. If we have information on all the units of a population we have a complete description of the variation in the data. Such a description of a population is a census. Characteristics of populations are parameters.A sample is an incomplete collection of units from the population. A sample necessarily provides incomplete information. Characteristics of samples are called (the word) statistics.

34NOT a unit

GDP per capita and LongevityCountry GDP / person Longevity (years)

Qatar $86006 75.6

U. S. $47440 78.2

Spain $30589 80.9

[world average] $10433 67.2

Haiti $1317 60.9

: :   :Parameters

Not statistics

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Purposes of variablesExplanatory and Response VariableChanging the value of the explanatory variable (EV) results in a change in the distribution of the response variable (RV).Loosely: A change in the explanatory variable alters the prediction of the response variable.

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Variable:Form of study.

Units:The (200) college students involved in the experiment.

Form of study varies from student to student.

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Variable:Score on the short answer test.

Units:The (200) college students involved in the experiment.

Score on the short answer test varies from student to student.

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Experimental studyThe explanatory variable is assigned (often by the people conducting the study).Units do not enter the study with a value for this variable.

Observational studyThe explanatory variable is a characteristic of the unit.