How do I propagate uncertainties through a calculation ...

[Pages:4]How do I propagate uncertainties through a calculation? (Step-by-step method)

You have made some measurements during your experiment, estimated the uncertainty in those measurements, and now you have to input those values into a formula. Of course you know what to do with the measured values ... stick the numbers in the formula and do the math! But what do you do with the uncertainties that accompany those measurements? It is easy - there are simple rules that help you tell you how to combine the uncertainties as you step through the formula, one mathematical step at a time.

Before we get to those simple rules and some examples, we first need to look at some definitions.

Some definitions for the Step-by-Step method of propagating uncertainties

Best estimate of the true value: this is our best estimate of the value we are trying to measure or calculate. We call it the best estimate because we do not know the true value, but can only estimate it. If we make multiple measurements of the same thing, it is usually the average. If you see a value listed with it's uncertainty, the best estimate is the number to the left of the "?".

Absolute uncertainty: this is the largest likely difference between the best estimate and the true value. This number has the same units as the measured or calculated "best" value. It is located to the right of the "?".

Relative uncertainty: this is sometimes called the fractional uncertainty and we often express it as a percent to remind ourselves that it is a relative uncertainty rather than the absolute uncertainty. Simply, it is the absolute uncertainty divided by measured ("best") value.

Experimental range: this is the range of values that we believe bracket the true value. We think that our best estimate is closest to the true value, but we acknowledge because of uncertainty, that it is likely to lie anywhere in the range. To find the minimum of the range take the best estimate and subtract the absolute uncertainty. To find the maximum of the range, take the best estimate and add the absolute uncertainty.

Definitions Example: x = (15.5 ? 0.1) cm

What is the best estimate of x?

15.5 cm

What is the absolute uncertainty of x?

0.1 cm

What is the relative uncertainty of x?

0.1/15.5 = 0.006452 = 0.6452%

What is the range of x?

15.4 to 15.6 cm

CSU Pomona

Updated 1/20/19

Dr. Julie J. Nazareth

Rules for combining uncertainties during the step-by-step method of propagating uncertainty

The rules below tell you how to combine the uncertainties in each step of the calculation.

Rule #1 ? Addition and/or Subtraction of numbers with uncertainty Add the absolute uncertainties.

Rule #2 ? Multiplication and/or Division of numbers with uncertainty Add the relative uncertainties.

Rule #3 ? Powers applied to numbers with uncertainty (like squared or square root) Multiply the relative uncertainty by the power.

(Unofficial) Rule #4 ? Multiplying by a number without uncertainty (like ? or ) Multiply the absolute uncertainty by the number without uncertainty

Things to remember when propagating uncertainty through a calculation

? Do the calculation step by step following the order of mathematical precedence. ? In each step, do what the calculation says to do to the best estimates. ? Follow the appropriate rule to combine the uncertainties in each step. ? Keep 2 extra significant digits in each intermediate step to reduce rounding error and

only round severely in the very last step for the final.

? Only convert between absolute and relative uncertainties when you have to, to reduce rounding error.

? Remember, the uncertainty in the final answer must be an absolute uncertainty, so if needed, you must convert a relative uncertainty to an absolute uncertainty before the final severe rounding step.

? Include units. ? ALWAYS SHOW WORK.

Examples of Single (mathematical) step calculations

Ex 1: Addition (Use Rule #1) (10 ? 2) cm + (5 ? 1) cm = (15 ? 3) cm

Ex 2: Subtraction (Use Rule #1) (10 ? 2) cm - (5 ? 1) cm = (5 ? 3) cm

Ex 3: Multiplication (Use Rule #2)

(10 ? 2) cm x (5 ? 1) cm

relative uncertainties: 2/10 = 0.20 = 20%;

= (10 cm ? 20%) x (5 cm ? 20%) = 50 cm2 ? 40%

1/5 = 0.20 = 20%

= (50 ? 20) cm2

convert relative to absolute uncertainty for final answer 40% of 50 cm2 = (0.40)(50 cm2) = 20 cm2

CSU Pomona

Updated 1/20/19

Dr. Julie J. Nazareth

Ex 4: Division (Use Rule #2) [(10 ? 2) cm]/[(5 ? 1) s] = (10 cm ? 20%)/(5 s ? 20%) = 2.0 cm/s ? 40%

= (2.0 ? 0.8) cm/s

relative uncertainties: 2/10 = 0.20 = 20%; 1/5 = 0.20 = 20%

convert relative to absolute uncertainty for final answer 40% of 2.0 cm/s = (0.40)(2.0 cm/s) = 0.8 cm/s

Ex 5: Powers (Use Rule #3) [(100.0 ? 2.0) cm2]? = (100.0 cm2 ? 2.00%)? = 10.0 cm ? ? (2.00%) = 10.0 cm ? 1.00%

= (10.0 ? 0.1) cm

relative uncertainty: 2.0/100.0 = 0.0200 = 2.00%;

convert relative to absolute uncertainty for final answer 1.00% of 10.0 cm = (0.010)(10.0 cm) = 0.1 cm

Ex 6: Powers (Use Rule #3) [(10.0 ? 2.0) cm]3 = (10.0 cm ? 20%)3 = 1000 cm3 ? 3(20%) = 1000 cm3 ? 60%

= (1000 ? 600) cm3 = (1.0 ? 0.6) x103 cm3

relative uncertainty: 2.0/10.0 = 0.200 = 20%;

convert relative to absolute uncertainty for final answer 60% of 1000 cm3 = (0.60)(1000 cm3) = 600 cm3

Ex 7: Multiplying by a number without uncertainty (Use Unofficial Rule #4) ? (4.2 ? 0.2) cm = ? (4.2 cm) ? ? (0.2 cm) = (2.1 ? 0.1) cm

Please note that all of the single step calculations result in a final answer that is converted back to absolute uncertainty for the final step. Remember, the uncertainty in the final answer must be an absolute uncertainty. In a multistep calculation, only convert between absolute and relative uncertainties when you have to, to reduce rounding error.

Example of a Multiple (mathematical) step calculation Do the calculation step by step following the order of mathematical precedence.

Ex 8: multiple step calculation involving subtraction, powers and division A = [(26.72 ? 0.05) cm ? (13.8 ? 0.2) cm] / [(4.11 ? 0.03) s]2 = ?

Step 1: subtraction ? Use Rule #1 to combine uncertainties. A = [(26.72 - 13.8) cm ? (0.05 + 0.2) cm] / [(4.11 ? 0.03) s]2

CSU Pomona

Updated 1/20/19

Dr. Julie J. Nazareth

A = [(12.92) cm ? (0.25) cm] / [(4.11 ? 0.03) s]2

Step 2: Square ? Use Rule #3 to combine uncertainties.

Convert the absolute uncertainty to relative uncertainty to use Rule #3 (0.03 s)/(4.11 s) = 0.0072992 ? 0.730%

A = [(12.92) cm ? (0.25) cm] / [4.11 s ? 0.730%]2 A = [(12.92) cm ? (0.25) cm] / [(4.11 s)2 ? (2)(0.730%)] A = [(12.92) cm ? (0.25) cm] / [16.8921 s2 ? 1.46%]

Step 3: Division ? Use Rule #2 to combine uncertainties.

Convert the absolute uncertainty in the numerator to relative uncertainty to use Rule #2. The uncertainty in the denominator is already a relative uncertainty from the end of step 2. (0.25 cm)/(12.92 cm) = 0.0193498 ? 1.93%

A = [12.92 cm ? 1.93%] / [16.8921 s2 ? 1.46%] A = [(12.92 cm)/( 16.8921 s2)] ? (1.93% + 1.46%) A = 0.764855 cm/s2 ? 3.39%

Step 4: Final step ? rounding

If our uncertainty is not an absolute uncertainty, we have to convert for the final answer. 3.39% of 0.764855 cm/s2 = (0.0339)( 0.764855 cm/s2) = 0.025929 cm/s2

Now do the severe rounding for the final answer. See Do's and Don'ts #26 for help.

A = (0.764855 ? 0.026)cm/s2

DD #26 part a)

A = (0.765 ? 0.026)cm/s2

DD #26 part b)

A = (0.765 ? 0.026)cm/s2 {Best}

{A = (0.76 ? 0.03)cm/s2 is OK answer}

CSU Pomona

Updated 1/20/19

Dr. Julie J. Nazareth

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