Oneway

This table displays the descriptive statistics for each group and for the entire data set.
The left-hand column rows show the grade levels from 7th to 10th grades.
N indicates that there were 50 respondents in each group.
The Mean column shows the averages by grade level.
One-way ANOVA compares these sample estimates to determine if the population means differ.
For ANOVA to be appropriate, the standard deviations should be similar to each other, and this appears to be the case.
(Equaltiy could be inspected by using the Levene test.)
The lower and upper bounds contain the true value of the population mean 95% of the time.

The results of the analysis are present in an ANOVA table above.
The total variation is partitioned into two components - that between groups, and that within groups.
Between groups represents variation of the group means around the overall mean.
Within groups represents variation of the individual scores around their respective group means.
(As you can see in this instance, there was much more variation within, than between groups.)
Sig. indicates the significance level of the F test.  (Alpha for this problem was equal to .05)
Small significance values (<.05) indicate group differences, (as is found in this problem).  At least one of the grade levels differs from the others.

Planned contrasts, or Post Hoc comparisons are methods to determine which group or groups differ.

This table lists the pairwise comparisons of the group means for all selected post hoc procedures.
Mean difference lists the differences between the sample means.
Sig. lists the probability that the population mean difference is zero.
A 95% confidence interval is constructed for each difference.  If this interval contains zero, the two groups do not differ.

Notice that when 7th graders are compared with 8th graders, the Tukey test does not contain zero, but the Sheffe does contain zero.
The Sheffe test is regarded as the more conservative test.

The asterisk shows those mean differences found to be significant at the .05 level.
For example, Tukey shows that 7th graders differ significantly with 8th and with 10th graders over perception of math as fun.
Scheffe, however, says that 7th graders differ significantly only with 10th graders over perception of math as fun.

Said differently, 10th graders consider math less fun than 9th and 7th graders.  Data from both Tukey and Scheffe found these to be significant.

Seen from the 7th graders point of view, Scheffe found it significant that 7th graders scored higher than 10th graders on Math is Fun.
But Tukey in examining the same 7th graders, found it significant that 7th graders scored higher than both 8th and 10th graders.  We conclude, that the Scheffe test is more conservative, and "finds" fewer activities to be significant according to the laws of chance.

 

In this figure, for all selected post hoc procedures, homogeneous groups are defined.
These homogeneous groups are in columns 1, 2, and 3.
The means for each level of the independent variable are listed in their corresponding group.

Homogeneous group 1 contains 10th and 8th graders in both the Tukey and Scheffe.
Homogeneous group 2 contains 8th and 9th graders in Tukey, but 7th, 8th, and 9th graders in Scheffe.
Homogeneours group 3 contains 9th and 7th graders in Tukey, but no grade levels in Scheffe.

So the Scheffe and Tukey tests differ in whether or not there are three homogeneous groups.  The Scheffe is more conservative in saying that there are but two homogeneous groups.

The textbook comments on the Scheffe method by saying:
"Overall, there is a difference among 7th, 8th, 9th and 10th graders in their perceptions of mathematics as a fun activity.  In particular, 10th grade students were less likely to view mathematics as a fun activity than than 7th and 9th grade students."  Notice that 10th and 8th graders showed up in the same group, but that 7th, 8th, and 9th graders showed up in the second group.  In neither of these grouping were there statistical differences that exceeded alpha = .05.