ANOVA 1-way - How to calculate SSF I'm studying and I can't find an answer to make the ANOVA table for this.
I have 4 groups, with mean and variance group. My N=60, n=15 for each group.
To find Sum Square Errors I did Sum( ni * variance i) but I don't know how to calculate SST or SSF having just this information.
I could have made SST = total variance * N
And then SSF=SST-SSE, but I don't know how to get to total variance from this information.
Thank you
 A: For your balanced design (15 replicates in each of four groups), a quick answer can be illustrated using fictitious data sampled in R:
set.seed(2021)
x1 = round(rnorm(15, 50, 7),2)
x2 = round(rnorm(15, 51, 7),2)
x3 = round(rnorm(15, 51, 7),2)
x4 = round(rnorm(15, 52, 7),2)
x = c(x1,x2,x3,x4)
g = as.factor(rep(1:4, each=15))

In R, there are several ways to get relevant parts of the ANOVA
table. One is as follows, giving Total Sum of Squares =  $3485.53.$
aov(x~g)
Call:
   aov(formula = x ~ g)

Terms:
                       g Residuals
Sum of Squares   113.405  3372.125
Deg. of Freedom        3        56

Residual standard error: 7.75993
Estimated effects may be unbalanced

The total sum of squares is $113.405 + 3372.125 =  3485.53.$
Another method gives the same answer:
anova(lm(x ~ g))

Analysis of Variance Table

Response: x
          Df Sum Sq Mean Sq F value Pr(>F)
g          3  113.4  37.802  0.6278 0.6001
Residuals 56 3372.1  60.217  

Also, 'mean squared' entries are 'sum of squared entries' divided by 'degrees of freedom' and 'mean squared entries' can be found from group means and variances. Thus,
$\mathrm{MSE = MS(Resid)} = 60.217$ and
$\mathrm{MSF = MS(Group)} = 37.80$ agree
with the outputs above.
MSE = mean(c(var(x1),var(x2),var(x3),var(x4)));  MSE
[1] 60.21651
MSF = 15*var(c(mean(x1),mean(x2),mean(x3),mean(x4)));  MSF
[1] 37.80172

