Populations vs. Samples Populations vs. Samples Who = Who = Population Population : : all individuals of interest all individuals of interest US Voters, Dentists, College students, US Voters, Dentists, College students, Children Children What = P What = P arameter arameter Characteristic of population Characteristic of population Problem: can Problem: can ’ ’ t study/survey whole t study/survey whole pop pop Solution: Use a Solution: Use a sample sample for the for the “ “ who who ” ” subset, selected from population subset, selected from population calculate a calculate a statistic statistic for the for the “ “ what what ” ”
Transcript
Populations vs. SamplesPopulations vs. Samples Who = Who = PopulationPopulation: :
all individuals of interestall individuals of interest US Voters, Dentists, College students, ChildrenUS Voters, Dentists, College students, Children
What = PWhat = Parameterarameter Characteristic of populationCharacteristic of population
Problem: canProblem: can’’t study/survey whole t study/survey whole poppop
Solution: Use a Solution: Use a samplesample for the for the ““whowho”” subset, selected from population subset, selected from population calculate a calculate a statisticstatistic for the for the ““whatwhat””
Types of SamplingTypes of Sampling
Simple Random SamplingSimple Random Sampling Stratified Random SamplingStratified Random Sampling Cluster SamplingCluster Sampling Systematic SamplingSystematic Sampling Representative SamplingRepresentative Sampling
(Can be stratified random or quota sampling)(Can be stratified random or quota sampling)
Convenience or Haphazard SamplingConvenience or Haphazard Sampling Sampling with Replacement vs. Sampling Sampling with Replacement vs. Sampling
without Replacementwithout Replacement
Representative SampleRepresentative Sample
Sample should be Sample should be representativerepresentative of the target populationof the target population so you can so you can generalizegeneralize to population to population
Random samplingRandom sampling All members of pop have equal chance All members of pop have equal chance
of being selectedof being selected Roll dice, flip coin, draw from hatRoll dice, flip coin, draw from hat
Types of StatisticsTypes of Statistics
DescriptiveDescriptive statistics: statistics: Organize and summarize scores Organize and summarize scores from from
samplessamples
InferentialInferential statistics: statistics: Infer information Infer information about the populationabout the population
based on what we know from sample based on what we know from sample datadata
Decide if an experimental manipulation Decide if an experimental manipulation has had an effecthas had an effect
THE BIG PICTURE OF THE BIG PICTURE OF STATISTICSSTATISTICSTheoryTheory
QuestionQuestion to answer / to answer / HypothesisHypothesis to test to test
Variable: reflects construct, but is directly measurable and can differ from subject to subject (not a constant). Variables can be Discrete or Continuous.
Operational Definition: concrete, measurable
Defines variable by specific operations used to measure it
WISCSAT Vocab
TestGrades
Some Definitions:Some Definitions:
Types of VariablesTypes of Variables
QuantitativeQuantitative Measured in Measured in
amountsamounts Ht, Wt, Test scoreHt, Wt, Test score
QualitativeQualitative Measured in categoriesMeasured in categories Gender, race, diagnosisGender, race, diagnosis
ContinuousContinuous::infinite values in infinite values in
betweenbetweenGPA GPA
Scales of MeasurementScales of Measurement
Nominal Scale:Nominal Scale: Categories, labels, data carry no numerical Categories, labels, data carry no numerical valuevalue
Ordinal Scale:Ordinal Scale: Rank ordered data, but no information about Rank ordered data, but no information about distance between ranksdistance between ranks
Interval Scale:Interval Scale: Degree of distance between scores can be Degree of distance between scores can be assessed with standard sized intervalsassessed with standard sized intervals
Ratio Scale:Ratio Scale: Same as interval scale with an absolute zero Same as interval scale with an absolute zero point.point.
Sigma NotationSigma Notation
= summation= summation X = VariableX = Variable What do each of these Mean?What do each of these Mean?
XX X + 2 versus X + 2 versus (X + 2)(X + 2) XX2 2 versus (versus (X)X)22
(X + 2)(X + 2)2 2 versus versus (X(X2 2 + 2) + 2)
Types of ResearchTypes of Research
Correlational MethodCorrelational Method
No manipulation: just observe 2+ No manipulation: just observe 2+ variables, then measure relationshipvariables, then measure relationship
Also called:Also called: DescriptiveDescriptive Non-Non-
ManipulateManipulate one variable (independent) one variable (independent) MeasureMeasure effects on another (dependent) effects on another (dependent) Evaluate the relationshipEvaluate the relationship
Values of DV (presumed to be)Values of DV (presumed to be) ““dependentdependent”” on the level of the IV on the level of the IV
How do we know DV depends on IV?How do we know DV depends on IV? ControlControl Random assignment Random assignment equivalent groups equivalent groups
Experimental ValidityExperimental Validity
Internal ValidityInternal Validity:: Is the experiment free from Is the experiment free from
confoundingconfounding? ?
External ValidityExternal Validity: : How well can the results be generalized How well can the results be generalized
Still study effects of IV on DV, Still study effects of IV on DV, but:but:
IV not manipulated IV not manipulated groups/conditions/levels are differentgroups/conditions/levels are different natural groups (sex, race)natural groups (sex, race) ethical reasons (child abuse, drug use)ethical reasons (child abuse, drug use)
Participants Participants ““selectselect”” themselves into themselves into groupsgroups
IV not manipulated IV not manipulated
no random assignmentno random assignment
What Do we mean by the term What Do we mean by the term ““Statistical Significance?Statistical Significance?””
Control over situation (over IV, random Control over situation (over IV, random assignment, etc)assignment, etc)
Provides sensitive measurement of DVProvides sensitive measurement of DV
Disadvantages:Disadvantages: Subjects know they are being studied, leading to Subjects know they are being studied, leading to
demand characteristicsdemand characteristics Limitations on manipulationsLimitations on manipulations Questions of reality & generalizability Questions of reality & generalizability
Research SettingsResearch Settings
Field Studies:Field Studies: Advantages:Advantages:
Minimizes suspicionMinimizes suspicion Access to different subject populationsAccess to different subject populations Can study more powerful variablesCan study more powerful variables
Disadvantages:Disadvantages: Lack of control over situationLack of control over situation Random Assignment may not be possibleRandom Assignment may not be possible May be difficult to get pure measures of the DVMay be difficult to get pure measures of the DV Can be more costly and awkwardCan be more costly and awkward
Frequency DistributionsFrequency Distributions
Frequency DistributionsFrequency Distributions
After collecting data, the first task for a After collecting data, the first task for a researcher is to organize and simplify the researcher is to organize and simplify the data so that it is possible to get a general data so that it is possible to get a general overview of the results. overview of the results.
One method for simplifying and organizing One method for simplifying and organizing data is to construct a data is to construct a frequency frequency distributiondistribution. .
Frequency Distribution Frequency Distribution TablesTables
A A frequency distribution tablefrequency distribution table consists of at consists of at least two columns - one listing categories on the least two columns - one listing categories on the scale of measurement (X) and another for scale of measurement (X) and another for frequency (f). frequency (f).
In the X column, values are listed from the In the X column, values are listed from the highest to lowest, without skipping any. highest to lowest, without skipping any.
For the frequency column, tallies are determined For the frequency column, tallies are determined for each value (how often each X value occurs in for each value (how often each X value occurs in the data set). These tallies are the frequencies the data set). These tallies are the frequencies for each X value. for each X value.
The sum of the frequencies should equal N. The sum of the frequencies should equal N.
Frequency Distribution Tables Frequency Distribution Tables (cont.)(cont.)
A third column can be used for the A third column can be used for the proportion (p) for each category: p = f/N. proportion (p) for each category: p = f/N. The sum of the p column should equal The sum of the p column should equal 1.00. 1.00.
A fourth column can display the A fourth column can display the percentage of the distribution percentage of the distribution corresponding to each X value. The corresponding to each X value. The percentage is found by multiplying p by percentage is found by multiplying p by 100. The sum of the percentage column is 100. The sum of the percentage column is 100%. 100%.
Regular Frequency Regular Frequency DistributionDistribution
When a frequency distribution table lists all When a frequency distribution table lists all of the individual categories (X values) it is of the individual categories (X values) it is called a called a regular frequency distributionregular frequency distribution. .
Grouped Frequency Grouped Frequency DistributionDistribution
Sometimes, however, a set of scores Sometimes, however, a set of scores covers a wide range of values. In these covers a wide range of values. In these situations, a list of all the X values would situations, a list of all the X values would be quite long - too long to be a be quite long - too long to be a ““simplesimple”” presentation of the data. presentation of the data.
To remedy this situation, a To remedy this situation, a grouped grouped frequency distributionfrequency distribution table is used. table is used.
Grouped Frequency Distribution Grouped Frequency Distribution (cont.)(cont.)
In a grouped table, the X column lists groups of In a grouped table, the X column lists groups of scores, called scores, called class intervalsclass intervals, rather than , rather than individual values. individual values.
These intervals all have the same width, usually These intervals all have the same width, usually a simple number such as 2, 5, 10, and so on. a simple number such as 2, 5, 10, and so on.
Each interval begins with a value that is a Each interval begins with a value that is a multiple of the interval width. The interval width multiple of the interval width. The interval width is selected so that the table will have is selected so that the table will have approximately ten intervals.approximately ten intervals.
Example of a Grouped Frequency Example of a Grouped Frequency DistributionDistribution
Choosing a width of 15 we have the following frequency Choosing a width of 15 we have the following frequency distribution.distribution.
Class Interval FrequencyRelative Frequency
100 to <115 2 0.025115 to <130 10 0.127130 to <145 21 0.266145 to <160 15 0.190160 to <175 15 0.190175 to <190 8 0.101190 to <205 3 0.038205 to <220 1 0.013220 to <235 2 0.025235 to <250 2 0.025
79 1.000
Frequency Distribution Frequency Distribution GraphsGraphs
In a In a frequency distribution graphfrequency distribution graph, the , the score categories (X values) are listed on score categories (X values) are listed on the X axis and the frequencies are listed the X axis and the frequencies are listed on the Y axis. on the Y axis.
When the score categories consist of When the score categories consist of numerical scores from an interval or ratio numerical scores from an interval or ratio scale, the graph should be either a scale, the graph should be either a histogram or a polygon. histogram or a polygon.
HistogramsHistograms
In a In a histogramhistogram, a bar is centered above , a bar is centered above each score (or class interval) so that the each score (or class interval) so that the height of the bar corresponds to the height of the bar corresponds to the frequency and the width extends to the frequency and the width extends to the real limits, so that adjacent bars touch. real limits, so that adjacent bars touch.
PolygonsPolygons
In a In a polygonpolygon, a dot is centered above , a dot is centered above each score so that the height of the dot each score so that the height of the dot corresponds to the frequency. The dots corresponds to the frequency. The dots are then connected by straight lines. An are then connected by straight lines. An additional line is drawn at each end to additional line is drawn at each end to bring the graph back to a zero frequency. bring the graph back to a zero frequency.
Bar graphsBar graphs
When the score categories (X values) are When the score categories (X values) are measurements from a nominal or an measurements from a nominal or an ordinal scale, the graph should be a bar ordinal scale, the graph should be a bar graph. graph.
A A bar graphbar graph is just like a histogram except is just like a histogram except that gaps or spaces are left between that gaps or spaces are left between adjacent bars. adjacent bars.
Smooth curveSmooth curve
If the scores in the population are If the scores in the population are measured on an interval or ratio scale, it is measured on an interval or ratio scale, it is customary to present the distribution as a customary to present the distribution as a smooth curvesmooth curve rather than a jagged rather than a jagged histogram or polygon. histogram or polygon.
The smooth curve emphasizes the fact The smooth curve emphasizes the fact that the distribution is not showing the that the distribution is not showing the exact frequency for each category.exact frequency for each category.
Frequency distribution Frequency distribution graphsgraphs
Frequency distribution graphs are useful Frequency distribution graphs are useful because they show the entire set of because they show the entire set of scores. scores.
At a glance, you can determine the highest At a glance, you can determine the highest score, the lowest score, and where the score, the lowest score, and where the scores are centered. scores are centered.
The graph also shows whether the scores The graph also shows whether the scores are clustered together or scattered over a are clustered together or scattered over a wide range. wide range.
ShapeShape
A graph shows the A graph shows the shapeshape of the distribution. of the distribution. A distribution is A distribution is symmetricalsymmetrical if the left side of if the left side of
the graph is (roughly) a mirror image of the right the graph is (roughly) a mirror image of the right side. side.
One example of a symmetrical distribution is the One example of a symmetrical distribution is the bell-shaped normal distribution. bell-shaped normal distribution.
On the other hand, distributions are On the other hand, distributions are skewedskewed when scores pile up on one side of the when scores pile up on one side of the distribution, leaving a "tail" of a few extreme distribution, leaving a "tail" of a few extreme values on the other side. values on the other side.
Positively and Negatively Positively and Negatively Skewed DistributionsSkewed Distributions
In a In a positively skewedpositively skewed distribution, the distribution, the scores tend to pile up on the left side of scores tend to pile up on the left side of the distribution with the tail tapering off to the distribution with the tail tapering off to the right. the right.
In a In a negatively skewednegatively skewed distribution, the distribution, the scores tend to pile up on the right side and scores tend to pile up on the right side and the tail points to the left.the tail points to the left.
Percentiles, Percentile Ranks, Percentiles, Percentile Ranks, and Interpolationand Interpolation
The relative location of individual scores The relative location of individual scores within a distribution can be described by within a distribution can be described by percentiles and percentile ranks. percentiles and percentile ranks.
The The percentile rankpercentile rank for a particular X for a particular X value is the percentage of individuals with value is the percentage of individuals with scores equal to or less than that X value. scores equal to or less than that X value.
When an X value is described by its rank, When an X value is described by its rank, it is called a it is called a percentilepercentile. .
Transforming back and forth Transforming back and forth between X and zbetween X and z
The basic z-score definition is usually The basic z-score definition is usually sufficient to complete most z-score sufficient to complete most z-score transformations. However, the definition transformations. However, the definition can be written in mathematical notation to can be written in mathematical notation to create a formula for computing the z-score create a formula for computing the z-score for any value of X. for any value of X.
X – X – μμz = z = ────────
σσ
Transforming back and forth Transforming back and forth between X and z (cont.)between X and z (cont.)
Also, the terms in the formula can be Also, the terms in the formula can be regrouped to create an equation for regrouped to create an equation for computing the value of X corresponding to computing the value of X corresponding to any specific z-score.any specific z-score.
X = X = μμ + z + zσσ
Characteristics of z ScoresCharacteristics of z Scores
Z scores tell you the number of standard deviation Z scores tell you the number of standard deviation units a score is above or below the meanunits a score is above or below the mean
The mean of the z score distribution = 0The mean of the z score distribution = 0 The SD of the z score distribution = 1The SD of the z score distribution = 1 The shape of the z score distribution will be exactly The shape of the z score distribution will be exactly
the same as the shape of the original distributionthe same as the shape of the original distribution z = 0z = 0 zz22 = SS = N = SS = N σσzz22/N)/N)
Sources of Error in Probabilistic ReasoningSources of Error in Probabilistic Reasoning
The Power of the ParticularThe Power of the Particular Inability to Combine ProbabilitiesInability to Combine Probabilities Inverting Conditional ProbabilitiesInverting Conditional Probabilities Failure to Utilize sample Size informationFailure to Utilize sample Size information The GamblerThe Gambler’’s Fallacys Fallacy Illusory Correlations & Confirmation BiasIllusory Correlations & Confirmation Bias A Tendency to Try to Explain Random EventsA Tendency to Try to Explain Random Events Misunderstanding Statistical RegressionMisunderstanding Statistical Regression The Conjunction FallacyThe Conjunction Fallacy
Characteristics of the Normal Characteristics of the Normal DistributionDistribution
It is ALWAYS unimodal & symmetricIt is ALWAYS unimodal & symmetric The height of the curve is maximum at μThe height of the curve is maximum at μ For every point on one side of mean, there is an exactly For every point on one side of mean, there is an exactly
corresponding point on the other sidecorresponding point on the other side The curve drops as you move away from the meanThe curve drops as you move away from the mean Tails are asymptotic to zeroTails are asymptotic to zero The points of inflection always occur at one SD above The points of inflection always occur at one SD above
and below the mean.and below the mean.
The Distribution of Sample MeansThe Distribution of Sample Means
A distribution of the means from all possible samples of A distribution of the means from all possible samples of size nsize n
The larger the n, the less variability there will beThe larger the n, the less variability there will be The sample means will cluster around the population The sample means will cluster around the population
meanmean The distribution will be normal if the distribution of the The distribution will be normal if the distribution of the
population is normalpopulation is normal Even if the population is not normally distributed, the Even if the population is not normally distributed, the
distribution of sample means will be normal when n > 30distribution of sample means will be normal when n > 30
Properties of the Distribution of Properties of the Distribution of Sample MeansSample Means
The mean of the distribution = μThe mean of the distribution = μ The standard deviation of the distribution = σ/√nThe standard deviation of the distribution = σ/√n The mean of the distribution of sample means is called The mean of the distribution of sample means is called
the the Expected Value of the MeanExpected Value of the Mean The standard deviation of the distribution of sample The standard deviation of the distribution of sample
means is called the means is called the Standard Error of the Mean Standard Error of the Mean (σ(σMM))
Z scores for sample means can be calculated just as we Z scores for sample means can be calculated just as we
did for individual scores. Z = M-μ/σdid for individual scores. Z = M-μ/σMM
What is a Sampling What is a Sampling Distribution?Distribution?
It is the distribution of a statistic from all It is the distribution of a statistic from all possible samples of size npossible samples of size n
If a statistic is unbiased, the mean of the If a statistic is unbiased, the mean of the sampling distribution for that statistic will sampling distribution for that statistic will be equal to the population value for that be equal to the population value for that statistic.statistic.
Introduction to Hypothesis TestingIntroduction to Hypothesis Testing
We use a sample to estimate the likelihood that our We use a sample to estimate the likelihood that our hunch about a population is correct.hunch about a population is correct.
In an experiment, we see if the difference between the In an experiment, we see if the difference between the means of our groups is so great that they would be means of our groups is so great that they would be unlikely to have been drawn from the same population unlikely to have been drawn from the same population by chance.by chance.
Formulating HypothesesFormulating Hypotheses The Null Hypothesis (HThe Null Hypothesis (H00))
Differences between means are due only to chance Differences between means are due only to chance fluctuationfluctuation
Alternative Hypotheses (HAlternative Hypotheses (Haa))
Criteria for rejecting a null hypothesisCriteria for rejecting a null hypothesis Level of Significance (Alpha Level)Level of Significance (Alpha Level)
Traditional levels are .05 or .01Traditional levels are .05 or .01
Region of distribution of sample means defined by Region of distribution of sample means defined by alpha level is known as the alpha level is known as the ““critical regioncritical region””
No hypothesis is ever No hypothesis is ever ““provenproven””; we just fail to reject ; we just fail to reject nullnull
When the null is retained, alternatives are also When the null is retained, alternatives are also retained.retained.
Obtained Difference Between Obtained Difference Between Means/Difference due to chance or errorMeans/Difference due to chance or error
This is the z ratio, and it is the basis for This is the z ratio, and it is the basis for most of the hypothesis tests we will most of the hypothesis tests we will discussdiscuss
Errors in Hypothesis TestingErrors in Hypothesis Testing
Type I ErrorsType I Errors You reject a null hypothesis when you You reject a null hypothesis when you
shouldnshouldn’’tt You conclude that you have an effect when You conclude that you have an effect when
you really do notyou really do not The alpha level determines the probability of The alpha level determines the probability of
a Type I Error (hence, called an a Type I Error (hence, called an ““alpha erroralpha error””))
Type II ErrorsType II Errors Failure to reject a false null hypothesisFailure to reject a false null hypothesis Sometimes called a Sometimes called a ““BetaBeta”” Error. Error.
Statistical PowerStatistical Power
How sensitive is a test to detecting real How sensitive is a test to detecting real effects?effects?
A powerful test decreases the chances of A powerful test decreases the chances of making a Type II Errormaking a Type II Error
Ways of Increasing Power:Ways of Increasing Power: Increase sample sizeIncrease sample size Make alpha level less conservativeMake alpha level less conservative Use one-tailed versus a two-tailed testUse one-tailed versus a two-tailed test
Assumptions of Parametric Assumptions of Parametric Hypothesis Tests (z, t, anova)Hypothesis Tests (z, t, anova)
Random sampling or random assignment Random sampling or random assignment was usedwas used
Independent ObservationsIndependent Observations Variability is not changed by experimental Variability is not changed by experimental
treatment (homogeneity of variance)treatment (homogeneity of variance) Distribution of Sample Means is normalDistribution of Sample Means is normal
Measuring Effect SizeMeasuring Effect Size
Statistical significance alone does not imply a substantial Statistical significance alone does not imply a substantial effect; just one larger than chanceeffect; just one larger than chance
CohenCohen’’s s dd is the most common technique for assessing is the most common technique for assessing effect sizeeffect size
CohenCohen’’s s d d = Difference between the means divided by = Difference between the means divided by the population standard deviation.the population standard deviation.
d > .8 means a large effect!d > .8 means a large effect!
Introduction to the t StatisticIntroduction to the t Statistic
Since we usually do not know the population variance, Since we usually do not know the population variance, we must use the sample variance to estimate the we must use the sample variance to estimate the standard errorstandard error Remember? SRemember? S22 = SS/n-1 = SS/df = SS/n-1 = SS/df
Estimated Standard Error = SEstimated Standard Error = SMM = √S = √S22/n/n
t = M – μt = M – μ00/S/SMM
Differences between the distribution of Differences between the distribution of the t statistic and the normal curvethe t statistic and the normal curve
t is only normally distributed when n is very t is only normally distributed when n is very large. Why?large. Why? The more statistics you have in a formula, the more sources of The more statistics you have in a formula, the more sources of
sampling fluctuation you will have.sampling fluctuation you will have. M is the only statistic in the z formula, so z will be normal M is the only statistic in the z formula, so z will be normal
whenever the distribution of sample means is normalwhenever the distribution of sample means is normal In In ““tt”” you have things fluctuating in both the numerator and the you have things fluctuating in both the numerator and the
denominatordenominator Thus, there are as many different t distributions as there are Thus, there are as many different t distributions as there are
possible sample sizes. You have to know the degrees of possible sample sizes. You have to know the degrees of freedom (df) to know which distribution of t to use in a problem.freedom (df) to know which distribution of t to use in a problem.
All t distributions are unimodal and symmetrical around zero.All t distributions are unimodal and symmetrical around zero.
Comparing Differences between Comparing Differences between Means with t TestsMeans with t Tests
There are two kinds of t tests:There are two kinds of t tests: t Tests for Independent Samplest Tests for Independent Samples
Also known as a Also known as a ““Between-SubjectsBetween-Subjects”” Design Design Two totally different groups of subjects are compared; Two totally different groups of subjects are compared;
randomly assigned if an experiment randomly assigned if an experiment
t Tests for related Samplest Tests for related Samples Also known as a Also known as a ““Repeated MeasuresRepeated Measures”” or or ““Within-SubjectsWithin-Subjects””
or or ““Paired SamplesPaired Samples”” or or ““Matched GroupsMatched Groups”” Design Design A group of subjects is compared to themselves in a different A group of subjects is compared to themselves in a different
conditioncondition Each individual in one sample is matched to a specific Each individual in one sample is matched to a specific
individual in the other sampleindividual in the other sample
Advantages of Independent Sample Advantages of Independent Sample DesignsDesigns
Independent Designs have no carryover effectsIndependent Designs have no carryover effects Independent designs do not suffer from fatigue or Independent designs do not suffer from fatigue or
practice effectspractice effects You do not have to worry about getting people to You do not have to worry about getting people to
show up more than onceshow up more than once Demand characteristics may be stronger in repeated Demand characteristics may be stronger in repeated
measure studies than in independent designsmeasure studies than in independent designs Since more individuals participate in independent Since more individuals participate in independent
design studies, the results may be more design studies, the results may be more generalizeable generalizeable
Disadvantages of Independent Disadvantages of Independent Sample DesignsSample Designs
Usually requires more subjects (larger n)Usually requires more subjects (larger n) The effect of a variable cannot be assessed for each The effect of a variable cannot be assessed for each
individual, but only for groups as a wholeindividual, but only for groups as a whole There will be more individual differences between There will be more individual differences between
groups, resulting in more variabilitygroups, resulting in more variability
Advantages of Paired-Sample Advantages of Paired-Sample DesignsDesigns
Requires fewer subjectsRequires fewer subjects Reduces variability/more statistically efficientReduces variability/more statistically efficient Good for measuring changes over timeGood for measuring changes over time Eliminates problems caused by individual Eliminates problems caused by individual
differencesdifferences Effects of variables can be assessed for each Effects of variables can be assessed for each
individualindividual
Disadvantages of Paired Sample Disadvantages of Paired Sample DesignsDesigns
Carryover effects (2Carryover effects (2ndnd measure influenced by 1 measure influenced by 1stst measure)measure)
Progressive Error (Fatigue, practice effects)Progressive Error (Fatigue, practice effects) Counterbalancing is a way of controlling carryover and practice Counterbalancing is a way of controlling carryover and practice
effectseffects
Getting people to show up more than onceGetting people to show up more than once Demand characteristics may be strongerDemand characteristics may be stronger
What is really going on with t Tests?What is really going on with t Tests?
Essentially the difference between the means of the two Essentially the difference between the means of the two groups is being compared to the estimated standard groups is being compared to the estimated standard error.error.
t = difference between group means/estimated standard t = difference between group means/estimated standard errorerror
t = variability due to chance + independent t = variability due to chance + independent variable/variability due to chance alonevariable/variability due to chance alone
The t distribution is the sampling distribution of The t distribution is the sampling distribution of differences between sample means. (comparing differences between sample means. (comparing obtained difference to standard error of differences)obtained difference to standard error of differences)
Assumptions underlying t TestsAssumptions underlying t Tests
Observations are independent of each other (except Observations are independent of each other (except between paired scores in paired designs)between paired scores in paired designs)
Homogeneity of VarianceHomogeneity of Variance Samples drawn from a normally distributed Samples drawn from a normally distributed
populationpopulation At least interval level numerical dataAt least interval level numerical data
Analysis of Variance (anova)Analysis of Variance (anova)
Use when comparing the differences between means Use when comparing the differences between means from more than two groupsfrom more than two groups
The independent variable is known as a The independent variable is known as a ““FactorFactor”” The different conditions of this variable are known as The different conditions of this variable are known as
““levelslevels”” Can be used with independent groupsCan be used with independent groups
Completely randomized single factor anovaCompletely randomized single factor anova
Can be used with paired groupsCan be used with paired groups Repeated measures anovaRepeated measures anova
The F Ratio (anova)The F Ratio (anova)
F = variance between groups/variance within groupsF = variance between groups/variance within groups F = Treatment Effect + Differences due to F = Treatment Effect + Differences due to
chance/Differences due to chancechance/Differences due to chance F = Variance among sample means/variance due to F = Variance among sample means/variance due to
chance or errorchance or error The denominator of the F Ratio is known as the The denominator of the F Ratio is known as the ““error error
termterm””
Evaluation of the F RatioEvaluation of the F Ratio
Obtained F is compared with a critical valueObtained F is compared with a critical value If you get a significant F, all it tells you is that at If you get a significant F, all it tells you is that at
least one of the means is different from one of least one of the means is different from one of the othersthe others
To figure out exactly where the differences are, To figure out exactly where the differences are, you must use Multiple Comparison Testsyou must use Multiple Comparison Tests
Multiple Comparison TestsMultiple Comparison Tests The issue of The issue of ““Experimentwise ErrorExperimentwise Error””
Results from an accumulation of Results from an accumulation of ““per comparison per comparison errorserrors””
Planned ComparisonsPlanned Comparisons Can be done with t tests (must be few in number)Can be done with t tests (must be few in number)
Unplanned Comparisons (Post Hoc tests)Unplanned Comparisons (Post Hoc tests) Protect against experimentwise errorProtect against experimentwise error Examples:Examples:
TukeyTukey’’s HSD Tests HSD Test The Scheffe TestThe Scheffe Test FisherFisher’’s LSD Tests LSD Test Newman-Keuls TestNewman-Keuls Test
Measuring Effect Size in AnovaMeasuring Effect Size in Anova
Most common technique is Most common technique is ““rr22”” Tells you what percent of the variance is due Tells you what percent of the variance is due
to the treatmentto the treatment rr22 = SS between groups/SS total = SS between groups/SS total
Single Factor AnovaSingle Factor Anova(One-Way Anova)(One-Way Anova)
Can be Independent MeasuresCan be Independent Measures Can be Repeated MeasuresCan be Repeated Measures
Do you know population SD?
Yes
Use ZTest
No
Are there only 2 groups toCompare?
Yes - Only 2
No - More than
2
Do you have Independent data?
Yes - Only 2Groups
No - More than2 Groups
UseANOVA
If F notSignificant, Retain Null
If F isSignificant, Reject Null
Yes No
Do you have Independent data?
Yes No
If F notSignificant, Retain Null
If F isSignificant, Reject Null
Compare MeansWith Multiple
Comparison TestsUseIndependent
Sample T test
Use PairedSampleT test
UseIndependent
Sample T test
Use PairedSampleT test
Is t testSignificant?
Yes No
Retain Null
Hypothesis
RejectNull
Hypothesis
Compare means
Correlational MethodCorrelational Method
No manipulation: just observe 2+ No manipulation: just observe 2+ variables, then measure relationshipvariables, then measure relationship
Also called:Also called: DescriptiveDescriptive Non-Non-
DISADVANTAGE: No control over extraneous variablesDISADVANTAGE: No control over extraneous variables
The Uses of CorrelationThe Uses of Correlation
Predicting one variable from anotherPredicting one variable from another Validation of TestsValidation of Tests
Are test scores correlated with what they say Are test scores correlated with what they say they measure?they measure?
Assessing ReliabilityAssessing Reliability Consistency over time, across raters, etcConsistency over time, across raters, etc
Hypothesis TestingHypothesis Testing
Correlation CoefficientsCorrelation Coefficients Can range from -1.0 to +1.0Can range from -1.0 to +1.0 The DIRECTION of a relationship is indicated by the sign The DIRECTION of a relationship is indicated by the sign
of the coefficient (i.e., positive vs. negative)of the coefficient (i.e., positive vs. negative) The STRENGTH of the relationship is indicated by how The STRENGTH of the relationship is indicated by how
closely the number approaches -1.0 or +1.0closely the number approaches -1.0 or +1.0 The size of the correlation coefficient indicates the The size of the correlation coefficient indicates the
degree to which the points on a scatterplot approximate degree to which the points on a scatterplot approximate a straight linea straight line As correlations increase, standard error of estimate gets smaller As correlations increase, standard error of estimate gets smaller
& prediction becomes more accurate& prediction becomes more accurate
The closer the correlation coefficient is to zero, the The closer the correlation coefficient is to zero, the weaker the relationship between the variables.weaker the relationship between the variables.
Types of Correlation CoefficientsTypes of Correlation Coefficients The Pearson rThe Pearson r
Most common correlationMost common correlation Use with scale data (interval & ratio)Use with scale data (interval & ratio) Only detects linear relationshipsOnly detects linear relationships The coefficient of determination (rThe coefficient of determination (r22) measures proportion of ) measures proportion of
variability in one variable accounted for by the other variable.variability in one variable accounted for by the other variable. Used to measure Used to measure ““effect sizeeffect size”” in ANOVA in ANOVA
The Spearman CorrelationThe Spearman Correlation Use with ordinal level dataUse with ordinal level data Can assess correlations that are not linearCan assess correlations that are not linear
The Point-Biserial CorrelationThe Point-Biserial Correlation Use when one variable is scale data but other variable is Use when one variable is scale data but other variable is
nominal/categoricalnominal/categorical
Problems with Interpreting PearsonProblems with Interpreting Pearson’’s rs r
Cannot draw cause-effect conclusionsCannot draw cause-effect conclusions Restriction of rangeRestriction of range
Correlations can be misleading if you do not Correlations can be misleading if you do not have the full range of scoreshave the full range of scores
The problem of outliersThe problem of outliers Extreme outliers can disrupt correlations, Extreme outliers can disrupt correlations,
especially with a small n.especially with a small n.
Introduction to RegressionIntroduction to Regression In any scatterplot, there is a line that provides the “best In any scatterplot, there is a line that provides the “best
fit” for the datafit” for the data This line identifies the “central tendency” of the data and it can This line identifies the “central tendency” of the data and it can
be used to make predictions in the following form:be used to make predictions in the following form: Y = bx + a Y = bx + a ““b” is the slope of the line, and a is the Y intercept (the value of Y when X = b” is the slope of the line, and a is the Y intercept (the value of Y when X =
0)0)
The statistical technique for finding the best fitting line is The statistical technique for finding the best fitting line is called “linear regression,” or “regression”called “linear regression,” or “regression”
What defines whether a line is the best fit or not?What defines whether a line is the best fit or not? The “least squares solution” (finding the line with the smallest The “least squares solution” (finding the line with the smallest
summed squared deviations between the line and data points)summed squared deviations between the line and data points)
The Standard Error of EstimateThe Standard Error of Estimate Measure of “average error;” tells you the precision of your Measure of “average error;” tells you the precision of your
predictionspredictions As correlations increase, standard error of estimate gets smallerAs correlations increase, standard error of estimate gets smaller
Simple RegressionSimple Regression
Discovers the regression line that provides Discovers the regression line that provides the best possible prediction (line of best fit)the best possible prediction (line of best fit)
Tells you if the predictor variable is a Tells you if the predictor variable is a significant predictorsignificant predictor
Tells you exactly how much of the Tells you exactly how much of the variance the predictor variable accounts variance the predictor variable accounts forfor
Multiple RegressionMultiple Regression
Gives you an equation that tells you how Gives you an equation that tells you how well multiple variables predict a target well multiple variables predict a target variable in combination with each other.variable in combination with each other.
Nonparametric StatisticsNonparametric Statistics
Used when the assumptions for a Used when the assumptions for a parametric test have not been met:parametric test have not been met: Data not on an interval or ratio scaleData not on an interval or ratio scale Observations not drawn from a normally Observations not drawn from a normally
distributed populationdistributed population Variance in groups being compared is not Variance in groups being compared is not
homogeneoushomogeneous Chi-Square test is the most commonly used Chi-Square test is the most commonly used
when nominal level data is collected when nominal level data is collected
