Review
Hints for Final
Descriptive Statistics:
Describing a data set
How Does One Describe A Variable?
•Scale of Measurement
•Central Tendency
•Variability
•Shape of Distribution
Describing SampleDescribing Sample
Choice of Statistic Depends on Scale of MeasurementChoice of Statistic Depends on Scale of Measurement
Interval
Ordinal or interval that isskewed or open-ended
Mode
Semi-Interquartile
Range; Range
Stan. Dev.
Mean
Central
Tendency
Median
Nominal
Variability
Graphs
Histogram;
Freq. Poly.
Bar Graph
Bar Graph
Why Do We Need Descriptive Statistics?
1.To describe a group, as just reviewed
2.To describe where a person’s score falls ina group
•z score
•Percentile rank
3.To test assumptions of statistical tests
•Normality by graphs (recognize criteria)
•Homogeneity of variances
Inferential Statistics
Psychology’s Truth Tool
Hypothesis Testing
•We want to know whether, say, two groupsare different.
•The truth lies in the population values,which we do not have.
•We have to guess the population valuesfrom samples.
1
Is the truth that
2
Men and women are different in competitiveness.
Women
Men
Competitiveness
1
Or
Men and women are NOT different in competitiveness.
Women
2
Men
If M Always Equaled
We wouldn’t need statistics.
1
2
Sample Means Would Reflect theDifferences in Population ’s
Women
Men
See how sample means lined up with population means.
1
2
When Population ’s Are the Same
Women
Men
Sample M’s would also be the same.
Alas, M May Be Greater Than
. . .Or M May Be Smaller Than
1
2
Such Random Differences MayMislead Us
Women
Men
Our sample means are quite different, but only bychance. We would falsely conclude there is adifference, which would be a Type __ error.
I
1
2
The Opposite is Also Possible
We would conclude incorrectly that men and women donot differ in competitiveness. What kind of error is that?
Type II
Solution: Hypothesis Testing
•Need to distinguish between chance variation andreal differences.
•We do so by estimating how likely it is that theresult* is simply a random chance variation.
* In t and ANOVA the result is difference betweenmeans; in correlation/regression the result isrelationship between IV(s) and DV.
Estimate of Chance:Distribution of Sample Means
The variance (SD) of the sampling distributions (thestandard error) provides an estimate of how much thesample means might vary from the population mean ifonly chance is operating on scores.
We Compare Obtained Mean Differenceto the Estimate of Random Variability
•If mean difference is large enough compared tochance variation in means, we decide thedifference is real.
•We need a criterion for “large enough.”
•The criterion is in terms of probability--how likelya difference that large is likely to happen bychance.
We Set the Probability by Alpha
we decide that it is likely to be due to our experimentalmanipulation rather than due to chance.
H0: M = F = 0
For alpha = .05
If
is so extreme that it will only occur lessthan 5% of the time
.025
.025
Statistical Decision-Making Steps
1.State the null and alternative hypotheses.
2.Find the critical value (t, F, r, R, ) . (Todo so we need to choose alpha =.05 or .01,nondirectional or directional, and to figureout the degrees of freedom.)
3.Collect data and calculate obtained (t, F, r,R, ).
4.Make a decision.If obtained (t, F, r, R, ) is in the criticalrejection region, reject H0.
Step 1 Nondirectional
•H0=no effect (no difference betweenmeans or no relationship)H1=is some effect (is a difference betweenmeans or is a relationship)
•All tests this semester--t, F, r, R,
Step 1 Directional
•Difference between meansH1= mean 1 > mean 2H0= mean 1 not > mean 2 (smaller orequal)
•RelationshipH1= relationship >0H0= relationship not >0 (inverse or equal)
•Can not use directional with F or R or
Step 2 Set Alpha and FindCritical Value
•Alpha (, p) is probability of Type I error(reject H0 when it is false)
•Traditional procedure
–Set alpha at .05
–Look up critical value in relevant Table
•(Alternative--use exact sig level SPSSprovides)
Collect Data and CalculateObtained Statistic
•Know how to calculate
–One sample t
–F from source table
•Know how to get SPSS to give you
–Independent samples and related samples ts
–Independent groups Fs
–Two-way ANOVA, independent groups
Collect Data and CalculateObtained Statistic 2
•Know how to read SPSS output
–Independent samples and related samples ts
–Independent samples Fs
–Two-way ANOVA, independent groups
–Standard Multiple Regression (R) with two IVs
–Chi Square ()
Step 4 Make a Decision
•Traditional
–Reject H0 if obtained statistic in criticalrejection region (i. e., more extreme fornondirectional test)
–All statistics this semester--z, t, F, r,
•Exact significance level
–Reject H0 if exact sig level is smaller than .05
–Can use when SPSS gives exact sig. level
Additional Tasks
•Step 0 Check assumptions and conditions
•Step 5 Follow-on tests
–>2 groups--post hoc tests
–Interaction--simple effect
•Step 6 Determine effect size
Assumptions
•All tests-- independence of observations
–Determined when experimental procedure developed
•All parametric tests--distribution of sample meansis normal
–Will be with large samples (more than 30 per cell)
–With small samples can only check sample distribution--if it is symmetric, we guess population distribution isnormal and therefore distribution of sample means willbe normal.
Assumptions, cont.
•Homogeneity of variance
–Levene’s in SPSS output (if you can cope withexact sig. Levels)
–Fmax
Conditions, Problems
•Quasi-experiment
•Repeated measures
•Correlation/regression
•Chi-square
Effect Size
•Is estimate of how big effect is.
•Usefulness
–If the effect is significant, effect size tells howbig the effect is.
–If the effect is nonsignificant, it gives a clue asto whether increasing power would lead tosignificance (i. e., whether result is trulynonsignificant or there is a Type II error).
–Know how to calculate for correlation (r2 andR2).
Are Results Likely to Be Replicated?
•Type I error = alpha
•Type II error
•How to increase power
–Larger sample size
–Smaller variability (error)
–Larger effect (e. g., difference between means)
Describing Results in APA Style
The study was . . .
The result was significant (not significant), statistic(df) =
obtained value, p < (.05 or .01 or p = exact value).
The nature of the differences was (which group(s) better,
with post hoc test if necessary, or whether relationship
positive or negative).
The means and standard deviations were___________.
The effect size, ____ = _____, which was __________.
The effect size shows (for significant results, how big
effect; for nonsignificant results, whether there might be aType II error.
Choosing a Statistical TestChoosing a Statistical Test
Is the independent variable a nominal or interval scale of measurement?Is the independent variable a nominal or interval scale of measurement?
Interval
interval
Nominal
Repeated Measures or
Independent Samples?
Independent
Ind. t
Rep. t
Rep.
ANOVA
2
Ind. ANOVA
>2
2
Rep. Meas.
Chi-square
How many IVs?
Scale of dependent variable?
nominal
interval
How manygroups in IV?
Scale of
DV?
One-way
Two-way
1 IV
2 IV
>2
r
Correlation
R
Multiple
Regression
1
>1
How manygroups in IV?
Structure Out of Chaos
H: Tests
Main
H: Tests
Followup
Effect Sizes
Assumptions
t
t
Cohen’s d
Cohen’s d
r2
r2
Fmax
Fmax
F
F
Post-hoc
Post-hoc
Levene’s
Levene’s
Simple E.
Simple E.
r
r
R
R
R2
R2
skew
skew
Conditions
Scatterplot
2
2
The End
•I still love statistics--do you know why?