Error & Uncertainty Propagation & Reporting
Absolute Error or Uncertainty is the total uncertainty in a measurement reported as a ± with the
measurement.
Absolute uncertainty.V ± ΔV = (13.8 ± 0.2) mLV falls within 13.6 – 14.0 mL
Fractional or RelativeΔV / V = 0.2 / 13.8 ≈ 0.0145
Percentage ΔV % = (ΔV / V) × 100 = (0.2 / 13.8) × 100 ≈ 1.4 %
Three styles for expressing uncertainties:
Absolute Uncertainty show a range
0.1 kg 4.5 kg
Absolute Uncertainty or Error
The absolute error should be 1 SF only. Its place must agree with the measurement’s place.
This says the actual value lies between 4.4 – 4.6 kg.
Where absolute error come from? How do you know the correct range?
• Measure the diameter of a ball with the ruler. Report your measurement.
• 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.
• 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.
Instrument Uncertainty
How do we report this measurement?
1.36 cm ± 0.05 cm or
1.4 ± 0.1 cm
There must be agreement between the uncertainty place & the last digit.
Ways of reporting uncertainty
• Fractional or Relative uncertainty
• % uncertainty/error
• % difference/discrepancy
• Absolute error of mean
Relative/fractional Uncertainty or Error gives idea of what fraction of the measure the uncertainty represents. It is calculated as:
Absolute Uncertainty Measurement
.5.4
1.0
kg
kg
0.1 kg 4.5 kgFor the measure find relative and % uncertainty
Relative Error/Uncert. This does not get a ± . It can be more than 1 SF.
0.022 or 2.2%
% 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.
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.
When adding or subtracting measurements, the total absolute error is the sum of the absolute errors of each measurement!.
2.61 0.05 cm
2.82 0.05 cm+
5.4 0.1 cm
5.6 0.1 cm
2.1 0.1 cm-
3.5 0.2 cm
Decimal Agreement
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.
• 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.
1.
Find the relative/percent error of each measurement:
Sum the relative errors:
Multiply relative error by the answer to find abs uncert.
0.5 ÷ 2.6 = 0.1920.5 ÷ 2.8 = 0.179
0.192 + 0.179 = 0.371 or 37%
0.371 x 7.28 cm2 = 2.70 cm2.
This is the ± giving the range on your measurement.
It means 7.28 ± 2.70cm2.
Round uncertainty (not meas) 1 SF &report2.70 cm2 becomes ± 3 cm2.
7.28 cm2 = becomes 7 cm2 to agree with 3 cm2.
Answer gets rounded to the same place as ± .
Report: 7 cm2 ± 3 cm2.
• Add the sides 32.0 m = perimeter.
• Add the abs uncert. 0.3 +0.3 + 0.2 +0.2 = ±1.0 m.
• 32.0 m ±1.0 m.
• Round to abs uncert to 1 SF 32 ± 1 m.
10.0 ± 0.3 m
10.0 ± 0.3 m
Raising measurements to power n
• Solve equation
• Find relative uncertainty
• Multiply relative uncertainty by n (power).
Ex 2: find volume of cube with side length of 2.5 0.1 cm.
• Volume = (2.5 cm)3 = 15.625 cm3.
•Relative uncertainty for each side =
0.1 cm = 0.04
2.5 cm
0.04 x 3 (nth power) = 0.12
This is the fraction of uncertainty in the volume measure.
0.12 (15.625 cm3.) = 1.875 cm3.
Round to uncertainty to 1 sig fig ± 2 cm3.
Finish
• Round last digit of answer to same place as abs uncertainty. Uncertainty to 1 SF was 2cm3 (one’s place).
• Ans was 15.625 cm3.
• So 16 cm3 ± 2cm3.
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.
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:
Uncertainty in Series of 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 = 35.5 ± 2.5 ≈ 36 ± 3.
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.
Make minimum and maximum calculations for the uncertainty range then round to a positive or negative symmetrical value.
θ ± Δθ = (13 ± 1)°sin13° = 0.22495
sin14° = 0.24192 sin12° = 0.20791
sin(13 ± 1)° = 0.22495 (+0.01704 and –0.01697)sin(13 ± 1)° = 0.22 ± 0.02
Reciprocals, logarithms, & trigonometric functions. Uncertainties are not usually symmetrical.
Uncertainty Tutorialhttps://www.youtube.com/watch?v=0lt-9qimLf4