Uncertainty in a Single Measurement

Uncertainty in a single measurement

Bob weighs himself on his bathroom scale. The smallest divisions on the scale are 1-pound marks, so the least count of the instrument is 1 pound.

Bob reads his weight as closest to the 142-pound mark. He knows his weight must be larger than 141.5 pounds (or else it would be closer to the 141-pound mark), but smaller than 142.5 pounds (or else it would be closer to the 143-pound mark). So Bob's weight must be

weight = 142 +/- 0.5 pounds

In general, the uncertainty in a single measurement from a single instrument is half the least count of the instrument.

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Fractional and percentage uncertainty

What is the fractional uncertainty in Bob's weight?

uncertainty in weight

fractional uncertainty = ------------------------

value for weight

0.5 pounds

= ------------- = 0.0035

142 pounds

What is the uncertainty in Bob's weight, expressed as a percentage of his weight?

uncertainty in weight

percentage uncertainty = ----------------------- * 100%

value for weight

0.5 pounds

= ------------ * 100% = 0.35%

142 pounds

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Combining uncertainties in several quantities: adding or subtracting

When one adds or subtracts several measurements together, one simply adds together the uncertainties to find the uncertainty in the sum.

Dick and Jane are acrobats. Dick is 186 +/- 2 cm tall, and Jane is 147 +/- 3 cm tall. If Jane stands on top of Dick's head, how far is her head above the ground?

combined height = 186 cm + 147 cm

= 333 cm

uncertainty in combined height = 2 cm + 3 cm

= 5 cm

combined height = 333 cm +/- 5 cm

Now, if all the quantities have roughly the same magnitude and uncertainty -- as in the example above -- the result makes perfect sense. But if one tries to add together very different quantities, one ends up with a funny-looking uncertainty. For example, suppose that Dick balances on his head a flea (ick!) instead of Jane. Using a pair of calipers, Dick measures the flea to have a height of 0.020 cm +/- 0.003 cm. If we follow the rules, we find

combined height = 186 cm + 0.020 cm

= 186.020 cm

uncertainty in combined height = 2 cm + 0.003 cm

= 2.003 cm

??? combined height = 186.020 cm +/- 2.003 cm ???

But wait a minute! This doesn't make any sense! If we can't tell exactly where the top of Dick's head is to within a couple of cm, what difference does it make if the flea is 0.020 cm or 0.021 cm tall? In technical terms, the number of significant figures required to express the sum of the two heights is far more than either measurement justifies. In plain English, the uncertainty in Dick's height swamps the uncertainty in the flea's height; in fact, it swamps the flea's own height completely. A good scientist would say

combined height = 186 cm +/- 2 cm

because anything else is unjustified.

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Combining uncertainties in several quantities: multiplying and dividing

When one multiplies or divides several measurements together, one can often determine the fractional (or percentage) uncertainty in the final result simply by adding the uncertainties in the several quantities.

Jane needs to calculate the volume of her pool, so that she knows how much water she'll need to fill it. She measures the length, width, and height:

length L = 5.56 +/- 0.14 meters

= 5.56 m +/- 2.5%

width W = 3.12 +/- 0.08 meters

= 3.12 m +/- 2.6%

depth D = 2.94 +/- 0.11 meters

= 2.94 m +/- 3.7%

To calculate the volume, she multiplies together the length, width and depth:

volume = L * W * D = (5.56 m) * (3.12 m) * (2.94 m)

= 51.00 m^3

In this situation, since each measurement enters the calculation as a multiple to the first power (not squared or cubed), one can find the percentage uncertainty in the result by adding together the percentage uncertainties in each individual measurement:

percentage uncertainty in volume = (percentage uncertainty in L) +

(percentage