Data CollectionMATH 130, Elements of Statistics I

J. Robert Buchanan

Department of Mathematics

Fall 2017

Big Definition

DefinitionStatistics is the science of

I collecting,I organizing,I summarizing, andI analyzing

information to draw conclusions or answer questions. Statisticsalso provides a measure of confidence in any conclusions.

Process of Statistics

1. Identify the research objective (the question to beanswered).The group to be studied is called a population which iscomposed of individuals.

2. Collect the data needed to answer the question.Data are collected from a subset of the population called asample. A statistic is a numerical summary of a sample.

3. Organize, summarize, and describe the data (calleddescriptive statistics).

4. Perform an inference or draw conclusions about thepopulation from the information (called inferentialstatistics). A parameter is a numerical summary of apopulation.

Process of Statistics

1. Identify the research objective (the question to beanswered).The group to be studied is called a population which iscomposed of individuals.

2. Collect the data needed to answer the question.Data are collected from a subset of the population called asample. A statistic is a numerical summary of a sample.

3. Organize, summarize, and describe the data (calleddescriptive statistics).

4. Perform an inference or draw conclusions about thepopulation from the information (called inferentialstatistics). A parameter is a numerical summary of apopulation.

Process of Statistics

1. Identify the research objective (the question to beanswered).The group to be studied is called a population which iscomposed of individuals.

2. Collect the data needed to answer the question.Data are collected from a subset of the population called asample. A statistic is a numerical summary of a sample.

3. Organize, summarize, and describe the data (calleddescriptive statistics).

4. Perform an inference or draw conclusions about thepopulation from the information (called inferentialstatistics). A parameter is a numerical summary of apopulation.

Process of Statistics

1. Identify the research objective (the question to beanswered).The group to be studied is called a population which iscomposed of individuals.

2. Collect the data needed to answer the question.Data are collected from a subset of the population called asample. A statistic is a numerical summary of a sample.

3. Organize, summarize, and describe the data (calleddescriptive statistics).

4. Perform an inference or draw conclusions about thepopulation from the information (called inferentialstatistics). A parameter is a numerical summary of apopulation.

Example (1 of 2)The AP-National Constitution Center conducted a poll August11-16, 2010, to learn how adult Americans feel existinggun-control laws infringe on the second amendment to the U.S.Constitution.

1. Identify the research objective.

The researchers wished todetermine the percentage of adult Americans who believegun-control laws infringe on the public’s right to bear arms.The population being studied was adult Americans.

2. Collect the data.

It is unreasonable to expect to surveymore than 200 million adult Americans, so the researcherssurveyed a sample of 1007 adult Americans. Of thosesurveyed, 514 stated they believe existing gun-control lawsinfringe on the public’s right to bear arms.

3. Describe the data.

Of the 1007 individuals in the survey,51% ( 514

1007 · 100%) believe existing gun-control laws infringeon the public’s right to bear arms. This is a statisticbecause its value was determined from a sample.

Example (1 of 2)The AP-National Constitution Center conducted a poll August11-16, 2010, to learn how adult Americans feel existinggun-control laws infringe on the second amendment to the U.S.Constitution.

1. Identify the research objective. The researchers wished todetermine the percentage of adult Americans who believegun-control laws infringe on the public’s right to bear arms.The population being studied was adult Americans.

2. Collect the data.

It is unreasonable to expect to surveymore than 200 million adult Americans, so the researcherssurveyed a sample of 1007 adult Americans. Of thosesurveyed, 514 stated they believe existing gun-control lawsinfringe on the public’s right to bear arms.

3. Describe the data.

Of the 1007 individuals in the survey,51% ( 514

1007 · 100%) believe existing gun-control laws infringeon the public’s right to bear arms. This is a statisticbecause its value was determined from a sample.

Example (1 of 2)The AP-National Constitution Center conducted a poll August11-16, 2010, to learn how adult Americans feel existinggun-control laws infringe on the second amendment to the U.S.Constitution.

1. Identify the research objective. The researchers wished todetermine the percentage of adult Americans who believegun-control laws infringe on the public’s right to bear arms.The population being studied was adult Americans.

2. Collect the data. It is unreasonable to expect to surveymore than 200 million adult Americans, so the researcherssurveyed a sample of 1007 adult Americans. Of thosesurveyed, 514 stated they believe existing gun-control lawsinfringe on the public’s right to bear arms.

3. Describe the data.

Of the 1007 individuals in the survey,51% ( 514

1007 · 100%) believe existing gun-control laws infringeon the public’s right to bear arms. This is a statisticbecause its value was determined from a sample.

Example (1 of 2)The AP-National Constitution Center conducted a poll August11-16, 2010, to learn how adult Americans feel existinggun-control laws infringe on the second amendment to the U.S.Constitution.

1. Identify the research objective. The researchers wished todetermine the percentage of adult Americans who believegun-control laws infringe on the public’s right to bear arms.The population being studied was adult Americans.

2. Collect the data. It is unreasonable to expect to surveymore than 200 million adult Americans, so the researcherssurveyed a sample of 1007 adult Americans. Of thosesurveyed, 514 stated they believe existing gun-control lawsinfringe on the public’s right to bear arms.

3. Describe the data. Of the 1007 individuals in the survey,51% ( 514

1007 · 100%) believe existing gun-control laws infringeon the public’s right to bear arms. This is a statisticbecause its value was determined from a sample.

Example (2 of 2)

4. Perform inference.

The researchers at the AP-NationalConstitution Center wanted to extend the results of thesurvey to all adult Americans. Remember, whengeneralizing results from a sample to a population, theresults are uncertain. To account for this uncertainty,researchers reported a 3% margin of error. This meansthe researchers feel fairly certain (in fact 95% certain) thatthe percentage of all adult Americans who believe existinggun-control laws infringe on the public’s right to bear armsis somewhere between 48% and 54%.

Example (2 of 2)

4. Perform inference. The researchers at the AP-NationalConstitution Center wanted to extend the results of thesurvey to all adult Americans. Remember, whengeneralizing results from a sample to a population, theresults are uncertain. To account for this uncertainty,researchers reported a 3% margin of error. This meansthe researchers feel fairly certain (in fact 95% certain) thatthe percentage of all adult Americans who believe existinggun-control laws infringe on the public’s right to bear armsis somewhere between 48% and 54%.

Variables

DefinitionA variable is a characteristic of interest about the individuals ina population.

Example

I heightI weightI hair colorI incomeI zip code

Variables

DefinitionA variable is a characteristic of interest about the individuals ina population.

Example

I heightI weightI hair colorI incomeI zip code

Qualitative vs. Quantitative Variables

DefinitionQualitative (or categorical) variables allow for classification ofindividuals based on some attribute or characteristic.

DefinitionQuantitative variables provide numerical measures ofindividuals. The values of a quantitative variable can be addedor subtracted and provide meaningful results.

Example

Identify each of the following as an example of a qualitative orquantitative variable.

1. The breaking strength of a piece of string.2. The number of stop signs in towns of less than 500 people.3. The hair color of children auditioning for a play.4. Whether or not a faucet is defective.5. The number of questions answered correctly on a

standardized test.6. The length of time spent on hold to have a question

answered by the help desk via telephone.

Continuous vs. Discrete Variables

DefinitionA discrete variable is a quantitative variable that has either afinite number of possible values or a countable number ofpossible values. The term countable means that the valuesresult from counting 0, 1, 2, 3, and so on. Discrete variablevalues generally have gaps between them.

DefinitionA continuous variable is a quantitative variable that has aninfinite number of possible values that are not countable. Thereare generally no gaps between the values that a continuousvariable may take on.

Example

Identify each of the following as an example of a discrete orcontinuous variable.

1. A poll of registered voters about which candidate they willsupport.

2. The length of time required for a wound to heal after abandage is applied.

3. The number of telephone calls received by a help desk in a10-minute period.

4. The distance freshmen can kick a football.5. The number of pages in term papers written for ENGL 110.6. The kind of tree used as a Christmas tree.

Variables, Values, and Data

The list of observed values for a variable is data.I Qualitative data are observations corresponding to a

qualitative variable.I Quantitative data are observations corresponding to a

quantitative variable.I Discrete data are observations corresponding to a

discrete variable.I Continuous data are observations corresponding to a

continuous variable.

Example

Country Government Type Life Expectancy Population(years) (in millions)

Australia Federal parliamentarydemocracy 81.81 21.8

Canada Constitutional monarchy 81.38 34.0France Republic 81.19 65.3Morocco Constitutional monarchy 75.90 32.0Poland Republic 76.05 38.4Sri Lanka Republic 75.73 21.3United States Federal republic 78.37 313.2

Identify the individuals, variables, and data in this table.

Level of Measurement of a VariableI A variable is at the nominal level of measurement if the

values of the variable name, label, or categorize.I A variable is at the ordinal level of measurement if it has

the properties of the nominal level of measurement and thevalues of the variable can be ranked.

I A variable is at the interval level of measurement if it hasthe properties of the ordinal level of measurement anddifferences in the values of the variable have meaning. Avalue of zero does not mean the absence of the quantity.Arithmetic operations such as addition and subtraction canbe performed on the values of the variable.

I A variable is at the ratio level of measurement if it has theproperties of the interval level of measurement and ratiosof the values of the variables have meaning. A value ofzero means the absence of the quantity. Arithmeticoperations such as multiplication and division can beperformed on the values of the variable.

Determining the Level of MeasurementIdentify the level of measurement for each of the following typesof data.

1. In a sample of stereo speakers, 12 were rated “good”, 16were rated “better”, and 8 were rated “best”.

2. Zip codes.3. Weights of plastic discarded by households.4. The years 1000, 2000, 1776, 1941, and 1996.5. Movies listed according to their genre, such as comedy,

adventure, romance, martial arts.6. Room temperatures (measured in ◦C).7. Math courses College Algebra, Elements of Statistics, and

Business Calculus.8. Lengths (measured in minutes) of movies.9. Social security numbers.

10. Differences in skiing times between skiers in the down-hillslalom event and the current world record.