# multiplying Standard error of means

#### frymor

##### New Member
Hi guys,

I am new here (and also to statistics ).

I have a data set of many measurements. For each one of them I calculated the mean and the SD (deviation). From that I calculated separately for each of the data set a SEM.
Now I would like to multiply, divide add and subtract this data samples from/with each other.

Doing so for the actual values is quite trivial, but what do I do with the SEM-values.
How do this SEM Values changed according the applied calculation?

Example( with data from the internet):
set 1: 46,42,44,45,43 => mean 44 ; SD= 1.6 ==> SEM : 1.6
set 2: 52,80,22,30,36 => mean 44 ; SD = 22.9 ==> SEM : 10.3

What do I do with the two sets if I want to combine them together?

I would appreciate any help I can get.

Thanks
Assa

#### SPR

##### New Member
Hello Frymor,

I do not fully understand what do you mean by doing arithmetic with samples. For example, addition could be considered as combining data from both samples having one with double number of observations or adding paired observations of two samples giving a sample of the same size.

In the first case standard deviation of combined sample equals (in case of large number of observations)

s=SQRT(0.5*s1+0.5*s2+0.25*(m1-m2)**2)

and its mean

m=0.5*(m1+m2)

sem=s/SQRT(n) so you can derive formulas for combined sample yourself.

Sincerely,
SPR

#### frymor

##### New Member
Hi SPR,

thanks for the answer. I'll try to explain what I meant with multiplications.

We have many observations for the number of people in a city. so we calculated the number and the SEM, which gave us 2200 +- 150. Than we have the number of cities in each country with 120 +- 17 (SEM).

Now I would like to calculate the number of persons in each country including the standard error of the means. T calculate the number per each country I just do 2200/120 = 18.333, but how do I include here the SEM? do I also just divide them?

The same question apply for the multiplicaton of the data, if i want to find out how many people do I have in total. I can multiply 2200*120,but what about the SEM?

I hope this clarify my question a bit more.

Thanks
Assa

#### SPR

##### New Member
Hello Frymor,

I've got what you need. Let x1=x+/-Dx (from that point I'll omit minus for simplicity) and y1=y+Dy then x1=x(1+Dx/x)=x(1+dx) and the same for y1.

z+Dz=y1/x1=y/x*(1+dy)/(1+dx)={according to Taylor's expansion 1/(1+dx) ~ 1-dx for small dx}=y/x*(1+dy)*(1-dx)=y/x(1+dx-dy+dx*dy). And neglecting dx*dy as small we are getting the result z+DZ=y/x(1+dx-dy) and to be more conservative z+Dz=y/x*(1+dx+dy) so Dz=y/x*(DX/x+Dy/y).

In your case x=120 Dx=17 y=2200 Dy=150 we get
dx=0.142 dy=0.0681 (dx*dy=0.0096 << 0.0681) and
Dz=18.33+/-3.85

The same way for multiplication. If dx or dy is large enough then you need more terms in Taylor's expansion.

Sincerely,
SPR

#### fattydog

##### New Member
Hi, there!

I am a new to statistics. Just follow this thread to this forum.
My question is: Just as above, what about if I want to do X to the power of Y, or Y to the power of X, what kind of formula I should know or any hints?