1. Measurement Guidelines
Precision, Accuracy, and Estimating Uncertainty

2. 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.

3. Accuracy vs. Precision

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

5. 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.

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

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

9. 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 kg ± 2 kg or cm ± 0.05 cm
Percent Uncertainty
Uncertainty as a ratio to the measurement
Ex. 2 𝑘𝑔 235 𝑘𝑔 = or 0.85%

10. 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”.

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

12. 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

13. 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.

14. 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
cm ± cm
30.70 cm ± 0.18 cm = cm ± 0.2 cm
Only one place past decimal because of 30.2 cm
Precision matches that of the measurement

15. 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 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

16. 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 sf

17. 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 sf