Chapter 1Chapter 1

Introduction to StatisticsIntroduction to Statistics

Statistical MethodsStatistical Methods

Were developed to serve a purposeWere developed to serve a purpose The purpose for each statistical The purpose for each statistical

procedure provides a background or procedure provides a background or context for the details of the context for the details of the formulas and calculationsformulas and calculations

If you understand why the procedure If you understand why the procedure is needed, you will find it easier to is needed, you will find it easier to learn the procedure.learn the procedure.

Definition of StatisticsDefinition of Statistics

Facts and figuresFacts and figures Refers to a set of methods and rules Refers to a set of methods and rules

for organizing, summarizing, and for organizing, summarizing, and interpreting informationinterpreting information

StatisticsStatistics

Statistical procedures help ensure Statistical procedures help ensure that the information or observations that the information or observations are presented and interpreted in an are presented and interpreted in an accurate and informative way.accurate and informative way.

Statistics provide researchers with a Statistics provide researchers with a set of standardized techniques that set of standardized techniques that are recognized and understood are recognized and understood throughout the scientific community.throughout the scientific community.

PopulationPopulation

The set of all individuals of interest The set of all individuals of interest for a particular study.for a particular study.

Populations are usually largePopulations are usually large Ex. The number of women on the planetEx. The number of women on the planet

Populations can be limitedPopulations can be limited The number of female registered voters The number of female registered voters

in the U.S.in the U.S.

Population (cont.)Population (cont.)

The population can vary in size from The population can vary in size from large to smalllarge to small

The population should always be The population should always be identified by the researcheridentified by the researcher

Populations do not have to be peoplePopulations do not have to be people

Population (cont.)Population (cont.)

It is impossible to examine every It is impossible to examine every individual in a populationindividual in a population

Researchers typically select a Researchers typically select a smaller more manageable group – smaller more manageable group – called a samplecalled a sample

SampleSample

A sample is intended to represent a A sample is intended to represent a populationpopulation

A sample should always be identified A sample should always be identified in terms of the population from which in terms of the population from which it was selectedit was selected

Samples can vary in sizeSamples can vary in size

Sample of ScoresSample of Scores

The book will discuss a sample of The book will discuss a sample of scoresscores

Each sample of individuals produces Each sample of individuals produces a corresponding sample (or a corresponding sample (or population) of scorespopulation) of scores

Occasionally a set of scores is called Occasionally a set of scores is called a statistical population or a statistical a statistical population or a statistical sample to differentiate from a sample to differentiate from a population or sample of individualspopulation or sample of individuals

Parameters and StatisticsParameters and Statistics

Always distinguish if the data come Always distinguish if the data come from a population or a samplefrom a population or a sample

A characteristic that describes a A characteristic that describes a population is called a parameterpopulation is called a parameter Example: the population averageExample: the population average

A characteristic that describes a A characteristic that describes a sample is called a statisticsample is called a statistic Example: the average scoreExample: the average score

Parameters and StatisticsParameters and Statistics

Typically the research process begins Typically the research process begins with a question about a population with a question about a population parameter parameter

The actual data come from a sample The actual data come from a sample and are used to compute sample and are used to compute sample statisticsstatistics

ParameterParameter

A parameter is a value, usually a A parameter is a value, usually a numerical value, that describes a numerical value, that describes a populationpopulation

May be obtained from a single May be obtained from a single measurementmeasurement

Or may be derived from a set of Or may be derived from a set of measurements from the populationmeasurements from the population

StatisticStatistic

A statistic is a value, usually a A statistic is a value, usually a numerical value that describes a numerical value that describes a samplesample

May be obtained from a single May be obtained from a single measurement, or it may be derived measurement, or it may be derived from a set of measurments from the from a set of measurments from the samplesample

Descriptive and Inferential Descriptive and Inferential Statistical MethodsStatistical Methods

After data are obtained, statistical After data are obtained, statistical methods are used to organize and methods are used to organize and interpret the datainterpret the data

Methods are categorized into two Methods are categorized into two general methods descriptive and general methods descriptive and inferentialinferential

Descriptive StatisticsDescriptive Statistics

Statistical procedures that are used Statistical procedures that are used to summarize, organize, and simplify to summarize, organize, and simplify datadata

Inferential StatisticsInferential Statistics

Consist of techniques that allow us to Consist of techniques that allow us to study samples and then make study samples and then make generalizations about the generalizations about the populations from which they were populations from which they were selectedselected

One problem with using samples – One problem with using samples – provide only limited information provide only limited information about the populationabout the population

Inferential Statistics (cont.)Inferential Statistics (cont.)

Samples should be representative of its Samples should be representative of its populationpopulation General characteristics should be General characteristics should be

consistent with the characteristics of the consistent with the characteristics of the populationpopulation

There is usually some discrepancy or There is usually some discrepancy or sampling error that needs to be sampling error that needs to be addressed when using inferential addressed when using inferential statisticsstatistics

Sampling ErrorSampling Error

The discrepancy, or amount of error, The discrepancy, or amount of error, that exists between a sample statistic that exists between a sample statistic and the corresponding population and the corresponding population parameterparameter The statistic obtained using a sample will The statistic obtained using a sample will

not be the same if you measured the entire not be the same if you measured the entire populationpopulation

Two or more samples within a population Two or more samples within a population will also find different statisticswill also find different statistics

Margin of ErrorMargin of Error

The margin of error is the sampling The margin of error is the sampling errorerror

FigureFigure 1.1 1.1

The role of statistics in researchThe role of statistics in research

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Step 1: Data Collection Step 2: Descriptive Statistics

Step 3: Inferential Statistics: Interpret the results

The Scientific Method and the The Scientific Method and the Design of Research StudiesDesign of Research Studies

Variable Variable A characteristic or condition that changes A characteristic or condition that changes

or has different values for different or has different values for different individualsindividuals

Example 1 – height, weight, gender, SESExample 1 – height, weight, gender, SES Example 2 – Environmental conditions – Example 2 – Environmental conditions –

temperature, time of day, size of the roomtemperature, time of day, size of the room When variables are measured, the resulting When variables are measured, the resulting

values are often identified by letters (i.e. values are often identified by letters (i.e. X,Y)X,Y)

ConstantConstant

A value that does not change is A value that does not change is called a constantcalled a constant

A constant is a characteristic or A constant is a characteristic or condition that does not vary but is condition that does not vary but is the same for every individualthe same for every individual

Science involves a search for Science involves a search for relationships between variablesrelationships between variables

CorrelationsCorrelations

The simplest way to look for The simplest way to look for relationships between variables is to relationships between variables is to make observations of the two make observations of the two variables as they exist naturally for a variables as they exist naturally for a set of individualsset of individuals

This is called the correlational This is called the correlational methodmethod

Correlational MethodCorrelational Method

With the correlational method, two variables With the correlational method, two variables are observed to see whether there is a are observed to see whether there is a relationshiprelationship

The correlational method involves measuring The correlational method involves measuring two different variables for each individualtwo different variables for each individual

After measurements are obtained, the After measurements are obtained, the researcher examines the data to see researcher examines the data to see whether there are any consistent trends or whether there are any consistent trends or patternspatterns

Correlational Method (cont.)Correlational Method (cont.)

It is often tempting to conclude that It is often tempting to conclude that one variable is causing changes in the one variable is causing changes in the other variableother variable

However, this conclusion is not justifiedHowever, this conclusion is not justified A A limitationlimitation of the correlational method of the correlational method

is that it simply describes the is that it simply describes the relationship – it does not explain the relationship – it does not explain the cause and effect mechanism of the cause and effect mechanism of the relationshiprelationship

To establish a cause and effect To establish a cause and effect relationship, it is necessary to exert a relationship, it is necessary to exert a much greater level of control over much greater level of control over the variables being studied. the variables being studied.

This is accomplished by the This is accomplished by the experimental methodexperimental method Highly structuredHighly structured Systematic approach to the study of the Systematic approach to the study of the

relationships between variablesrelationships between variables

The Experimental MethodThe Experimental Method

Goal is to establish a cause-and-Goal is to establish a cause-and-effect relationship between two effect relationship between two variablesvariables

Method is intended to show that Method is intended to show that changes in one variable are caused changes in one variable are caused by changes in the other variableby changes in the other variable

The Experimental Method The Experimental Method (cont.)(cont.)

Two Distinguishing CharacteristicsTwo Distinguishing Characteristics The researcher manipulates on of the The researcher manipulates on of the

variables and observes the second variables and observes the second variable to determine whether or not the variable to determine whether or not the manipulation causes changes to occurmanipulation causes changes to occur

The researcher must exercise some The researcher must exercise some control over the research situation to control over the research situation to ensure that other, extraneous variables ensure that other, extraneous variables do not influence the relationship being do not influence the relationship being examinedexamined

FigureFigure 1.2 1.2

An example of the experimental methodAn example of the experimental method

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The Experimental Method The Experimental Method (cont.)(cont.)

The researcher must rule out any The researcher must rule out any other possible explanations for the other possible explanations for the difference to say that temperature difference to say that temperature affects memory performanceaffects memory performance Random assignment of individualsRandom assignment of individuals

Helps assure that subjects are not very Helps assure that subjects are not very different from the other groupdifferent from the other group

Treatment conditions must be identicalTreatment conditions must be identical Hold all other variables constantHold all other variables constant

Independent VariableIndependent Variable

The variable that is manipulated by The variable that is manipulated by the researcher. In behavioral the researcher. In behavioral research, the independent variable research, the independent variable usually consists of the two (or more) usually consists of the two (or more) treatment conditions to which treatment conditions to which subjects are exposed.subjects are exposed.

Dependent VariableDependent Variable

The one that is observed for changes The one that is observed for changes in order to assess the effect of the in order to assess the effect of the treatmenttreatment In psychological research, the In psychological research, the

dependent variable is typically a dependent variable is typically a measurement or score obtained for each measurement or score obtained for each subject.subject.

Experiment and Experiment and CorrelationalCorrelational

In an experimental study, one In an experimental study, one variable is actually measuredvariable is actually measured

In a correlational study, where both In a correlational study, where both variables are measured, the data variables are measured, the data consist of two separate scores for consist of two separate scores for each individualeach individual

Control and Experimental Control and Experimental ConditionCondition

Individuals in a control condition do Individuals in a control condition do not receive the experimental not receive the experimental treatment. They receive either no treatment. They receive either no treatment, or a control or placebo treatment, or a control or placebo treatmenttreatment

In an experimental condition, In an experimental condition, individuals receive the experimental individuals receive the experimental treatmenttreatment

Quasi-Experimental MethodQuasi-Experimental Method

Instead of using an independent variable Instead of using an independent variable to create treatment conditions, a quasi-to create treatment conditions, a quasi-experimental research study uses a non-experimental research study uses a non-manipulated variable to define the manipulated variable to define the conditions that are being compared.conditions that are being compared.

The non-manipulated variable is usually a The non-manipulated variable is usually a subject variable (male vs. female) or a subject variable (male vs. female) or a time variable (before vs. after treatment).time variable (before vs. after treatment).

FigureFigure 1.3 1.3

Examples of the quasi-experimental methodExamples of the quasi-experimental method

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Scales of MeasurementScales of Measurement

The distinctions among the scales The distinctions among the scales are important because they are important because they underscore the limitation of certain underscore the limitation of certain types of measurements and because types of measurements and because certain statistical procedures are certain statistical procedures are appropriate for data collected on appropriate for data collected on some scales but not on others.some scales but not on others.

Nominal ScaleNominal Scale

A nominal scale consists of a set of categories A nominal scale consists of a set of categories that have different names.that have different names.

Measurements on a nominal scale label and Measurements on a nominal scale label and categorize observations but do not make any categorize observations but do not make any quantitative distinctions between quantitative distinctions between observations.observations.

A nominal scale consists of qualitative A nominal scale consists of qualitative differences. It does not provide information differences. It does not provide information about quantitative differences between about quantitative differences between individuals.individuals.

Ordinal ScaleOrdinal Scale

An ordinal scale consists of a set of An ordinal scale consists of a set of categories that are organized in an categories that are organized in an ordered sequence.ordered sequence.

Measurements on an ordinal scale Measurements on an ordinal scale rank observations in terms of size or rank observations in terms of size or magnitude.magnitude.

However, the data do not reveal how However, the data do not reveal how much better.much better.

Interval ScalesInterval Scales

An interval scale of measurement An interval scale of measurement consists of an ordered set of consists of an ordered set of categories (like an ordinal scale), categories (like an ordinal scale), with the additional requirement that with the additional requirement that the categories form a series of the categories form a series of intervals that are all exactly the intervals that are all exactly the same size.same size.

Allows you to measure how much Allows you to measure how much difference there is between two difference there is between two individual scores.individual scores.

Ratio ScalesRatio Scales

A ration scale is an interval scale A ration scale is an interval scale with the additional feature of an with the additional feature of an absolute zero point.absolute zero point.

In a ratio scale, ratios of numbers In a ratio scale, ratios of numbers reflect ratios of magnitude.reflect ratios of magnitude.

Discrete VariablesDiscrete Variables

A discrete variable consists of A discrete variable consists of separate, indivible categories.separate, indivible categories.

No values can exist between two No values can exist between two neighboring categories.neighboring categories. Ex. Numbers of students in a classEx. Numbers of students in a class 18 or 19 students, not 18.25 students18 or 19 students, not 18.25 students

Continuous VariablesContinuous Variables

A continuous variable is divisible into A continuous variable is divisible into an infinite number of fractional parts.an infinite number of fractional parts.

There are an infinite number of There are an infinite number of possible values that fall between any possible values that fall between any two observed values.two observed values. Ex. Amount of time to complete a task.Ex. Amount of time to complete a task.

FigureFigure 1.4 1.4

Representing time on a continuous number lineRepresenting time on a continuous number line

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Real LimitsReal Limits Real limits are the boundaries of intervals Real limits are the boundaries of intervals

for scores that are represented on a for scores that are represented on a continuous number line. continuous number line.

The real limit separating two adjacent The real limit separating two adjacent scores is located exactly halfway scores is located exactly halfway between the scores.between the scores.

Each score has two real limits – The Each score has two real limits – The upper real limit and the lower real limit.upper real limit and the lower real limit. Ex. Used for constructing graphs and for Ex. Used for constructing graphs and for

various calculations with continuous scales.various calculations with continuous scales.

Figure 1.5Figure 1.5

An illustration of real limitsAn illustration of real limits

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Summation NotationSummation Notation

The Greek letter sigma, or The Greek letter sigma, or E, E, is used is used to stand for summation.to stand for summation.

The expression The expression EEX means to add all X means to add all the scores for variable X. the scores for variable X.

Read as “the sum of.”Read as “the sum of.”