How to test differences of means of 3 independent groups I'm new to statistics and the only software i know how to use is SPSS. I need help....
I have 3 independent groups with sizes: 24, 27, 37. I wanted to test the difference between the means of their ages using ANOVA.
                                *mean age ± SD*    
                       *Group1:   40.7297 ± 6.01225  
                       *Group2:   31.5926 ± 4.93231 
                       *Group3:   32.125  ± 4.4557
I have no problem with variances because Levene's test says they have equal. When I tested the NORMALITY of each group (with Shapiro-Wilk), one of them is NOT (p=0.022). I tried transforming my data using natural log and checked for normality of each.This time the other group gave p=0.031. Is it correct if i still use ANOVA? or is there any other test I can use with this situation?
Any help would be much appreciated.
 A: The ANOVA's parametric F-Test is fairly robust to the normality-assumption violation (Maxwell and Delaney, 2004), and you have one group barely reaching the 5% significance level, so it should be OK to perform the F-test. You could run modified-parametric F-test like Welch's or Brown/Forsythe Test, if SPSS still has these options. If the results differ between parametric and modified-parametric tests, you may want to stick with the result based on the latter. If you are trying to testing the equality of medians across the groups, I would not recommend the Kruskal-Wallis test unless the three groups' distributions are similarly shaped. This should not be the case, since only one group was deemed non-normal by the SW test.
By the way, you should not rely on Levene's test for testing heterogeneity of variance because the test is not powerful (i.e., tends to produce more type II errors). Regardless, your data seem fine since the sample sizes and SDs are not too far off of each other: Max(SS)/Min(SS) < 4 (Maxwell and Delaney, 2004).
