ReviewHints 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 is skewed or open-ended

Mode

Semi-InterquartileRange; Range

Stan. Dev.

MeanCentralTendency

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 in a group

• z score • Percentile rank

3. To test assumptions of statistical tests• Normality by graphs (recognize criteria)• Homogeneity of variances

Inferential StatisticsPsychology’s Truth Tool

Hypothesis Testing

• We want to know whether, say, two groups are different.

• The truth lies in the population values, which we do not have.

• We have to guess the population values from samples.

1

Is the truth that

2

Men and women are different in competitiveness.

Women MenCompetitiveness

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 the Differences in Population ’s

Women Men

See how sample means lined up with population means.

1

2

When Population ’s Are the Same

WomenMen

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 May Mislead Us

Women Men

Our sample means are quite different, but only by chance. We would falsely conclude there is a difference, which would be a Type __ error.I

1

2

The Opposite is Also Possible

We would conclude incorrectly that men and women do not differ in competitiveness. What kind of error is that?

Type II

Solution: Hypothesis Testing • Need to distinguish between chance variation and

real differences.• We do so by estimating how likely it is that the

result* is simply a random chance variation.

* In t and ANOVA the result is difference between means; in correlation/regression the result is relationship between IV(s) and DV.

Estimate of Chance: Distribution of Sample Means

The variance (SD) of the sampling distributions (the standard error) provides an estimate of how much the sample means might vary from the population mean if only chance is operating on scores.

We Compare Obtained Mean Difference to the Estimate of Random Variability

• If mean difference is large enough compared to chance variation in means, we decide the difference is real.

• We need a criterion for “large enough.”• The criterion is in terms of probability--how likely a

difference that large is likely to happen by chance.

We Set the Probability by Alpha

we decide that it is likely to be due to our experimental manipulation rather than due to chance.

H0: M = F = 0For alpha = .05

If is so extreme that it will only occur less than 5% of the time

.025 .025

Statistical Decision-Making Steps1. State the null and alternative hypotheses. 2. Find the critical value (t, F, r, R, ) . (To do so

we need to choose alpha =.05 or .01, nondirectional or directional, and to figure out 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 critical rejection region, reject H0.

Step 1 Nondirectional

• H0=no effect (no difference between means or no relationship)H1=is some effect (is a difference between means or is a relationship)

• All tests this semester--t, F, r, R,

Step 1 Directional

• Difference between meansH1= mean 1 > mean 2 H0= mean 1 not > mean 2 (smaller or equal)

• RelationshipH1= relationship >0 H0= relationship not >0 (inverse or equal)

• Can not use directional with F or R or

Step 2 Set Alpha and Find Critical 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 SPSS provides)

Collect Data and Calculate Obtained 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 Calculate Obtained 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 critical

rejection region (i. e., more extreme for nondirectional 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 means is 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 is normal and therefore distribution of sample means will be normal.

Assumptions, cont.

• Homogeneity of variance– Levene’s in SPSS output (if you can cope with

exact 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 how big the effect is.

– If the effect is nonsignificant, it gives a clue as to whether increasing power would lead to significance (i. e., whether result is truly nonsignificant or there is a Type II error).

– Know how to calculate for correlation (r2 and R2).

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 StyleThe 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 relationshippositive or negative).The means and standard deviations were___________.The effect size, ____ = _____, which was __________. The effect size shows (for significant results, how bigeffect; for nonsignificant results, whether there might be a Type II error.

Choosing a Statistical TestChoosing a Statistical TestIs the independent variable a nominal or interval scale of measurement? Is the independent variable a nominal or interval scale of measurement?

Interval

interval

NominalRepeated Measures or

Independent Samples?

Independent

Ind. t

Rep. tRep.

ANOVA2

Ind. ANOVA

>22

Rep. Meas.

Chi-squareHow many IVs?

Scale of dependent variable?

nominalinterval

How many groups in IV?

Scale of DV?

One-way Two-way

1 IV 2 IV

>2

rCorrelation

RMultiple

Regression

1 >1

How many groups in IV?

Structure Out of Chaos

H: TestsMain

H: TestsFollowup

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?