How to obtain the variance from an ANOVA table I was under the expression that the mean square error (MS) of a 1-way anova table is the estimated variance, and that the between and within group variance add to the total variance. However, using an example I get an result which puts me into doubts. Here the example:
I have a dataset in matlab.
data = [1.5377    0.6923    1.6501    3.7950    5.6715
    2.8339    1.5664    6.0349    3.8759    3.7925
   -1.2588    2.3426    3.7254    5.4897    5.7172
    1.8622    5.5784    2.9369    5.4090    6.6302
    1.3188    4.7694    3.7147    5.4172    5.4889];

If I use the anova1 function I obtain the anova table

From this I expected that the variance of the columns is 13.4309 and the variance of the random error is 2.2204. I also expected to obtain that the two MS values add to the total variance. However, if I use the variance function, I get the result
var(data(:)) 
>> 4.0888

This result differens significantly from the MS sum.
I know that

*

*the sum-of-squares add up to the total sum-of-square (first column in picture)
$$SS_{total} = SS_{columns} + SS_{error}$$

*that the total variance is given by
$$var_{total} = SS_{total}/df_{total}$$
However, I often read the following: "we are divide the total variance into a bewteen and a within group variance". I expected that this is actually true. However, from the above example and the two pieces of knowledge I conclude that it's actually the total sum-of-squares, which is seperated and not the total variance.  Is this true or is there a way to seperate the total variance into two groups?

Using R as a calculator, the variance of the 25 observations (without regard to groups)
is $4.088775.$
x = c(1.5377,    0.6923,    1.6501,    3.7950,    5.6715, 
      2.8339,    1.5664,    6.0349,    3.8759,    3.7925,
     -1.2588,    2.3426,    3.7254,    5.4897,    5.7172,
      1.8622,    5.5784,    2.9369,    5.4090,    6.6302,
      1.3188,    4.7694,    3.7147,    5.4172,    5.4889)
var(x)
[1] 4.088775

Also, SS(Total)/DF(Total) $= 4.088792,$ which is the same
as above, except for slight rounding of
SS(Total) = SS(Group) + SS(Resid) in your table.
53.723 + 44.408
[1] 98.131
98.131/24
[1] 4.088792

