Theory:  Measurement Uncertainty

 

Uncertainty in a single laboratory measurement is a product of two variable factors:

 

·        resolution of the measuring device (size of the scale increments)

·        visual image of the scale reading (what the operator sees)

 

If the operator uses the measuring device properly and understands how to read the scale, no special skill is required to evaluate and report the uncertainty of a single measurement.  With a digital (discrete-scale) display, the human component of uncertainty can be reduced, but may also involve some trade-off in resolution, cost, or design complexity. 

 

For an analog (continuous-scale) reading (such as a fish scale, metal ruler, or micrometer), one reads the quantity by means of some pointing indicator positioned against a numerical scale.  The resolution of the scale and thickness of the pointer both limit the certainty of the measurement.  If the pointer appears blurred or warped to the operator, the actual uncertainty of the measurement is increased.

 

In any case, the job of the operator is to use the device carefully and report honestly what she or he sees on the indicator.  The uncertainty is specified by an interval.  For a discrete-scale measurement, the interval may contain a range of variation in the least significant digit.  For a continuous-scale measurement, the interval is correctly defined by the apparent “width” (to include visible blurring or vibration) of the pointer against the scale; i.e., the range of numerical values on the scale subtended by the pointer (with all its apparent “width”).

 

In reporting uncertainty, this objective characteristic of a single measurement defines its value in scientific work.  An interval containing numbers that clearly lie outside the range of the pointer’s “width” can not specify the actual uncertainty of the measurement.  Similarly, an interval that excludes numbers clearly subtended by pointer “width” fails to specify the measurement uncertainty.[1]  Any tendency for the operator to specify either excessive or inadequate intervals in repeated measurements will introduce bias (systematic error) to the process of determining maximum error in a quantity derived from multiple measurements.

Theory:  Error Propagation

 

Having specified the uncertainty of a single measurement as an interval, we may express the maximum measurement error by naming the range to either side of the midpoint of the specified interval, and contained within the interval.  In class we learned how to account for the effects of propagating this error through calculations involving other quantities that each have a specified measurement error.  We treat the calculation as a multivariable function, and use a differential approximation as an estimate of maximum error propagated from its variables.[2]

 

RECTANGULAR SOLID (Aluminum)

 

Volume from Length Measurements Using Three Devices

 

	x [mm]	delta_x	y [mm]	delta_y	z [mm]	delta_z	V [mm3]	delta_V	Reported Volume
scale	25.1	0.2	12.2	0.2	47.2	0.2	1.45E+04	4.E+02	(1.45 ± 0.04) x 101 cm3
vernier	25.45	0.05	12.70	0.05	47.75 *	0.05 *	1.543E+04	1.1E+02	(1.543 ± 0.011) x 101 cm3
micrometer	25.390	0.001	12.695	0.001	47.75 *	0.05 *	1.539E+04	2.E+01	(1.539 ± 0.002) x 101 cm3

 

* (vernier caliper measurement used in volume calculation)

 

Density from Mass Measurement and Volume Calculation

 

	mass [g]	delta_m	V [mm3]	delta_V	rho [g/mm3)	delta_rho	Reported Density	Uncertainty
mass scale	42.9398	0.0001	1.539E+04	2.E+01	2.790E-03	3.E-06	(2.790 ± 0.003) g/cm3	± 0.1%

 

 

 

 

 

ONE SHEET OF PAPER

 

Volume From Length Measurements

 

	x [mm]	delta_x	y [mm]	delta_y	z [mm]	delta_z	V [mm3]	delta_V	Reported Volume	Uncertainty
scale	280.1	0.2	216.0	0.2	(10 sheets)	(10 sheets)	(1 sheet)	(1 sheet)	(1 sheet)
micrometer	-	-	-	-	0.989	0.001	5.98E+03	2.E+01	(5.98 ± 0.02) cm3	± 0.3 %

 

Density of the Aluminum Block vs. Table Value

 

Appendix F in our text lists the density of aluminum as 2.699 g/cm³ at 20^oC. Our reported density (2.790 g/cm³) exceeds this value by more than 3% – well outside the range of our measurement uncertainty (0.1%).  Some systematic error might have biased our mass reading too high.  One or more of our length measurements might have been subject to systematic error, yielding too low volume.

 

Another possibility is that the block was not pure aluminum.  In fact, aluminum is commonly alloyed with zinc and manganese to form a structurally useful material.[3]  Appendix F in our text lists the densities of zinc and manganese as 7.44 and 7.133 g/cm³ respectively.  A quick calculation shows that if our block happened to be composed of exactly 1% zinc, 1% manganese, and 98% aluminum, then the density of the block would be 2.791 g/cm³ – within the range of our measurement uncertainty.