Here are fake data simulated to have roughly the means you
report. Notice that, 'not significantly different' is not the
same as 'equal'.
We have approximate means as follows:
A2 A1 A3
85.9 87.3 87.6
A1 and A2 differ with P-value .001;
A2 and A3 differ with P-value only .026; and
A1 and A3 do not differ (P-value .71).
As @GordonSmyth has commented, this pattern is partly due to the high variability of A3.
Additionally, in the
case of my fake data, the sample size of only $n = 30$ is
not sufficient to guarantee that the same mean and standard
deviations I used in the simulation will always give this
same pattern of differences. If you use R, you can try
changing seeds to see that somewhat different patterns can
occur at random. You don't say what $n$ you have, but if it
small relative to the variances, you might see a different
pattern upon repeating your experiment.
Here is R code for the simulated dataset:
e = rnorm(30, 0, 1)
x1 = rnorm(30,87, 2) + e
x2 = rnorm(30,86, 1) + e
x3 = rnorm(30,88, 4) + e
mean(x1); mean(x2); mean(x3)
Correlations amon the three vectors
x3 are relatively small. Real paired data often have larger correlations. To increase correlation for the fake data:
increase the SD of
e in the code above.
x1 x2 x3
x1 1.0000000 0.1848602 0.1912384
x2 0.1848602 1.0000000 0.2327309
x3 0.1912384 0.2327309 1.0000000
Notes: (1) The third parameter of
rnorm is the population standard deviation (not variance). (2) To see the complete printout of the paired t tests,
remove the suffix
$p.val at the end of each