# Velocity Measurement Error Estimate

I have 2 position estimates (along with their measurement error) and a difference in time between estimates. I estimate velocity using

Velocity = (PosA - PosB)/DeltaT


I am trying to estimate the error in my velocity estimate, but I can't seem to find any ways to calculate this. I assume it has to use Sigma_PosA and Sigma_PosB. I would also assume it's relative to DeltaT and/or abs(PosA - PosB). What is the velocity measurement variance/standard deviation?

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What do you know about the error in the two positions? Do you have an explicit distribution, or just an error term? –  templatetypedef Aug 31 '11 at 21:23
Are the errors in the positions measurements uncorrelated? (If you don't understand that question, the answer is probably "yes".) –  Beta Aug 31 '11 at 21:30
templatetypedef: I am assuming a gaussian distribution with a standard deviation of Sigma_Pos –  user858146 Aug 31 '11 at 21:55
Beta: I'm not sure if they are. They are the same object, but that's the only relationship between the 2 measurements. –  user858146 Aug 31 '11 at 21:57

## migrated from stackoverflow.comAug 31 '11 at 22:09

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sigmav = sqrt((sigmaA)2 + (sigmaB)2) / (DeltaT)

EDIT:

(Corrected an error above-- DeltaT should not be squared.)

It sounds as if the measurements are independent, so the errors are uncorrelated. We want the standard deviation of a linear combination of (two) variables:

$V = \frac{(B-A)}{\Delta_t} = \frac{1}{\Delta_t}B - \frac{1}{\Delta_t}A$

$\sigma_V^2= \sum_i^n a_i^2\sigma_i^2 = (\frac{1}{\Delta_t})^2\sigma_B^2 + (\frac{1}{\Delta_t})^2\sigma_A^2 = (\frac{1}{\Delta_t})^2(\sigma_A^2 + \sigma_B^2)$

$\sigma_V = \sqrt {(\frac{1}{\Delta_t})^2(\sigma_A^2 + \sigma_B^2)} = \frac{\sqrt{\sigma_A^2 + \sigma_B^2}}{\Delta_t}$

(I'm new to math.stackexchange-- gotta say I'm lovin' the MathJax.)

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Can you explain where this comes from? –  templatetypedef Aug 31 '11 at 21:41
It looks like that would work, but I would like to see some explanation. –  Charles L. Sep 1 '11 at 2:18
sqrt(sigma_posA * sigma_posA + sigma_posB * sigma_posB)