# total deviation of the sum of unrelated means

#### blewsman

##### New Member
As a starter - I am a newbie to this forum (and forums in general). Also, I have a very (very) basic understanding of stats/prob. Thank you in advance for your patience and please don't hesitate to provide feedback that will help me, help myself more efficiently!

I am trying to compute the expected standard deviation of a sum of sample means that are independent. Let me offer an example:

• Larry, Curly, & Mo are each weighed "N" times
• the mean weight is L (for Larry), C (for Curly), M (for Mo)
• The standard deviation for each is Sl, Sc, Sm

How would I compute the mean combined weight of the three and the std dev of the combined mean.

The combined mean seems trivial and I am assuming is L+C+M, but the std dev seems a little trickier. I've searched countless wiki articles, forums, etc and keep arriving at the square root of the sum of the variances:

Stotal = SQRT(Sl^2+Sc^2+Sm^2)

But I'm not convinced. For example, if I run through this exercise with Larry, Curly & Mo and 1) compute the std deviations Sl, Sc, Sm and combine them using the equation above, I get a different answer than if I 2) sum each (N) measurement of Larry, Curly, and Mo, then compute the std deviation.

Thanks in advance for the help!

David

Last edited:

#### Dragan

##### Super Moderator
As a starter - I am a newbie to this forum (and forums in general). Also, I have a very (very) basic understanding of stats/prob. Thank you in advance for your patience and please don't hesitate to provide feedback that will help me, help myself more efficiently!

I am trying to compute the expected standard deviation of a sum of sample means that are independent. Let me offer an example:

• Larry, Curly, & Mo are each weighed "N" times
• the mean weight is L (for Larry), C (for Curly), M (for Mo)
• The standard deviation for each is Sl, Sc, Sm

How would I compute the mean combined weight of the three and the std dev of the combined mean.

The combined mean seems trivial and I am assuming is L+C+M, but the std dev seems a little trickier. I've searched countless wiki articles, forums, etc and keep arriving at the square root of the sum of the variances:

Stotal = SQRT(Sl^2+Sc^2+Sm^2)

But I'm not convinced. For example, if I run through this exercise with Larry, Curly & Mo and 1) compute the std deviations Sl, Sc, Sm and combine them using the equation above, I get a different answer than if I 2) sum each (N) measurement of Larry, Curly, and Mo, then compute the std deviation.

Thanks in advance for the help!

David
You are asking two different questions and thus different computations.

#### blewsman

##### New Member
Dragan, huge thanks for the link! While I'm still not sure I have my answer, it will greatly aid in crystalizing my question.

If I understand correctly, you detail two different questions & the resulting calcs of each:
• the std deviation of the "combined samples" (your 2nd eqn)
• the average std deviation of the individual samples (your 1st eqn)

Think we can cross the 2nd eqn off the list, as I am not looking to measure the std error between the means - and, as you point out, the diff in the means will impact the computed SD using eqn 2. Think of it more as quantifying the error in the measurement equipment, rather than the difference in the measured value.

I believe the 1st equation is a derivative of the eqn I posted above. Difference being avg vs sum of the std dev. I think this calc is closer to what I am looking for, but the numbers just don't seem to jive. In the example below, the computed SD using SQRT(Sl^2+Sc^2+Sm^2) is ~2.4...which is quite different the 1.9 computed directly.

What am I missing?