Transcript Slide 1

Error & Uncertainty
Propagation & Reporting
Absolute Error or Uncertainty is the
total uncertainty in a measurement
reported as a ± with the measurement.
Kerr p 10 -11
Absolute, Relative, & Percent
Uncertainty Error
4.5 kg

0.1 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.
Where absolute error come from?
• 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.
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.
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
Ways of reporting uncertainty
•
•
•
•
Relative uncertainty
% uncertainty/error
% difference/discrepancy
Absolute error of mean (not needed for SL)
Relative/fractional Uncertainty or Error gives
idea of what fraction of the measure the
uncertainty represents. It is calculated as:
Absolute Error/Uncertainty
Measurement
For the measure 4.5 kg

0.1 kg
0.1kg
 . 0.022
4.5kg
Relative Error/Uncert.
This does not get a ± . It can be more than 1 SF.
% Error or Uncertainty
Turn relative or fractional into a %
=
Relative Error x 100 %
0.1kg
 0.022x100% = 2.2%
4.5kg
Relative Error
% Error
% 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.
Absolute Error of Mean
Sometimes you use mean to calculate absolute
uncertainty. Say you take 5 measures all ± 0.1. It looks
like the uncertainty is larger than the instrument error.
You can find the absolute error from the mean.
3.11
2.91
3.0
2.95
3.21
= 15.19
15.18 ÷ 5 =
3.036 ~ 3.0
3.0 is mean
Absolute error is largest
deviation from mean.
Calculate the deviation of the mean from
each measure.
3.11 – 3.0

0.11
2.91 – 3.0

0.09
3.01 – 3.0

0.01
2.95 – 3.0

0.05
3.21 – 3.0

0.21
largest
Absolute Error is 
Largest deviation
3.0  0.21
True value lies between mean ±0.49 Round it
off to 1 SF.
3.0  0.2
So the abs uncertainty is 5x larger than the instrument
uncertainty.
Propagation of Error
• Measure height of counter in cm with a meter-stick.
• Measure height of student with meter-stick.
• Which has more uncertainty?
• If you need to do math on 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
5.6
 0.1 cm
+ 2.82  0.05 cm
- 2.1
 0.1 cm
5.4
 0.1 cm
Decimal Agreement
3.5
 0.2 cm
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.
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.
Find the relative/percent error of each measurement:
0.5 ÷ 2.6 = 0.192
0.5 ÷ 2.8 = 0.179
Sum the relative errors: 0.192 + 0.179
= 0.371 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.
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.
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
2.5 cm
= 0.04
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.
10.0 ± 0.3 m
10.0 ± 0.3 m
• 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.
• Activity.
Abs error graphed as error bars.
Outliers ignored.