Absolute Uncertainty – Relative Uncertainty

Uncertainties and Significant Figures

Absolute Uncertainty ? the absolute uncertainty is the number which, when combined with a reported value, gives the range of true values. For instance, a length may be reported as 7.3 mm ? 0.2 mm. Here, the reported value is 7.2 mm and the absolute uncertainty is 0.2 mm; the range of true values is 7.1 mm to 7.5 mm. Absolute uncertainties always have the same units as the reported value with which that are associated. When you are told to report "the uncertainty" with your results, you should give the absolute uncertainty.

Relative Uncertainty ? The relative uncertainty is the ratio of the absolute uncertainty to the reported value. A length of 100 cm ? 1 cm has a relative uncertainty of 1 cm/100 cm, or 1 part per hundred (= 1% or 1 pph). Relative uncertainties are always unitless. Multiplying the relative uncertainty by the reported value yields the absolute uncertainty. For instance, a mass of 2.042 g which has a relative uncertainty of 3 parts per thousand (3 ppt) should be reported as 2.042 g ? 0.006 g.

The rules given below show the relation between uncertainty and the number of digits with which an answer should be reported.

RULE 1: Any value reported as the result of a measurement should contain only digits that are known exactly plus a final digit which is associated with some uncertainty. If the uncertainty associated with a value is not specified, it is assumed to be ? 1 in the final digit.

Correct: The mass of my pencil is 10.94 g ? 0.03 g.

Incorrect: The length of my desk is 1.88239 m ? 0.3049 g.

If the uncertainty is not specified, the number of significant figures can usually be determined by the manner in which the value is written.

Value 305 305.0 5.00 3.0 ? 105 3.00 ? 105 30, 000

# Significant Digits 3 4 3 2 3

1 or 2 or 3 or 4 or 5

RULE 2: When data points are combined, the uncertainties associated with those points accumulate.

When data are added or subtracted, absolute uncertainties are added. (72 g ? 2 g ) + (68 g ? 1 g) = 140 g ? 3 g (72 g ? 2 g ) ? (68 g ? 1 g) = 4 g ? 3 g

When data are added or subtracted, relative uncertainties are added.

(72 m ? 2 m ) ? (68 m ? 1 m) = 4900 m2 ? 200 m2 (relative uncertainty = (2 m/72 m) + (1 m/68 m) = 0.042; absolute uncertainty = 4896 m2 ? 0.042 = 206 m2)

(72 g ? 2g) ? (68 mL ? 1 mL) = 1.06 g/mL ? 0.04 g/mL

(relative uncertainty = (2 g/72 g) + (1 mL/68 mL) = 0.042;

absolute uncertainty = 1.06 g/mL ? 0.042 = 0.04 g/mL)

Approximation to RULE 2: When adding or subtracting, report the answer with the same number of place values as the data point which is least precise.

Correct: 78.074 g 26.1 g 18.62 g 122.794g = 122.8 g

Incorrect: 78.074 g 26.1 g 18.62 g 122.794g

When multiplying or dividing, report the answer with the same number of digits as there are in the least precise data point. Correct: 2.53 cm ? 0.143 cm ? 18.96 cm = 6.86 cm3 Incorrect: 2.53 cm ? 0.143 cm ? 18.96 cm = 6.8595384 cm3 Exercise: A block of metal which measures 8.0 cm by 10.0 cm by 3.0 cm has a mass of 1.92 kg. Assume the uncertainty in each length is 0.1 cm and the uncertainty in the mass is 0.03 kg. Find the density of the metal sample. Does the density of zinc (7.14 g/cm3) fall within the range of the true values of the density of the block?

Answer: d = 8.0 g/cm3 ? 0.6 g/cm3; the density of zinc does not fall in this range of values.

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