Uncertainties in Measurement

Laboratory investigations involve taking measurements of physical quantities. All measurements will involve some degree of experimental uncertainty.QUESTIONS

1. How does one express the uncertainty in an experimental measurement ?

2. How does one determine the uncertainty in an experimental measurement ?

3. How does one compare an experimental measurement with an accepted (or published) value ?

4. How does one determine the uncertainty in a quantity that is computed from uncertain measurements ?

Expressing Uncertainty

We will express the results of measurements in this laboratory as

(measured value uncertainty) units

For example

g g = (9.803 0.008) m/s2

Types of Experimental Uncertainty

Random, Indeterminate or Statistical

– Results from unknown and unpredictable variations that arise in all experimental situations.

– Repeated measurements will give slightly different values each time.

– You cannot determine the magnitude (size) or sign of random uncertainty from a single measurement.

– Random errors can be estimated by taking several measurements.

– Random errors can be reduced by refining experimental techniques.

Types of Experimental Uncertainty

Systematic or Determinate

– Associated with particular measurement instruments or techniques.

– The same sign and nearly the same magnitude of the error is obtained on repeated measurements.

– Commonly caused by improperly “calibrated” or “zeroed” instrument or by experimenter bias.

Accuracy and Precision

Accuracy

Is a measure of how close an experimental result is to the “true” (or published or accepted) value.

Precision

Is a measure of the degree of closeness of repeated measurements.

Accuracy and Precision

Consider the two measurements:

A = (2.52 ± 0.02) cm

B = (2.58 ± 0.05) cm

Which is more precise ?

Which is more accurate ?

Accuracy and Precision

_____ accuracy

_____ precision

_____ accuracy

_____ precision

_____ accuracy

_____ precision

Answer with GOOD or POOR ...

Implied Uncertainty

The uncertainty in a measurement can sometimes be implied by the way the result is written. Suppose the mass of an object is measured using two different balances.

Balance 1 Reading = 1.25 g

Balance 2 Reading = 1.248 kg

Significant FiguresIn a measured quantity, all digits are significant except any zeros whose sole purpose is to show the location of the decimal place.

123

123.0

0.0012

0.0001203

0.001230

1000

1000.

150

g

g

m

cm

s

cm

cm

1.23 x 102 g

1.230 x 102 g

1.2 x 10-3 m

1.203 x 10-4 s

1.230 x 10-4 s

1 x 103 cm

1.000 x 103 cm

150

_________

_________

_________

_________

_________

_________

_________

_________

Rounding

• If the digit to the right of the position you wish to round to is < 5 then leave the digit alone.

• If the digit to the right of the position you wish to round to is >= 5 then round the digit up by one.

• For multiple arithmetic operations you should keep one or two extra significant digits until the final result is obtained and then round appropriately.

• Proper rounding of your final result will not introduce uncertainty into your answer. ROUNDING DURING CALCULATIONS IS NOT A VALID SOURCE OF ERROR.

Expressing Uncertainty

When expressing a measurement and its associated uncertainty as

(measured value uncertainty) units

• Round the uncertainty to one significant digit, then

• round the measurement to the same precision as the uncertainty.

For example, round 9.802562 0.007916 m/s2 to

g g = (9.803 0.008) m/s2

Significant Figures in Calculations

When multiplying or dividing physical quantities, the number of significant digits in the final result is the same as the factor (or divisor…) with the fewest number of significant digits.

6.273 N 0.0204 m

* 5.5 m 21 C°

34.5015 N·m 0.00097142857 m/C°

________ N·m _________ m/C°

Multiplication and Division

Significant Figures in Calculations

When adding or subtracting physical quantities, the precision of the final result is the same as the precision of the least precise term.

132.45 cm

0.823cm

+ 5.6 cm

138.873cm --> _______ cm

Addition and Subtraction

Comparing Experimental and Accepted Values

E ± E = An experimental value and its uncertainty.

A = An accepted (published) value.

%100yDiscrepancPercent AAE

Percent Discrepancy quantifies the __________ of a measurement.

%100yUncertaintPercent EE

Percent Uncertainty quantifies the __________ of a measurement.

Comparing Two Experimental Values

E1 and E2 = Two different experimental values.

%100

221

12

EEEE

DifferencePercent

Average (Mean) Value

Let x1, x2,… xN represent a set of N measurements of a quantity x.

The average or mean value of this set of measurements is given by

NN

ii xxxN

xN

x

...11

211

Frequency Distribution (N=10)

data14222018202318191923

hist.x hist.d10 011 012 013 014 115 016 017 018 219 220 221 022 123 224 025 026 027 028 029 030 0

2.0

1.5

1.0

0.5

0.0

Frequency

3025201510

Value

Mean = 19.6

Frequency Distribution (N=100)

Mean = 19.89

hist.x hist.d10 011 012 013 014 015 316 517 818 1119 1520 1521 1922 1323 624 325 226 027 028 029 030 0

15

10

5

0

Frequency

3025201510

Value

Frequency Distribution (N=1,000)

Mean = 19.884

hist.x hist.d10 011 012 113 514 1115 2016 4917 7418 12719 15220 16121 14822 10023 8124 4425 1726 627 328 129 030 0

160

140

120

100

80

60

40

20

0

Frequency

3025201510

Value

Frequency Distribution (N=10,000)

Mean = 19.9879

hist.x hist.d10 011 112 713 3214 9615 23616 46617 78118 118419 143520 154121 145122 115923 79624 45325 24026 9827 1828 329 330 0

1400

1200

1000

800

600

400

200

0

Frequency

3025201510

Value

Expressing Uncertainty

N

iix xx

Ns

1

2)(1

1

The Standard Deviation of a set of N measurements of x is given by:

N

sSEMs x

x

The Standard Deviation of the Mean (or Standard Error of the Mean) of a set of N measurements of x is given by:

Expressing Uncertainty

N Mean S.D. SEM Result

10 19.6 2.71 0.857

100 19.89 2.26 0.226

1,000 19.884 2.48 0.0784

10,000 19.9879 2.52 0.0252

19.6 0.9

19.9 0.2

19.88 0.08

19.99 0.03

Combining Uncertainties:Propagation of Uncertainty

Let A ± A and B ± B represent two measured quantities.

The uncertainty in the sum S = A + B is

S = A + B

The uncertainty in the difference D = A - B is ALSO

D = A + B

Combining Uncertainties:Propagation of Uncertainty

Let A ± A and B ± B represent two measured quantities.

The uncertainty in the product P = A * B is

The uncertainty in the quotient Q = A / B is ALSO

BB

AA

PP

BB

AA

QQ