# Error propagation and Standard Deviation

I never quite got the hang of this during my entry stat course and it has been bugging me for a long time now.

Lets assume I'm trying to find the focal length of a concave lens. Using a mockup formula for f, lets say:

f = D*x/y


where x , y, and D are measurements done with calipers with "precision" of 1mm each. Now I continue my experiment and make 10 different measurements for x, and 10 for y and compute 10 values for my f.

I want to present one final value for f and an "error" associated with that value. Do I use the mean value of my 10 f's as my final value? what do I use as a representation of "error"? Do I use the STD of mean for my 10 values or do I propagate the 1mm errors from my measurement and then use them to compute a weighed mean with associated error?

Right, at the first-year undergraduate level, this is how you typically do it.

• Firstly, if you were to only take one measurement for x,y and z; use the instrument uncertainty (half its smallest scale or whatever is stated by the manufacturer), and propagate its error to f. So your final answer would look like:

where

Now, there is a better way to estimate the uncertainty of f. This is done by taking multiple measuremenets instead of only 1 measurement. By taking multiple measurements, we can obtain the standard deviation and subsequently the standard error (aka standard deviation of the mean). There is a caveat: the minimum number of measurements that should be carried out to utilize the standard error is N >= 5.

• When multiple measurements are taken, calculate the mean of each variable alongside its standard error (by dividing the standard deviation of each variable by sqrt(N)). In the case of f, solve for it by using the mean of each variable. In the case of delta(f), solve for it by propagating the standard errors of each variable through the same manner as shown in the equation above. This will give you the best estimate of f +- delta(f) in the case of multiple readings.

Now, there is still the uncertainty of the instrument in hand! (Precision of 1mm as you stated). In the first-year undergraduate level, this instrument uncertainty is typically ignored (when taking multiple readings). However, if there is a large discrepancy between experimental and theoretical value, this uncertainty can be taken into account. This is done by considering the instrumental uncertainty as a form of systematic error.

• To include instrumental uncertainty, let delta(f) calulcated above be considered as delta(fRandom). To calculate delta(fSystematic), multiply the instrumental uncertainty with the mean values of each variable to get the systematic uncertainty for each variable. Then, propagate the error for delta(fSystematic). The resultant/net uncertainty can be expressed the sum of the terms in the quadrature:

Therefore the best way to describe an experimental value in this case would be

You need to propagate the error. fortunately your equation is simple and you can plug into a formula and avoid doing any calculus!

for multiplication/division like f = D*xy

(sf/f)^2 = (sD/D)2 + (sx/x)^2 + (sy/y)^2

where s is the standard deviation or error you calculated for each measured value, and sf being the error of the final value that you solve for.

for completeness, there's also a simple equation for addition/subtraction like x = a + b - c

(sx)^2 = (sa)^2 + (sb)^2 + (sc)^2

again where s is the error of the measured varaibles a b and c.

Constants without measurement error are simply carried through.