ANOVA: Grand mean from means, standard deviation and sample size (n)?

Question

I would like to compare several samples using analysis of variance (ANOVA).

My input data looks like:

Mean SD      n
0.6  +/-5.4  40
0.6  +/-0.2  7
0.4  +/-0.3  21

When I input this data in this ANOVA-online calculator, the result is

p = 0.984

My questions:

1. Since p > 0.05, does that mean "There were no statistically significant differences between group means"?
2. I haven't yet understood the formulas cited below that online calculator, but I read that ANOVA computes a "grand mean" first. Is a grand mean computed with the formulas on that online calculator? If so, which of them?

Thank you very much for any help to understand this better.

EDIT: I understand that the common way to get a grand mean is to have all observations for each group, not just the means, sd and n. But how does this calculator do it? Isn't it computing some sort of grand mean, to which it compares the groups?

EDIT2: Reading on the subject a bit more, I found a source which confirms that using an average of the means weighted by sample count is an alternative way if the single samples are not available.

Answer 1

1. The null hypothesis for your situation is:

$H_0:μ_1=μ_2=μ_3$

$H_A: not\,\, all\,\, µ_i \,\,are\,\, equal.$

Since the p-value for the $F$-test is $> 0.05$, you fail to reject your null hypothesis and conclude that there is not significant difference between any of the $µ_i$.

2. The steps for calculating different ANOVA quantities are as follows:

(i) The grand mean is simply an average over all the observations in all the groups. It is denoted by $\bar{Y}..$.

(ii) The treatment means (group means in your case) are averaged over the observations in specific groups. It is denoted by $\bar{Y}.$.

(iii) Once we have the grand mean and the treatment means, we calculate the overall variability in the data - **Total Sum of Squares: $\sum_i\sum_j(Y_{ij}−\bar Y..)^2$

(iv) Next, we find out how much of the variability is accounted for by different treatments by calculating the **Treatment Sum of Squares = $\sum_1^{n=i}(\bar Y_i.−\bar Y..)^2$

(v) Once we find out the variability caused by the treatments (groups, in your case), the rest of the variability in the data is being caused by noise. We calculate that by subtracting the treatment SS by total SS. $Error\,\,sum\,\, of\,\, squares = Total\,\,SS - Treatment\,\, SS$.

(vi) Next we find $MSTrt = SSTrt/df_{Trt}$ and $MSError = SSError/df_{Error}$.

(vii) The ratio of the $MSTrt$ and $MSErr$ gives us the $F$-statistic.

(viii) If the F-statistic is greater than your critical F, we reject the null hypothesis.

The link you have posted for the online calculator doesn't show any formulas.