1. Level 1 Laboratories University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Handling & Propagation of Errors : A simple approach 1

2. Every physical quantity has :  A value or size  Uncertainty (or ‘Error’)  Units Without these three things, no physical quantity is complete. When quoting your measured result, follow the simple rules : e.g. A = 1.71  0.01 m Never quote uncertainty to more than 1 or 2 significant figures (this would make no sense) Always quote main value to the same number of decimal places as the uncertainty Always include Units ! ! (but if the quantity is dimensionless, say so) General Remarks 2

3. A reminder of terminology: ‘Uncertainty’ and ‘Error’ The terms Uncertainty and Error are used interchangeably to describe a measured range of possible true values. The meaning of the term Error is : –NOT the DIFFERENCE between your experimental result & that predicted by theory, or an accepted standard result ! –NOT a MISTAKE in the experimental procedure or analysis ! Hence, the term Uncertainty is less ambiguous. Nevertheless, we still use terms like ‘propagation of errors’, ‘error bars’, ‘standard error’, etc. The term “human error” is imprecise - avoid using this as an explanation of the source of error. A reminder of terminology: ‘Uncertainty’ and ‘Error’ 3

4. If uncertainty in measured x is Δx, what is uncertainty in a derived quantity z (x) ? Error propagation is just calculus – you do this formally in the “Data Handling” course Basic principle is that, if (Δx)/x is small, then to first order: e.g., if z = x n, then : Hence, for this particular function, the percent (or fractional) error in z is :                x x n z z or...... just n times the percent error in x Error Propagation using Calculus Functions of one variable 4

5. Suppose uncertainties in two measured quantities x and y are : Δx and Δy, what is the uncertainty in some derived quantity z (x,y) ? For such functions of 2 variables we use partial differentiation But, combining errors ALWAYS INCREASES total error - so make sure terms add with the same sign : Later we will show that it turns out to be better to add in quadrature i.e. “the root of the sum of the squares” :  2 2 2 2 y y z x x z                      or.... We can usually always handle error propagation in this way by calculus, (see Lab Handbook, Section 6, for details) but next we give some short cuts for common cases... Error Propagation using Calculus Functions of more than one variable 5

6. Instead of differentiating  z/  x,  z/  y etc, a simpler approach is also acceptable : 1. In the derived quantity z, replace x by x + Δx, say 2. Evaluate Δz in the approximation that Δx is small Simplified Error Propagation A short-cut avoiding calculus! Ex. 1 : z = x + a, where a = constant Ex. 2 : z = bx, where b = constant Ex. 3 : z = bx 2, where b = constant (same as obtained earlier for z = x n, with n=2) 6

7. Ex. 4 : z = x + y + xx yy =   z Prob. But in many cases the actual probability distribution is not rectangular but Normal ( or Gaussian) Then, the above method would overestimate  z Prob. xx + yy = zz What is  z for such prob. distributions? Answer : the errors add in quadrature : yy xx zz xx yy i.e., this would mean : Simplified Error Propagation Functions of more than one variable 7 Mnemonic :

8. Ex. 5 : z = x  y For a more accurate estimate, use the quadrature formula : Aside :For complicated expressions for z, as a last resort, you can always try the “Min-Max” method to get  z : given any errors in x, y,....etc 1. Calculate max. possible value z can take: e.g.z max = z( x+  x, y-  y,....) 2. Similarly, calc. min. poss. value z can take, z min = z( x-  x, y+  y,....) 3. Then error can be estimated as :  z  ½ (z max – z min ) Remember : Even a crude estimate of an error is always better than no estimate at all ! i.e. percent error in z  sum of percent errors in x & y Exercise : Show that for z = x/y then (try using the “min-max” method for this example) y y x x z z      Simplified Error Propagation Functions of more than one variable 8