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Error & Uncertainty
Propagation & Reporting
Absolute Error or Uncertainty is the
total uncertainty in a measurement
reported as a ± with the measurement.
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 ≈ 0.0145
Compares Abs. to actual measure.
Percentage
ΔV % = (ΔV / V) × 100 = (0.2 / 13.8) × 100 ≈ 1.4 %
Absolute Uncertainty show a range
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.
ST called raw unc.
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.
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.
Absolute Uncertainty of Mean
• Unc. Shrinks w more trials.
• Generally experiment requires a 5 x 5 table. Five
trials, Five different values for independent variable.
• Take five different drop heights, 5 trials each.
Measure the time to hit the ground. Average times.
• For set of measurements, absolute uncert:
(Max – Min) / 2
1: Given Ball drop experiment with set of drop
times in seconds for 1 height:
•
•
•
•
•
0.23
0.22
0.26
0.34
0.18
• Find the mean and
the absolute
uncertainty of the
mean.
• Proper value =
• mean = 0.246 s
• unc. is
• (0.34 – 0.18)/2
• 0.08
• 0.25 ± 0.08 s
Ways of reporting uncertainty
•
•
•
•
Absolute uncertainty is a range.
Fractional / Relative uncertainty
% uncertainty/error
% difference/discrepancy
Relative/fractional Uncertainty or Error gives idea of
what fraction of the measure the uncertainty represents.
It is calculated as:
Absolute Uncertainty
Measurement
2. For the measure 4.5 kg  0.1 kg find
relative and % uncertainty
0.1kg
.
4.5kg
0.022 or 2.2%
Relative Error/Uncert.
This does not get a ± . It can be more than 1 SF.
% 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 height 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.
Adding or subtracting measurements, the total
absolute error is the sum of the absolute errors!
Data booklet reference:
• If y = a  b then y = a + b
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.
• If y = a · b / c then y / y = a / a + b / b + c / c
• Absolute Error is reported as fraction of the answer.
• 3. Find the area of a rectangle measuring:
2.6 cm ±0.5 by 2.8 ±0.5 cm?
Solve
A = l x w:
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%
If y = a · b / c then y / y = a / a + b / b + c / c
Abs Unc. multiply relative unc. by the answer.
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.
4. Find the perimeter and absolute uncertainty of a
rectangular floor measuring 10.0 ± 0.3 m by 6.0 ± 0.2 m.
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.
Raising measurements to power n
• Solve equation
• Find relative uncertainty.
• Multiply relative uncertainty by n (power).
• • If y = a n then y / y = | n · a / a |
5: 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.
6. 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.
Uncertainty Tutorial
https://www.youtube.com/watch
?v=0lt-9qimLf4
Exact/ pure numbers
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.
For multiplication by an exact number, multiply the
uncertainty by the same exact number. For division, divide
uncert by number.
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 = 35.5 ± 2.5 ≈ 36 ± 3.
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° = 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