2. Error & Uncertainty Propagation & Reporting Absolute Error or Uncertainty is the total uncertainty in a measurement reported as a ± with the measurement.

3. Three styles for expressing uncertainties:
Absolute uncertainty.
V ± ΔV = (13.8 ± 0.2) mL
V falls within 13.6 – 14.0 mL
Fractional or Relative
ΔV / V = 0.2 / 13.8 ≈
Percentage
ΔV % =  (ΔV / V) × 100 = (0.2 / 13.8) × 100 ≈ 1.4 %

4. Absolute Uncertainty show a range
0.1 kg
4.5 kg
Absolute Uncertainty or Error
This says the actual value lies between 4.4 – 4.6 kg.
The absolute error should be 1 SF only. Its place must agree with the measurement’s place.

5. Where absolute error come from? How do you know the correct range?
Measure the diameter of a ball with the ruler. Report your measurement.

6. At minimum it’s the instrument uncertainty.
Usu instrument uncertainty plus other uncertainty sources. Use your judgment but be logical.
Ball radius in drop height.
Meniscus in graduated cylinder.

7. Instrument Uncertainty
For scales where you can read between 2 divisions, you can report ½ the smallest or the actual smallest measure as your instrument uncertainty (I generally use the smallest increment).
For digital measures
just report the smallest unit.

8. How do we report this measurement?
There must be agreement between the uncertainty place & the last digit.
1.36 cm ± 0.05 cm or
1.4 ± 0.1 cm

9. Ways of reporting uncertainty
Fractional or Relative uncertainty
% uncertainty/error
% difference/discrepancy
Absolute error of mean

10. Relative/fractional Uncertainty or Error gives idea of what fraction of the measure the uncertainty represents. It is calculated as:
Absolute Uncertainty
Measurement
For the measure find relative and % uncertainty
0.1 kg
4.5 kg
0.022 or 2.2%
Relative Error/Uncert.
This does not get a ± . It can be more than 1 SF.

11. % Uncertainty/Error is different than % difference, deviation, discrepancy.
% Dif measures difference from accepted value: Accept val – meas val x 100% Accepted Val
% Error - amount of uncertainty in measurement.

12. Propagation of Error
Measure width of counter in cm with a meter-stick.
Measure height of student with meter-stick.
Which has more uncertainty?
If you do calculations with the measurements with uncertainties – the uncertainty will increase.

13. When adding or subtracting measurements, the total absolute error is the sum of the absolute errors of each measurement!.
2.61
0.05 cm
5.6
0.1 cm
2.82
0.05 cm
2.1
0.1 cm + -
5.4
0.1 cm
3.5
0.2 cm
Decimal Agreement

14. Multiplication & Division
1st – solve it! Find product or quotient normal way.
Must calculate relative or percent uncertainty for each individual measure.
Then add the relative/percent errors.
Absolute Error is reported as fraction of the answer.

15. What is the area of a rectangle measuring: 2.6 cm ±0.5 by 2.8 ±0.5 cm?1.
What is the area of a rectangle measuring:
2.6 cm ±0.5 by 2.8 ±0.5 cm?
Find the product:
7.28 cm2.

16. Find the relative/percent error of each measurement:
0.5 ÷ 2.6 = 0.192
0.5 ÷ 2.8 = 0.179
Sum the relative errors:
= or 37%
Multiply relative error by the answer to find abs uncert.
0.371 x 7.28 cm2 = 2.70 cm2.
This is the ± giving the range on your measurement.
It means 7.28 ± 2.70cm2.

17. Answer gets rounded to the same place as ± .
Round uncertainty (not meas) 1 SF &report
2.70 cm2 becomes ± 3 cm2.
Answer gets rounded to the same place as ± .
7.28 cm2 = becomes 7 cm2 to agree with 3 cm2.
Report: 7 cm2 ± 3 cm2.

18. Add the sides 32.0 m = perimeter.
10.0 ± 0.3 m
10.0 ± 0.3 m
Add the sides 32.0 m = perimeter.
Add the abs uncert = ±1.0 m.
32.0 m ±1.0 m.
Round to abs uncert to 1 SF 32 ± 1 m.

19. Raising measurements to power n
Solve equation
Find relative uncertainty
Multiply relative uncertainty by n (power).

20. Ex 2: find volume of cube with side length of 2.5 0.1 cm.
Volume = (2.5 cm)3 = cm3.
Relative uncertainty for each side =
0.1 cm = 0.04
2.5 cm

21. Round to uncertainty to 1 sig fig ± 2 cm3.
0.04 x 3 (nth power) = This is the fraction of uncertainty in the volume measure.
0.12 ( cm3.) = cm3.
Round to uncertainty to 1 sig fig ± 2 cm3.

22. Finish
Round last digit of answer to same place as abs uncertainty. Uncertainty to 1 SF was 2cm3 (one’s place).
Ans was cm3.
So cm3 ± 2cm3.

24. There are no uncertainties associated with pure numbers, the type of operation determines the uncertainty propagation where, for example:
If a quantity is divided by 2, the uncertainty in the 2 is zero.
If you multiply a quantity by π, the uncertainty in π is zero.
The only uncertainty in πr2 is in the measurement of r, where r is ± Δr.

25. Uncertainty in Series of Measurements
Take the average value and determine the uncertainty from the range. The range is the difference between the largest and smallest measurements. The uncertainty is ± one half the range. Given 4 measurements:
x1 = 32, x2 = 36, x3 = 33, x4 = 37
mean of x = (x1 + x2 + x3 + x4) / 4 = 35.5 (Average)
Abs uncert , Δx = ±(xmax – xmin) / 2 = (37 – 32) / 2 = 2.5
mean of x ± Δx = ±  2.5 ≈ 36 ± 3.

26. 4. A protractor is precise to ±1o
4. A protractor is precise to ±1o. A student obtains the following measurements for a refraction angle: 45, 47, 46, 45, and 44 degrees. How show he express the refractive angle with its uncertainty?
Mean = 45.4o.
Max – Min = 47 – 44 = 3o.
Half range = 1.5o.
With rounding:
Value = 45 ± 2o.

27. Reciprocals, logarithms, & trigonometric functions
Reciprocals, logarithms, & trigonometric functions. Uncertainties are not usually symmetrical.
Make minimum and maximum calculations for the uncertainty range then round to a positive or negative symmetrical value.
θ ± Δθ = (13 ± 1)°
sin13° =
sin14° = sin12° =
sin(13 ± 1)° = ( and – )
sin(13 ± 1)° = 0.22 ± 0.02 

28. Uncertainty Tutorial https://www.youtube.com/watch?v=0lt-9qimLf4

29. Abs error graphed as error bars. Outliers ignored.