Introduction To StatisticsIntroduction To Statistics

Statistics, Science, ad Statistics, Science, ad ObservationsObservations

What are statistics?What are statistics?• The term statistics refers to a set of mathematical The term statistics refers to a set of mathematical

procedures for organizing, summarizing, and interpreting procedures for organizing, summarizing, and interpreting information.information.

• Statistics does the job of assigning a numeric value to Statistics does the job of assigning a numeric value to events that have already taken place. This numeric value events that have already taken place. This numeric value can then be interpreted to mean something (description or can then be interpreted to mean something (description or inference).inference).

Why do we use statistics?Why do we use statistics?• To summarize large amounts of dataTo summarize large amounts of data• To support hypothesesTo support hypotheses• To understand relationshipsTo understand relationships• To answer research questionsTo answer research questions

What do we need in order to do What do we need in order to do statistics?statistics?

What is the elemental content of statistics?What is the elemental content of statistics? DATA!!!!DATA!!!! Data are measurements of observations. A Data are measurements of observations. A

data set is a collection of measurement or data set is a collection of measurement or observations. observations.

A datum is a single measurement or A datum is a single measurement or observation and is commonly called a score observation and is commonly called a score or a raw score.or a raw score.

Populations and SamplesPopulations and Samples

What do we want to do with data?What do we want to do with data? Summarize it!Summarize it!

• For this we use descriptive statisticsFor this we use descriptive statistics Statistical procedures used to summarize, organize and Statistical procedures used to summarize, organize and

simplify data.simplify data.

Use it for predictions, or test hypotheses.Use it for predictions, or test hypotheses.• For this we use inferential statisticsFor this we use inferential statistics

Techniques that allow us to study samples and then Techniques that allow us to study samples and then make generalizations about the populations from which make generalizations about the populations from which they were selected.they were selected.

Where do we get data?Where do we get data?

In general (especially in psychology) data In general (especially in psychology) data is collected from people, or groups of is collected from people, or groups of people.people. PopulationPopulation

• The set of all the individuals of interest in a The set of all the individuals of interest in a particular study.particular study.

Is it feasible to collect data on everyone in a Is it feasible to collect data on everyone in a particular population?particular population?• SampleSample

A set of individuals selected from a population, usually A set of individuals selected from a population, usually intended to represent the population in a research study.intended to represent the population in a research study.

If we are collecting data from samples If we are collecting data from samples of people within a population, can we of people within a population, can we

ever know the exact population values?ever know the exact population values? We cant usually get data from everyone in We cant usually get data from everyone in

a population, so we estimate population a population, so we estimate population values with sample values.values with sample values. ParameterParameter

• A value, usually numerical, that describes a A value, usually numerical, that describes a population. Represented with Greek letters.population. Represented with Greek letters.

StatisticStatistic• A value, usually numerical, that describes a A value, usually numerical, that describes a

sample. Represented with Roman letters.sample. Represented with Roman letters.

How do we take into account error?How do we take into account error?

Sampling errorSampling error The discrepancy, or amount of error, that The discrepancy, or amount of error, that

exists between a sample statistic and the exists between a sample statistic and the corresponding population parameter.corresponding population parameter.

Data Structures, Research Data Structures, Research Methods, and StatisticsMethods, and Statistics

What kinds of research questions (hypotheses) What kinds of research questions (hypotheses) might we have that we can collect data with and might we have that we can collect data with and run statistics on?run statistics on?

What are the components of these hypotheses?What are the components of these hypotheses? ConstructConstruct

• An abstraction that cannot be observed but is useful in An abstraction that cannot be observed but is useful in explaining behaviorexplaining behavior

VariableVariable• A characteristic or condition that changes or has different A characteristic or condition that changes or has different

values for different individualsvalues for different individuals MethodologyMethodology

• The method by which the experiment will answer the The method by which the experiment will answer the question. Experimental, quasi-experimental, correlationalquestion. Experimental, quasi-experimental, correlational

ConstructsConstructs

Can we name a few constructs?Can we name a few constructs? IntelligenceIntelligence FriendlinessFriendliness CheerfulnessCheerfulness

What do we do with constructs?What do we do with constructs? OperationalizationOperationalization

An operationalization (or operational definition) An operationalization (or operational definition) defines a construct in terms of behaviors that can be defines a construct in terms of behaviors that can be measured and observed.measured and observed.

How can we operationalize the constructs I listed How can we operationalize the constructs I listed before?before? IntelligenceIntelligence

• IQIQ FriendlinessFriendliness

• Number of people you smile at in a dayNumber of people you smile at in a day CheerfulnessCheerfulness

• How often you laugh during a dayHow often you laugh during a day

More components of the More components of the hypotheseshypotheses

VariableVariable What are the two types of variables?What are the two types of variables?

• IndependentIndependent The variable used to measure the CAUSAL construct. The variable used to measure the CAUSAL construct.

The independent variable is the variable that is The independent variable is the variable that is manipulated by the researcher.manipulated by the researcher.

• DependentDependent The variable used to assess the affected construct. The The variable used to assess the affected construct. The

dependent variable is the one that is observed in order to dependent variable is the one that is observed in order to assess the effect of the treatment.assess the effect of the treatment.

Our hypothesesOur hypotheses

Of the hypotheses we have listed on the Of the hypotheses we have listed on the board, which are experimental and which board, which are experimental and which are correlationalare correlational CorrelationalCorrelational

• A method where two variables are observed in A method where two variables are observed in order to determine whether there is a relationship order to determine whether there is a relationship between thembetween them

ExperimentalExperimental• A method where one variable is manupulated while A method where one variable is manupulated while

another variable is observed and measuredanother variable is observed and measured

What do you need for an What do you need for an experiment?experiment?

CausationCausation What are the three conditions for causationWhat are the three conditions for causation

• Time priorityTime priority• (co) Relation(co) Relation• Non-spurious relationship (eliminate alternative Non-spurious relationship (eliminate alternative

hypotheseshypotheses

Non-Spurious RelationshipsNon-Spurious Relationships

What are the three conditions to eliminate alternative What are the three conditions to eliminate alternative hypotheses?hypotheses?

Initial equivalence of groups (random assignment)Initial equivalence of groups (random assignment) A manipulated variable (Control over the IV)A manipulated variable (Control over the IV)

• What kinds of variables might we not be able to manipulate?What kinds of variables might we not be able to manipulate? When groups differ by a participant variable (e.g., gender) as opposed When groups differ by a participant variable (e.g., gender) as opposed

to an experimentally manipulated variable, the variable that determines to an experimentally manipulated variable, the variable that determines the groups is called a quasi-independent variable. This leads to a the groups is called a quasi-independent variable. This leads to a quasi-experiment.quasi-experiment.

Control groupsControl groups• What does a control group mean?What does a control group mean?

Individuals in a control condition do not receive the experimental Individuals in a control condition do not receive the experimental treatment. The purpose of a control condition is to provide a baseline treatment. The purpose of a control condition is to provide a baseline for comparison with the experimental conditionfor comparison with the experimental condition

Individuals in the experimental condition do receive the experimental Individuals in the experimental condition do receive the experimental treatment.treatment.

More on variablesMore on variables

What a the scales of measurement for What a the scales of measurement for variables?variables? Levels of measurement and scales have to do Levels of measurement and scales have to do

with the way we measure variables. There with the way we measure variables. There are four levels of measurement commonly are four levels of measurement commonly distinguished: Nominal scales, ordinal scales, distinguished: Nominal scales, ordinal scales, interval scales, and ratio scales. Basically, interval scales, and ratio scales. Basically, these scales differ on the specificity and these scales differ on the specificity and precision of measurement.precision of measurement.

Scales of MeasurementScales of Measurement

Nominal scalesNominal scales These consist of a set of categories that have These consist of a set of categories that have

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

Ordinal scalesOrdinal scales These scales consist of a set of categories that are These scales consist of a set of categories that are

organized in an ordered sequence. Measurements organized in an ordered sequence. Measurements on an ordinal scale rank observations in terms of size on an ordinal scale rank observations in terms of size or magnitudeor magnitude

Scales of Measurement Cont’dScales of Measurement Cont’d

Interval scalesInterval scales These scales consist of ordered categories that are all These scales consist of ordered categories that are all

intervals of exactly the same size. With an interval intervals of exactly the same size. With an interval scale, equal differences between numbers on the scale, equal differences between numbers on the scale reflect equal differences in magnitude. scale reflect equal differences in magnitude. However, ratios of magnitudes are not meaningful.However, ratios of magnitudes are not meaningful.

Ratio ScalesRatio Scales These scales are interval scales with the additional These scales are interval scales with the additional

feature of an absolute zero point. With a ratio scale, feature of an absolute zero point. With a ratio scale, ratios of numbers do reflect ratios of magnitude.ratios of numbers do reflect ratios of magnitude.

What are some examples of What are some examples of variables?variables?

Nominal ScalesNominal Scales Male versus femaleMale versus female Registered voter? (Yes/No)Registered voter? (Yes/No)

Ordinal ScalesOrdinal Scales How much autonomy do you have at work?How much autonomy do you have at work?

• 1 – A little; 2 – Some; 3 – A lot1 – A little; 2 – Some; 3 – A lot Interval ScalesInterval Scales

Temperature on Celsius of FahrenheitTemperature on Celsius of Fahrenheit Income as a measure of social statusIncome as a measure of social status

• Is the difference in status the same for 20 – 50k as it is for 1m – 1.03m?Is the difference in status the same for 20 – 50k as it is for 1m – 1.03m? Ratio ScalesRatio Scales

Temperature on KelvinTemperature on Kelvin Measures of income using salaryMeasures of income using salary Weight, heightWeight, height

Scales of Measurement Cont’dScales of Measurement Cont’d

What scale do they sell popcorn at a movie What scale do they sell popcorn at a movie (small, medium, large)?(small, medium, large)? OrdinalOrdinal

The SAT measures aptitude on what scale?The SAT measures aptitude on what scale? IntervalInterval

What color are your eyes?What color are your eyes? NominalNominal

How many peanuts have you eaten today?How many peanuts have you eaten today? RatioRatio

Statistical NotationStatistical Notation

If you are worried about your math skills, please If you are worried about your math skills, please review appendix Areview appendix A

Variables are represented as X (IV’s) and Y Variables are represented as X (IV’s) and Y (DV’s)(DV’s)

N’s will represent the number of scoresN’s will represent the number of scores Upper case N is the number of total scoresUpper case N is the number of total scores Lower case n is the number of group scoresLower case n is the number of group scores

Subscripts will not group differencesSubscripts will not group differences nn11 notes the number of observations in group 1, while notes the number of observations in group 1, while

nn22 notes the number of observations in group 2. notes the number of observations in group 2.

Summation notationSummation notation ΣΣ represents a summation. This means we will take the represents a summation. This means we will take the

sum of the mathematical expression that follows.sum of the mathematical expression that follows. For exampleFor example

ΣΣ X X ΣΣ (X-1) (X-1) ΣΣ (X2 – 1) (X2 – 1) ΣΣ X2 – 1 X2 – 1 ((ΣΣ X)2 X)2 ΣΣ (XY) (XY)

Play close attention to the order of operations and Play close attention to the order of operations and especially parenthesesespecially parentheses

Find the value for the above expressions where:Find the value for the above expressions where: X {1,3,4,5}X {1,3,4,5} Y {2,4,3,7}Y {2,4,3,7}