# Using ANOVA and HSD, I find A and B are not significantly different from C, but I find A is significantly different from B. What does this mean?

I'm learning a bit about ANOVA and Tukey Range test and my understanding is if there's significance between two samples, they don't come from the same distribution. However, in some tests I'm having a hard time understanding what the results imply.

I have three samples named BP, GA, PSO. After collecting results, I run an ANOVA test:

Df Sum Sq  Mean Sq F value   Pr(>F)
sample      2 0.4305 0.107626  7.6132 5.87e-06 ***
Residuals 495 6.9977 0.014137

Since $$p < 0.001$$, the test shows strong significance that not every sample comes from the same distribution, and with the $$F > 1$$, confidence in the test. Then, I run Tukey's test to do pairwise analysis:

GA-BP  -0.04500006 -0.091036564 0.001036444 0.0589717
PSO-BP -0.02848048 -0.074516984 0.017556024 0.4387137
PSO-GA  0.01651958 -0.029516924 0.062556084 0.8631683

These results suggest GA and BP come from different distributions, but PSO comes from the same distribution as both BP and GA.

Is there no transitive property here? If A and B come from the same distribution as C, but A and B don't come from the same distribution, what can I infer from this?

'Not significantly different from' is a long way from 'comes from the same distribution'. The latter is transitive and the former is not. (Transitivity: $$A = B$$ and $$B = C$$ imply $$A = C.)$$ Strictly speaking, your title should not say ?come from the same distribution".