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

Abs error graphed as error bars.Outliers ignored.