Topic 1: Physics and physical measurement1.2 Uncertainties and Errors

© Kari Eloranta2018

August 15, 2018

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Measurement

As a result of a measurement, we get an approximate value ofthe measured quantity.Consider, for example, the measurement of a pencil lengthwith a ruler divided into millimetres.

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Measurement

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We cannot tell exactly where the ends are.We do not know how accurate the rulerscale is (instrumental uncertainty).We do not even know from which part ofthe right end the length should bemeasured (problem of definition).0 1 2 3 4 5 6 7 8 9 10

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Instrumental Uncertainty

Instrumental UncertaintyThe accuracy of a measurement device is called the instrumentaluncertainty.

The most obvious factor affecting the measurement accuracyis the measurement device used.You will find the instrumental uncertainty of many devicesfrom an online manual. For example, the manuals of theVernier instruments are athttps://www.vernier.com/products/sensors/.In the absence of a manual, we may estimate the instrumentalerror to be the precision of the last digit in the device reading.

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Example: Instrumental Uncertainty of Digital Multimeterin Measuring Temperature

For example, according to the manual, theinstrumental uncertainty of a Finest 205Multimeter in measuring temperature is±4◦C, while the last digit rule gives ±1◦C.In this case, the last digit rule underestimatesthe real uncertainty by a factor of four!In many other cases, such as in digital scales,the last digit rule is a reasonable estimate.

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Uncertainty in Measurement

Uncertainty in MeasurementThe uncertainty in a measurement is the overall uncertainty in ameasurement considering all the error sources.

First, to determine the uncertainty in a measurement, youneed to determine the instrumental uncertainty of themeasurement device used. This is the lower limit of theuncertainty in measurement.Second, you should consider the other factors affecting themeasurement accuracy. The examples include the limit ofreading of an analog scale, the effect of parallax, and sourcesof random and systematic errors.Third, the uncertainty in measurement is estimated based onthe instrumental uncertainty and the other factors.

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Random Errors

Definition of Random ErrorRandom errors are sources of uncertainties in a measurementwhose effect can be reduced by a repeated experiment, and takingthe average of the results.

An example of a random error is the uncertainty in themeasurement of falling time by manual timing. Due mainly toreaction times at starting and ending the measurement, themeasured times vary around the real value of the falling timerandomly. The uncertainty in manual timing is about ±0.2s.The measurement value may vary randomly especially indigital devices.

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Systematic ErrorsDefinition of Systematic ErrorSystematic errors are sources of uncertainties in the measurementthat tend to shift the measurement values by a fixed amount. Arepeated experiment and taking the average does not reduce theeffect of a systematic error.

For example, if the bathroom scale has a zero offset error of0.2 kg, all the measured masses will be off by a fixed amountof 0.2 kg.Other examples of systematic errors are measuring the massof a sample with a container, forgetting to add atmosphericpressure to measured gauge pressure values, poorly calibratedinstruments, and systematic misreading of a scale.Systematic errors can be hard to detect.In some cases, the y intercept of a best fit line indicates thesystematic error in the measurement.

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Accuracy

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

Or, in the absence of an accepted value

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

For example, if a repeated measurement for the accelerationdue to gravity yields the values9.80ms−2,9.84ms−2,9.76ms−2, and 9.82ms−2, themeasurement results are accurate (the accepted value for theacceleration due to gravity being 9.81 ms−2).

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PrecisionDefinition of PrecisionAn indication of the agreement among a number of measurementsmade in the same way (a measure of exactness).

For example, if a repeated measurement yields the values1.235A,1.232A,1.229A, and 1.231A for a constant electriccurrent, the measurement results are precise (spread of data islow).Or, if a repeated measurement for the falling time yields0.69s,0.24s,0.47s, and 0.50s, the measurement results are notprecise (spread of data is high).Note that measurement values may be precise, but notaccurate. As an example, the measured values of9.41ms−2,9.44ms−2,9.39ms−2, and 9.38ms−2 for theacceleration due to gravity are precise, but not accurate.

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Propagation of Error

When measured values are used in calculations, the associateduncertainties affect the uncertainty in the calculated result.This is called the propagation of error.

Most of the time, we need only two simple rules for ourcalculations.

Propagation of Error in Multiplication and DivisionFor multiplication, division and powers, percentage uncertaintiesadd.

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

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Example of Propagation of Error

ExampleExample 1. To find the volume of a cube, a student measured itsside as (8.2±0.1) cm. Calculate the volume of the cube with itsuncertainty.

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% = ∆ll = 0.1��cm

8.2��cm ≈ 0.0122= 1.22%.

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Example of Propagation of ErrorExampleIn multiplication, percentage uncertainties add. The percentageuncertainty in the volume is

∆V% = 3×∆l% = 3×0.0122= 0.0366= 3.66%.

The absolute uncertainty in the volume is thus

∆V =∆V%×V = 0.0366×551cm3 ≈ 20cm3.

The volume with uncertainty is

V = (550±20)cm3.

Note! The absolute uncertainty in the volume has to be roundedto one significant figure, and the precision of volume has to matchthat of uncertainty.

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Example of Propagation of Error

ExampleExample 2. A group of students studied free fall by dropping atennis ball from a shelf. In a repeated measurement they found theaverage of falling times to be (0.6±0.1) s. By using thepropagation of error, calculate the height of the shelf withuncertainty.

Answer. tav = 0.6s, ∆tav = 0.1s, g = 9.8ms−2, h =?

In free fall, the distance fallen is h = 12gt2

av, where g = 9.8ms−2 isthe acceleration due to gravity, and t is falling time.The falling height is

h = 12gt2

av =12 ×9.8ms−2× (0.6s)2 ≈ 1.8m. (1)

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Example of Propagation of Error

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,% = 2×∆tav,% = 2×0.167= 0.334= 33.4%.

The absolute uncertainty in falling time squared is

∆t2av =∆t2

av,%× t2av = 0.334× (0.6s)2 ≈ 0.120s2.

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Example of Propagation of Error

ExampleAssuming that the uncertainty in the acceleration due to gravity is∆g ≈ 0, the absolute uncertainty in falling height is

∆h = 12g∆t2

av =12 ×9.8ms−2×0.120s2 ≈ 0.6m.

The falling height with uncertainty is

h = (1.8±0.6)m

Note! Because of the high percentage uncertainty in themeasurement of time, the uncertainty in the calculated heightbecomes exceptionally large.

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