Extremely memory consuming Expand

Try this:

a = (x[1] - x[11])^2 + (x[2] - x[12])^2 + (x[3] - x[13])^2 + (x[4] - x[14])^2; 
b = (x[1] - x[21])^2 + (x[2] - x[22])^2 + (x[3] - x[23])^2 + (x[4] - x[24])^2; 
c = (x[1] - x[31])^2 + (x[2] - x[32])^2 + (x[3] - x[33])^2 + (x[4] - x[34])^2; 
d = (x[21] - x[31])^2 + (x[22] - x[32])^2 + (x[23] - x[33])^2 + (x[24] - x[34])^2; 
e = (x[11] - x[31])^2 + (x[12] - x[32])^2 + (x[13] - x[33])^2 + (x[14] - x[34])^2; 
f = (x[11] - x[21])^2 + (x[12] - x[22])^2 + (x[13] - x[23])^2 + (x[14] - x[24])^2; 
g = (1/12)*Sqrt[4*a*b*c - a*(b + c - d)^2 -  b*(a + c - e)^2 + (b + c - d)*
(a + c - e)*(a + b - f) - c*(a + b - f)^2]; 
g = g /. x[i_] -> (z[i] - [Mu][i])*t + [Mu][i]; 
taylor = Normal[Series[g, {t, 0, 2}]] /. t -> 1;
mean = taylor //. z[i_]^2 -> [Sigma]^2 + [Mu][i]^2; 
mean = Simplify[mean //. z[i_] -> [Mu][i]]

just omit the expansion

To avoid confusion I post another answer. The OP is right: the Expand is needed for the correct replacement of the square z terms. You can do the given task with 64 GB of RAM. One way is to do Expand not on the whole expression but in 'chunks', that is on the parts appearing within square roots, nominators and denominators separately. Begin with above code:

a = (x[1] - x[11])^2 + (x[2] - x[12])^2 + (x[3] - x[13])^2 + (x[4] - x[14])^2;
b = (x[1] - x[21])^2 + (x[2] - x[22])^2 + (x[3] - x[23])^2 + (x[4] - x[24])^2;
c = (x[1] - x[31])^2 + (x[2] - x[32])^2 + (x[3] - x[33])^2 + (x[4] - x[34])^2;
d = (x[21] - x[31])^2 + (x[22] - x[32])^2 + (x[23] - x[33])^2 + (x[24] - x[34])^2;
e = (x[11] - x[31])^2 + (x[12] - x[32])^2 + (x[13] - x[33])^2 + (x[14] - x[34])^2;
f = (x[11] - x[21])^2 + (x[12] - x[22])^2 + (x[13] - x[23])^2 + (x[14] - x[24])^2;
g = 1/12*Sqrt[4*a*b*c - a*(b + c - d)^2 - b*(a + c - e)^2 - 
 c*(a + b - f)^2 + (b + c - d)*(a + c - e)*(a + b - f)];
g = g /. x[i_] -> (z[i] - [Mu][i]) t + [Mu][i];
taylor = (Series[g, {t, 0, 2}] // Normal) /. t -> 1

the resulting expression appears as short output, -> Show All. Then identify nine terms, copy them 'by hand' into variables t1,...,t9. I worked from the back. Denominators, nominators and brackets within square roots. The result has the following structure:

(1/6)*Sqrt[t9] + (Sqrt[t8]*t7)/t6 + (1/12)*Sqrt[t5]*(-(t4/t3) + t2/t1).

Treat the variables all like

t1 = (((t1 // Expand) //. z[i_]^2 -> [Sigma]^2 + [Mu][i]^2) //.z[i_] -> [Mu][i]);

The total result can be simplified as t9=t8=t5 to

Sqrt[t9]*(1/6 + t7/t6 + (1/12)*(t2/t1 - t4/t3))

and evaluate above expression. It will be VERY large, takes a long time to even display (the nb file will be about half a GB). I would not try Simplification on the whole expression, but maybe on the parts t1...t9 after the replacement of z^2 and z. So it can be done, but will it be useful?