What Is the Standard Deviation of the Volume?

[Roger44]

Hello
A manufacturer produces blocks of metal of
length 100 standard deviation 2,
width 200 standard deviation 1
depth 150 standard deviation 3

The mean volume will be 100x200x150 but what will its sd function be? (The figures I've given are just for clarity)

Thanks in advance, I've really no idea of how to tackle this problem.

 

[mfb]

sd function? Standard deviation? If you are interested in more details, you need more details how the individual lengths vary.
Simple error propagation in the product will work.

 

[mathman]

Assuming that the length, width, and depth are statistically independent, the calculation is straightforward.
Let V(X) (=E(X²) - E(X)²) be variance of X. Let L,W,D be the three dimensions.
E((LWD)²) = E(L²)E(W²)E(D²), E(LWD) = E(L)E(W)E(D) (independence).
E(L²) = V(L) + E(L)², etc.
Combine to get V(LWD) in terms of the individual varances and means.

In case you don't remember statistics, variance is square of standard deviation.
If the dimensions are not statistically independent, you have a much harder problem.

 

[Roger44]

Many thanks

Straightforward, straightforward, for somebody who doesn't know what variance means and who has never ever seen E(..) before? But more than willing to thrash through it.

I'll be back on this in a few days, or should I say a few late nights... Wish me luck!

Many thanks again.
Roger

 

[mathman]

Many thanks

Straightforward, straightforward, for somebody who doesn't know what variance means and who has never ever seen E(..) before? But more than willing to thrash through it.

I'll be back on this in a few days, or should I say a few late nights... Wish me luck!

Many thanks again.
Roger

E(X) is mean value. For your purposes, variance is V(X) = E(X²) - (E(X))²
Standard deviation is square root of variance.

I am a little surprised that you are not familiar with these terms, given the problem you presented. What do you use for a definition of standard deviation?

 

[Roger44]

Hello

Thanks but obviously I"ve googled a lot since and am now a little more familiar with these new terms. For me the SD was just a number linked to the width at half max. As such it represented the tightness of the distribution, no more, no less. That it should be the root of the variance, a term I didn't know either, was not a thing I needed to know.

Now I have a problem that requires this knowledge, for your help I thank you all.

 

[Roger44]

Hello, using a more familiar language for me the result must be :

Variance of (LWD) = [variance of L + (meanL)^2] * [variance of W + (meanW)^2] * [variance of D + (meanD)^2] - (meanL x meanW x meanD)^2

which should give for my block of metal as example
length 100 standard deviation 2,
width 200 standard deviation 1
depth 150 standard deviation 3

variance of (LWD) = (4 +10000) * (1 +4000) * (9 + 2250) - 100*200*150 = approx 90E9

SD = 300 000 for a volume of 3 million, about 10% whereas the SDs of each dimension <2%

To tell you the truth, I wasn't at all expecting a wider dispersion, on the contrary. By the way, I put this question on a French forum where I live, nobody answered except one person who was also interested in a solution, I'll link him to this topic.

Thanks again

 

[mfb]

Where does the formula come from? It does not look right.

I get ~2.9% as relative standard deviation. 10% is certainly too high.

 

[Roger44]

OK

V(X) =E(X²) - E(X)² standard formula  derived from the definition of variance which is E[(X-E(X))^2] 
so E(X²) = V(X) + E(X)² re-arrangement                                                                        
E((LWD)²) = E(L²)E(W²)E(D²) your formula
= (V(L) + E(L)² )* (V(W) + E(W)² )* (V(D) + E(D)² ) my expansion of your formula
V(LWD) =E((LWD)²) - E(LWD)² standard formula replacing X by LWD
so = (V(L) + E(L)² )* (V(W) + E(W)² )* (V(D) + E(D)² ) - E(LWD)²
( 2² + 100² ) * ( 1² + 200²) * (3² + 150²) - (100*200*150)²
which gives SD/vol = 2,873 %

Don't know where I went wrong.

 

[mfb]

Hmm interesting idea to re-write the variance.

Good, now the results agree.

 

[mathman]

Why do think its wrong? I got the same result. Your s.d. for the original dimensions are 2%, .5%, and 2% so the answer is certainly in the right ball park.

 

[Roger44]

Hello

"Hmm interesting idea to re-write the variance." What line(s) are you referring to? I just took the lines from your posting,
-adding a first line, for my oxn personal comprehension "derived from the definition of variance which is E[(X-E(X))^2"] "
-inserting "so E(X²) = V(X) + E(X)² re-arrangement" to make the subsequent substitution clearer

by "went wrong" I was referring to my first calculation of 10%

By the way, I didn't like the use of E( ) as a way of expressing averages. I, a newcomer, especially balked at the fact that:

E(X)² means the the square of the avarage value of X whereas 3(5+6)² does not mean the square of 3*(5+6)
But I quicky realized that it was the most precise way of expression, but you have to be very very careful with the placement of the brackets.

I'm also a bit surprised at not having learned this terminology at grammar and university in England back in the 60's
Roger

 

[mfb]

"Hmm interesting idea to re-write the variance." What line(s) are you referring to?

Post #9.

E(X)² means the the square of the avarage value of X whereas 3(5+6)² does not mean the square of 3*(5+6)
But I quicky realized that it was the most precise way of expression, but you have to be very very careful with the placement of the brackets.

They are different brackets - one is a function bracket, the other one a "grouping" bracket. But still, it can be confusing, yes.

 

[mathman]

I suppose for clarity you could write (E(X))².

The use of E(X) for mean of X is standard notation.