You *want* homogeneous variances.

The variance test will be significant when it detects that the variances are not homogeneous. So the (non-significant) result of the test on homogeneous variances is not indicating that you have a problem.

Tukeys test results show you that you can reject the hypotheses that the means of sampkes 1 and 3 are equal and that the means of samples 2 and 3 are equal while keeping your desired family-wise type-I error rate. This supports your hypothesis that sample 3 is different from the other two samples.

You *want* homogeneous variances.

The variance test will be significant when it detects that the variances are not homogeneous. So the (non-significant) result of the test on homogeneous variances is not indicating that you have a problem.

Tukeys test results show you that you can reject the hypotheses that the means of sampkes 1 and 3 are equal and that the means of samples 2 and 3 are equal while keeping your desired family-wise type-I error rate. This supports your hypothesis that sample 3 is different from the other two samples.

Dear Bokkasam Harish,

You have to know that all tests that compare means, needs information about Mode. Do you know why?

 Due to average be an abstract measure of location it needs being close to the mode for their interpretation be appropriate (mean is the center of gravity of the distribution). That is why it requires the random variable has normal distribution or at least symmetrical (required whenever n's

Hello,

I agree with Jochen and Helena, but would like to add a couple of points, more or less for the sake of clarity.

Since you use the terms 'sample 1', 'sample 2', and 'sample 3' to describe your experimental conditions, I assume you sampled from three different populations then subjected them all to the same treatment.  

Ideally, three conditions should be met to use ANOVA with this experimental design.

(1) INDEPENDENCE:  All observations should be independent from one another - the 'value' of any data point should have no impact on the value of any other data point.

(2) NORMAL DISTRIBUTIONS:  Values of the dependent variable should be normally distributed within the three populations from which you sample.

(3) HOMOGENEITY OF VARIANCES:  The variances of the three populations along the dimension of your dependent variable should be close to one another.

Of course we rarely see ideal experimental conditions; these assumptions are often violated to some degree, and AVOVA is more resilient to violations of (3) and (2) than to violations of (1).  It appears that you ran tests on homogeneity after ANOVA.  You might want to know about that going in to your analysis, particularly if you anticipate small experimental effects.  If normality or heterogeneity is seriously violated, you might need to transform your data to better approximate normality, or use comparison tests other that ANOVA, such as the Welsh test.

But the bottom line is that your ANOVA tests were significant and post hoc tests confirmed your hypothesis regarding to the source of the significance.  The fact that homogeneity was violated for one set of trials does not present a big problem.  It would only result in a loss of statistical power the sizes of your samples were drastically different.

Here's a link to a simulation that allows you to change 'normality' and 'homogeneity' to see the effect on Type 1 error rates.  

http://onlinestatbook.com/2/tests_of_means/robust_sim.html

Thanks everyone for the repies and very useful suggestions.

Dear Oliver thanks for the simulation