# Statistically significant means with small mean difference

I ran an ANOVA on 3 sample sets with means 7.8, 8, and 7.9. The sample with a mean of 8 was statistically different than the other two. The sample size for each mean=200 and 95 confidence level. Why are these results statistically significant when means are so close?

• Because the variability of each sample is small. A standard deviation of 0.28 for each sample with a sample size of 200 translates into a standard error of the sample mean of about 0.02, more than enough for 8 to be significantly different from 7.9. Jan 13 '20 at 15:09
• Thank you for replying so quickly and helping me understand the reason for the statistical significance :) Jan 13 '20 at 15:53

STATISTICAL SIGNIFICANCE $$\ne$$ PRACTICAL SIGNIFIANCE
You have a pretty large sample size, meaning that you have the power to reject a null hypothesis of equal means, since standard error is a function of both effect size and sample size. You don't dispute that $$7.9 \ne 8$$, do you? Consequently, if you have that one distribution has a mean of $$7.9$$ and another has a mean of $$8$$, you don't dispute that they have different means. End of discussion;$$^{\dagger}$$ the groups have different means.
$$^{\dagger}$$ Well...