topic 1: uncertainties and propagation of error

7/29/2019 Topic 1: Uncertainties and propagation of error

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Topic 1.2 Measurement and Uncertainties

Uncertainty and error in measurement

Random Errors

Definition of Random ErrorRandom errors are random variations in the measurement data, whose effect canbe reduced by a repeated experiment, and taking the average of the results.

An example of a random error is the uncertainty in the measurement of fallingtime by manual timing. Due mainly to reaction times at starting and ending themeasurement, the measured times vary around the real value of the falling time

randomly.

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Uncertainty and error in measurement

Systematic Errors

Definition of Systematic Error

Systematic errors are uncertainties that tend to shift the measurement values by afixed amount. A repeated experiment and taking the average does not reduce theeffect of a systematic error.

For example, if the bathroom scale has a zero offset error of 0.2 kg, all themeasured masses will be off by a fixed amount of 0.2 kg.

Other examples of systematic errors are measuring the mass of a sample with a

container, forgetting to add atmospheric pressure to measured over pressurevalues, poorly calibrated instruments, and systematic misreading of a scale.

Systematic errors can be hard to detect.

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Sub-Topic 1.2.7: Precision and Accuracy

Accuracy

Definition of AccuracyAn indication of how close a measured value is to the accepted value (a measure ofcorrectness).

Or, in the absence of an accepted value

Definition of AccuracyAn indication of how close a measured value is to the real value of the measuredquantity.

For example, if a repeated measurement for the acceleration due to gravity yieldsthe values 9.80m s2,9.84m s2,9.76m s2, and 9.82m s2, the measurementresults are accurate (the accepted value for the acceleration due to gravity being9.81m s2).

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Sub-Topic 1.2.7: Precision and Accuracy

Precision

Definition of PrecisionAn indication of the agreement among a number of measurements made in thesame way (a measure of exactness).

For example, if a repeated measurement yields the values

1.235A,1.232A,1.229A, and 1.231A for a constant electric current, themeasurement results are precise (spread of data is low).

Or, if a repeated measurement for the falling time yields 0.69s,0.24s,0.47s, and0.50s, the measurement results are not precise (spread of data is high).

Note that measurement values may be precise but not accurate. As an example,the measured values of9.41m s2,9.44m s2,9.39m s2, and 9.38m s2 for theacceleration due to gravity are precise, but not accurate.

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Uncertainties in calculated results

Propagation of Error

When measured values are used in calculations, the associated uncertaintiesaffect the uncertainty in the calculated result. This is called the propagation oferror.

Most of the time, we need only two simple rules for our calculations.Propagation of Error in Multiplication and DivisionFor multiplication, division and powers, percentage uncertainties add.

Propagation of Error in Subtraction and AdditionFor addition and subtraction, absolute uncertainties add.

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Example of Propagation of Error

Example

Example 1. To find the volume of a cube, a student measured its side as(8.20.1) cm. Calculate the volume of the cube with its uncertainty.

Answer. l= 8.2cm, l= 0.1cm, V,V=?

The volume of the cube is

V= l3= (8.2cm)3 551cm3.

The percentage uncertainty in a side of the cube is

l%=l

l=0.1cm

8.2cm 0.0122= 1.22%.

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Example

In multiplication, percentage uncertainties add. The percentage uncertainty in thevolume isV%= 3l%= 30.0122= 0.0366= 3.66%.

The absolute uncertainty in the volume is thus

V=V%V= 0.0366551cm3 20cm3.

The volume with uncertainty is

V= (55020)cm3

.

Note! The absolute uncertainty in the volume has to be rounded to one significantfigure, and the precision of volume has to match that of uncertainty.

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ExampleExample 2. A group of students studied free fall by dropping a tennis ball from ashelf. In a repeated measurement they found the average of falling times to be(0.60.1) s. By using the propagation of error, calculate the height of the shelfwith uncertainty.

Answer. tav= 0.6s, tav= 0.1s, g= 9.8m s2, h=?

In free fall, the distance fallen is h= 12g t2av, where g= 9.8m s

2 is the accelerationdue to gravity, and t is falling time.The falling height is

h=1

2g t2av=

1

29.8m s2 (0.6s)2 1.8m. (1)

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ExampleThe percentage uncertainty in falling time is

tav,%=tav

tav=0.1s

0.6s 0.167= 16.7%.

The percentage uncertainty in falling time squared is

t2av,%= 2tav,%= 20.167= 0.334= 33.4%.

The absolute uncertainty in falling time squared ist2av=t

2av,% t

2av= 0.334 (0.6s)

2 0.120s2.

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ExampleThe absolute uncertainty in falling height is

h=1

2gt2av=

1

29.8m s20.120s2 0.6m.

The falling height with uncertainty is

h= (1.80.6)m

Note! Because of the high percentage uncertainty in the measurement of time, theuncertainty in the calculated height becomes exceptionally large.

Kari Eloranta 2013

topic 1: uncertainties and propagation of error

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