Measurement Guidelines Precision, Accuracy, and Estimating Uncertainty

Accuracy vs. Precision What is the difference? FOCS 1, a continuous cold caesium fountain atomic clock in Switzerland, started operating in 2004 at an uncertainty of one second in 30 million years. Measure to nanoseconds.

Accuracy vs. Precision

Comparison to a standard Error Technique Precision accuracy Exactness Divisions on scale Reproducibility Uncertainty Significant digits Correctness Calibration Comparison to a standard Error Technique

Basic Guidelines for Measurement Estimate and record one digit past the smallest division on the scale when using a non-digital scale. If digital, record all numbers in the display.

What is the first number on a number line of whole numbers?

Checking for Understanding Use whiteboards to display responses to those problems assigned by McGreevey.

Some Vocabulary Uncertainty Absolute Uncertainty Percent Uncertainty Range within which you are certain the measurement lies. Absolute Uncertainty Uncertainty reported with the same units and precision as the measurement itself Ex. 235 kg ± 2 kg or 14.03 cm ± 0.05 cm Percent Uncertainty Uncertainty as a ratio to the measurement Ex. 2 𝑘𝑔 235 𝑘𝑔 = 0.0085 or 0.85%

Estimating Uncertainty For an individual non-digital measurement: Consider between 1 and 5 in the estimated place value (that’s one-tenth to one-half the smallest division) Example: Measured value: 1.04 cm Uncertainty: as low as ± 0.01 as high as ± 0.05 Use your judgment in each circumstance to increase or decrease the uncertainty. Don’t overestimate just to be “safe”.

Checking for Understanding Use whiteboards to display responses to those problems assigned by McGreevey.

Estimating Uncertainty For an individual digital measurement: Make use of any uncertainty provided by the manufacturer. If no uncertainty is provided, estimate ± 1 - ± 5 in the smallest place value displayed

Estimating Uncertainty For multiple measurements: Calculate the average, and then add ± : (Hi-Lo)/2 (see Physics Skills #10 packet) As the name implies, Highest value minus lowest value, divided by 2 Average Absolute Deviation (see Physics Skills #10 packet) The average of the absolute value of the differences (deviations) of each individual measurement from the average.

Propagating Uncertainty When adding or subtracting measurements… The result is limited in precision by the least precise measurement. Add the absolute uncertainties, if uncertainties are in included. Example 0.50 cm ± 0.08 cm + 30. 2 cm ± 0.1 cm 30.70 cm ± 0.18 cm = 30.7 cm ± 0.2 cm Only one place past decimal because of 30.2 cm Precision matches that of the measurement

Propagating Uncertainty When multiplying or dividing measurements… The result is limited by the measurement with the least significant digits. Add the percent uncertainties, if uncertainties are included. Example 30.5 m ± 2.0% x 2. 2 m ± 1.5% 67.1 m2 ± 3.5 % = 67 m2 ± 3.5 % ( = 2 m2) Only 2 sig. figs. because of 2.2 m If converted to an absolute uncertainty, precision matches that of the measurement

Propagating Uncertainty When multiplying or dividing a measurement by a number (as in an equation)… Maintain the same number of significant digits Multiply or divide the absolute uncertainty by the number OR maintain the same percent uncertainty. Ex: radius = 3.0 mm ± 0.2 mm Circumference = 2πr = 2π (3.0 mm ± 0.2 mm) = 19 mm ± 1mm 2 sf 2 sf

Propagating Uncertainty When raising a measurement to a power (as in an equation)… Maintain the same number of significant digits Multiply the percent uncertainty by the power Ex: radius = 3.0 mm ± 0.2 mm (6.7%) Area = πr2 = π (3.0 mm ± 6.7%)2 = 28 mm2 ± 13% 2 sf 2 sf