PSY 2030: One-Way Within-Groups ANOVA Formulas and Practice

What is a One-Way Within-Groups ANOVA?

An ANOVA used when the same participants provide scores at every level of one independent variable.

What type of data is used in a One-Way Within-Groups ANOVA?

Within-groups data, where the same participants are measured under every level/condition of the independent variable.

What is a Repeated-Measures Design?

A design in which the same participants are measured in multiple conditions/levels of the independent variable.

What is the main purpose of a Within-Groups ANOVA?

To determine whether there is a significant difference among the means of different conditions/levels when the same participants are measured in each condition.

What is a requirement for conducting a Within-Groups ANOVA?

All participants need to have scores for ALL levels of the independent variable.

How can you recognize a Within-Groups ANOVA?

If there are multiple levels of ONE independent variable and the SAME participants are measured in every condition.

What is the difference between Within-Groups ANOVA and Between-Groups ANOVA?

Within-Groups ANOVA partitions variance into BETWEEN-GROUPS, SUBJECTS, and WITHIN-GROUPS variance, while Between-Groups ANOVA only partitions into BETWEEN-GROUPS and WITHIN-GROUPS variance.

What is Subjects Variance?

Variance caused by differences between individual participants.

Why is it important to remove Subjects Variance in Within-Groups ANOVA?

To make the error term smaller and facilitate the detection of differences among conditions.

What happens to the F value when the error term gets smaller?

The F value becomes larger.

What is the Grand Mean?

The mean of ALL scores from ALL participants and ALL conditions.

What does the Grand Mean formula represent?

GM = ΣX / N, where ΣX is the sum of all scores and N is the total number of scores.

What is a Condition Mean?

The mean score for one level/condition of the independent variable.

What is the formula for SS Between?

SSBetween = Σn(M − GM)², where n is the number of participants/scores in each condition.

What does SS Subjects measure?

How much each participant's personal mean differs from the grand mean.

What is the formula for SS Total?

SSTotal = Σ(X − GM)², where X is an individual score.

What is SS Within?

The remaining variability after removing between-groups and subjects variability from total variability.

What is the relationship for SS Within in Within-Groups ANOVA?

SSWithin = SSTotal − SSBetween − SSSubjects.

What does the term 'Personal Mean' refer to?

The average of one participant's scores across ALL levels of the independent variable.

How is a Personal Mean calculated?

By adding a participant's scores across all conditions and dividing by the number of conditions.

What is the significance of calculating Personal Means?

To determine how much each participant's average differs from the grand mean.

What does the symbol GM represent?

The grand mean.

What does the symbol M represent in the context of Condition Mean?

The mean for one condition.

What does the symbol n represent in the Mean Formula?

The number of participants/scores in that particular condition.

What is the purpose of squaring M − GM in SS Between?

To make all deviations positive and prevent positive and negative differences from canceling.

What does k represent in SS Subjects?

The number of levels/conditions of the independent variable.

What is the importance of SS Subjects in Within-Groups ANOVA?

It represents individual differences among participants.

What is the formula for SSWITHIN?

SSWithin = SSTotal − SSBetween − SSSubjects.

Why is SSWITHIN calculated last?

Because we can calculate the between, subjects, and total variance and then solve for within.

What does SSWITHIN represent?

Error/residual variability that is not explained by differences among conditions or participants.

What is the formula for dfBetween?

dfBetween = k − 1, where k is the number of conditions/levels of the independent variable.

What does n represent in dfSubjects?

n represents the NUMBER OF PARTICIPANTS, not the number of scores.

What is the formula for dfWithin?

dfWithin = dfBetween × dfSubjects or dfWithin = (k − 1)(n − 1).

What is MSBETWEEN?

The between-groups sum of squares divided by its degrees of freedom.

What does MSBETWEEN represent?

The estimated variance between the condition means.

What is MSWITHIN?

The within-groups sum of squares divided by its degrees of freedom.

What does MSWITHIN represent?

The estimated error/within-groups variance.

What is the F statistic?

The ratio of between-groups variance to within-groups/error variance.

What does a large F value indicate?

The differences among condition means are large relative to the remaining within-groups/error variability.

What does a small F value indicate?

The differences among condition means are relatively small compared with within-groups/error variability.

Why can within-groups ANOVA have a larger F?

Because participant/subjects variance is removed from the error term, making the denominator smaller.

What is the F distribution?

The sampling distribution used for the F statistic.

Does F have a negative value?

No, F values are positive.

How many degrees of freedom does an F distribution use?

Two: numerator degrees of freedom and denominator degrees of freedom.

What is needed to find a critical F value?

Alpha, numerator df, and denominator df.

What is the critical F decision rule?

If Fobserved > Fcritical, reject H0.

What is the fail-to-reject rule?

If Fobserved ≤ Fcritical, fail to reject H0.

Is the F test two-tailed?

No, the F statistic is positive, so the rejection region is in the upper/right tail.

In the example with 3 conditions and 6 participants, what is k?

k = 3, because there are 3 conditions.

In the same example, what is n?

n = 6, because there are 6 participants.

What is the formula for calculating SSBetween?

SSBetween = Σn(M − GM)².

What does the null hypothesis (H0) state in ANOVA?

All population means are equal (H0: μ1 = μ2 = μ3).

What does the alternative hypothesis (H1) state in ANOVA?

Not all population means are equal.

How is the grand mean (GM) calculated?

GM = Total Score / Total Number of Scores.

What is the SSWithin formula?

SSWithin = SSTotal − SSBetween − SSSubjects.

What is the formula for calculating dfSubjects?

dfSubjects = n − 1, where n is the number of participants.

How is dfWithin calculated in a one-way within-groups ANOVA?

dfWithin = dfBetween × dfSubjects.

What is the formula for calculating the F statistic?

F = MSBetween / MSWithin.

What is the critical F value used for?

To compare against the observed F value to determine if the null hypothesis can be rejected.

What conclusion can be drawn if the observed F value is greater than the critical F value?

Reject H0, indicating that there is a significant difference among the conditions.

Why is SSWithin not directly calculated in a one-way within-groups ANOVA?

SSWithin is found by subtracting SSBetween and SSSubjects from SSTotal.

What distinguishes one-way between-groups ANOVA from one-way within-groups ANOVA?

Between-groups uses separate participants for each condition, while within-groups uses the same participants across all conditions.

What is the significance of dfBetween in ANOVA?

dfBetween = k − 1, where k is the number of levels/conditions of the independent variable.

What does SSTotal represent?

Total variation in all scores, calculated as SSTotal = Σ(X − GM)².

How is SSBetween calculated?

SSBetween = Σn(M − GM)², where M is the mean of each group and GM is the grand mean.

What does SSSubjects measure?

Variation associated with differences among participants, calculated as SSSubjects = Σk(MParticipant − GM)².

What does the variable 'n' represent in ANOVA?

The number of participants.

What does the variable 'k' represent in ANOVA?

The number of levels/conditions of the independent variable.

What is the purpose of calculating the grand mean (GM)?

To find the average of all scores across all conditions.

What is the difference between between-groups and within-groups variance?

Between-groups variance is due to differences among group means, while within-groups variance is due to error/unexplained variability.

What is the first step in the within-groups ANOVA procedure?

Find each condition mean.

What is the last step in the between-groups ANOVA procedure?

Interpret the results.

What is the biggest procedural difference between between-groups and within-groups ANOVA?

Within-groups ANOVA includes SSSubjects and dfSubjects, and SSWithin is found by subtraction.

What does the variable 'MParticipant' represent?

The average score of one participant across all conditions.

What is the significance level (α) typically used in hypothesis testing?

Commonly set at 0.05.

What does the term 'observed F' refer to?

The calculated F statistic from the ANOVA analysis.

How do you calculate the total number of scores (N) in a study?

N = n × k, where n is the number of participants and k is the number of conditions.

What happens if the null hypothesis (H0) is rejected?

It suggests that there is a significant difference among the conditions tested.

What is dfBetween in a study with 3 conditions?

dfBetween = k − 1 = 3 − 1 = 2.

How is dfSubjects calculated for 5 participants?

dfSubjects = n − 1 = 5 − 1 = 4.

If there are 6 participants and 4 conditions, what is dfBetween?

dfBetween = 4 − 1 = 3.

How do you calculate F if MSBetween = 20 and MSWithin = 5?

F = MSBetween / MSWithin = 20 / 5 = 4.

What decision is made if Fobserved = 5.20 and Fcritical = 4.10?

Reject H0 because Fobserved > Fcritical.

What is the first step in hypothesis testing?

Identify the appropriate statistical test.

What are the null and alternative hypotheses for three anxiety-reduction methods?

H0: μ1 = μ2 = μ3; H1: Not all population means are equal.

What is the critical value for dfBetween = 2 and dfWithin = 10 at α = .05?

Fcritical = 4.10.

What does a significant ANOVA result indicate?

Reject H0 → there is evidence that not all population means are equal.

What should you do if a significant ANOVA does not tell you which groups differ?

Use an appropriate post hoc test.

What is the difference between between-groups and within-groups ANOVA?

Between-groups: separate people in groups; Within-groups: same people in every condition.

What does SSSubjects account for?

Variability due to individual differences among participants.

What is the formula for calculating SSWithin?

SSWithin = SSTotal − SSBetween − SSSubjects.

What is the formula for MSBetween?

MSBetween = SSBetween / dfBetween.

What is the formula for dfWithin in a within-groups ANOVA?

dfWithin = dfBetween × dfSubjects.

What is the significance of Fobserved ≤ Fcritical?

Fail to reject H0.

What is the main identifying clue for a one-way within-groups ANOVA?

The same participants provide scores for all conditions.

What is the formula for dfSubjects?

dfSubjects = n − 1.

What is the interpretation if Fobserved = 2.50 and Fcritical = 4.10?

Fail to reject H0; not sufficient evidence of differences.

What is the key difference in error terms between between-groups and within-groups ANOVA?

Within-groups variance is the error term for between-groups ANOVA.

What is the implication of a non-significant ANOVA?

Fail to reject H0; not sufficient evidence that population means differ.

What is the formula for calculating F?

F = MSBetween / MSWithin.