Appendix 5

ERROR PROPAGATION & COMPARISON

A typical pattern for a sophomore physics experiment is to calculate a final result from measurements of several interacting physical quantities. This result is then compared to what is expected from theory (or another experimental method). If the two results differ, the theory is not necessarily faulty. Remember, there is no real disagreement if the difference between the two results is no larger than what might be expected from the probable random or systematic errors in the measurements.

The question at hand then is "How do you combine the uncertainties to give the uncertainty in a final result?" In science combining uncertainties is called propagating errors. (That's incredibly unscientific, it should be called "propagating uncertainties). The goal of error propagation is to predict how large an uncertainty can be expected in a result calculated from measurements which possess intrinsic uncertainties.

You might be tempted to guess that uncertainties add algebraically every time you combine numbers that have uncertainties. For example, if you wanted to add two lengths, l₁ = 50.3 + 0.2 cm and l₂ = 22.6 + 0.1 cm, you might well guess that the total length is: 1 = l₁ + l₂ = 72.9 + 0.3 cm. This is NOT correct. However it is a good approximation of a more rigorous method of propagating uncertainties.

I. Sketchy Derivation of the Error in a Final Result

This more rigorous method is derived by assuming that the uncertainty in the final result is just the sum of each individual quantity's uncertainty weighted by the amount each individual quantity affects the final result. The amount each quantity affects the final result is the partial derivative of the final result with respect to that given quantity. Using the above example of two lengths this translates into:

In general, consider the final result as a quantity, f, and assume f depends upon two independent, measured quantities, x and y. You may then write f as a function of these two quantities: f = f(x,y). In these more general terms the above propagated uncertainty becomes;

However, a second simplifying assumption can be made. To understand this, first square the above equation to get:

sf² = [∂f/∂x]²sx² + [∂f/∂y]²sy² + 2(∂f/∂x)(∂f/∂y)sxsy

The last term in the above equation is often called the correlation term because it represents the degree to which any fluctuations in both x and y affect sf. For example, this correlation term will increase if x and y vary so that sx, sy and f all increase simultaneously. These sort of correlations between two independent, measured quantities (x and y) are rare. Hence this correlation term will be neglected. This yields a useful equation for calculating the propagated error in a function f.

sf² = [∂f/∂x]²sx² + [∂f/∂y]²sy² (5.1)

When more than two measured quantities go into a calculation to produce a final result, then the above equation is expanded to take into account the other terms. For example, if f = f(x,y,z), then the above equation becomes:

sf² = [∂f/∂x]²sx² + [∂f/∂y]²sy² + [∂f/∂z]²sz² (5.2)

Equation (5.l) is useful, only if the individual uncertainties are known. In practice, these sx's and sy's are either estimated uncertainties from a few measurements of x and y OR they are the standard deviations of the means of x and y.

You are expected to use this relation (or the consequent relations that follow) in your Data Analysis section in all experiments in which you cannot make a good case for a less sophisticated comparison of results. However, sometimes a full scale error propagation is NOT needed. Only experience will allow you to determine which is the correct path to follow.

It is usually more convenient to use the equations in the following table which have been derived from equation (5.l) for several of the most frequently occurring functions. To show you how the following table of error propagation formulas have been derived, several examples are first presented.

II. Error Propagation For Some Common Functions

1. Addition and Subtraction.

If f = x ± y, then ∂f/∂x = 1 and ∂f/∂y = ±1. Since these partial derivatives are squared, the ± sign has no effect, so equation (5.1) yields:

sf² = s_x² + s_y²

The rule for addition and subtraction is therefore that uncertainties add "in quadrature", (which is a fancy word for the square root of the sum of the squares of the individual uncertainties), instead of simply adding uncertainties.

2. Multiplication and Division.

If f = xy, then ∂f/∂x = y, ∂f/∂y = x, so equation (5.1) yields

s_f² = y²s_x² + x²s_y².

This is usually put into a more convenient form by dividing through by f² = x²y² to obtain

[s_f/f]² = [s_x/x]² + [s_y/y]², (5.3)

whose square root gives the uncertainty as a fractional part of f, a most useful form. For f = x/y, exactly the same equation is found for s_f/f !

3. Powers.

Let f = axⁿ, where a is a known constant and x is the variable. Then [∂f/∂_x]² = [nax^n-1]² = [nf/x]², so equation (5.1) yields

s_f/f = n[s_x/x].

III A Table of Error Propagation Formulas

Given an arbitrary function (f), two independent variables (x and y), two constants (a and b) and an integer n, the following formulas may be used to compute the propagated error (uncertainty) in the function f. The uncertainties for x and y (s_x and s_y) can be either standard deviations or estimated uncertainties. If they are standard deviations, then x and y must be those corresponding

means.

f(x,y) s_f

-------------------------------------------------------------

x ± y 1. [s_x²+ s_y²]^1/2

xy or x/y 2. xy[(s_x/x)² + (s_y/y)²]^1/2

x² 3. 2x²[s_x/x] = 2x[s_x]

axⁿ 4. naxⁿ[s_x/x] = u figger it.

a(x/y²) 5. a(x/y²)[(s_x/x)² + (s_y/y)²]^1/2

ae^bx 6. abe^bx[s_x]

aln(bx) 7. [a/x]s_x

IV. Words To The Wise

Calculating standard deviations is generally the best way to evaluate uncertainties, however, you must exercise judgement as to when data warrant such calculations. Sometimes you may be able to decide upon a valid uncertainty by "eyeballing" the data or by plotting the data or by closely scrutinizing the scale from which the data was read. By considering how the uncertainties in several measured quantities will propagate, you can often conclude that the uncertainties in all but one or two of the quantities can be neglected in calculating the propagated error. Such a conclusion should, of course, be defended in your report.

In this course you are responsible for evaluating all systematic and random errors in all measured quantities as rigorously and quantitatively as practicable and to propagate them mathematically when necessary. Then, most importantly, you must compare the disagreements in your results with these uncertainties.

For a very readable treatment of error propagation, you may be interested in reading P. R. Bevington's Data Reduction and Error Analysis for the Physical Sciences, McGraw Hill, New York, 1969. This book is in your CACC library at this very moment!

V Error Comparison

Suppose you have just completed an experiment in which you measured several quantities, calculated their uncertainties, used these quantities to compute an experimental result (called x_e) and propagated these uncertainties to an uncertainty in the final result (called s_xe). Further, suppose that some theory predicts that you should have obtained a value of x_t+s_xt as your final result. There exists a well defined method of determining whether these two x's agree within the stated experimental and theoretical error bars. To do this you must first compute the disagreement between the theoretical and experimental result (called d below)

d = |x_e - x_t| (don't forget x_e and x_t are mean values).

Next compute the propagated uncertainty in this disagreement

s_d = [s_xe² + s_xt²]^l/2

If the disagreement (d) is larger than this propagated uncertainty (s_d) then x_e and x_t do not agree within the limits of experimental error. But if d < s_d then the two x's do agree and the experiment can be termed a success. This procedure is the best quantitative method for judging the success or failure of an experiment. You will sometimes be required to determine whether the difference between your final result and a theoretical predication is less than the propagated uncertainty in this difference.

Do not interpret "failure" here to mean that everything you've done is worthless. Many factors have gone into this sequence of calculations. For example, if you estimated some uncertainties in the initial measurements, then perhaps you overestimated the accuracy you could obtain. Also there might be systematic errors present in the experiment. It was stated at the outset that this quantitative method of calculating and propagating errors applied only to random errors. Therefore if systematic errors are present, this analysis has neglected them.

If sd is slightly larger than d, the next best thing is to see if d < (sxe + sxt) since sxe + sxt is the maximum difference you might expect to obtain. If the disagreement is smaller than this upper bound then the experiment can still be called marginally successful. However, if the disagreement is larger than

3(sxe + sxt) then you probably made some procedural mistake which has invalidated your results. You must make an attempt to uncover such mistakes or postulate some plausible experimental errors which could cause this failure. Without this sort of critical thought, any preceding pages of calculations are worthless and your report grade will reflect this.

Term	Definition	Symbol or Eqn.
the quantity	the universal unknown to be measured	x
mean	the average of a set of measurements.	µ
uncertainty	the intrinsic difference between a measured quantity and the "true" or "perfect" value of that quantity.	sx
fractional uncertainty	the uncertainty of a quantity divided by that quantity.	sx/x
percent uncertainty	the fractional uncertainty multiplied by 100.	100(sx/x)
discrepancy	the absolute value of the difference between a theoretical value and an experimental value.	d = |xt - xe|
percent error	the difference between a theoretical value and an experimental value, expressed as a percent of the theoretical value.
percent difference	the difference between two experimental expressed as a percent of their average.

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