ERRORS ON MEASUREMENT. DIRECT MEASUREMENTS

1 INTRODUCTION. ABSOLUT AND RELATIVE ERRORS AND UNCERTAINTIES

Always we carry out a measurement of some physical quantity, such measurement is

subjected to an error depending on several factors, but the error is inherent to any

measurement of a physical quantity. The difference between the measurement of a quantity,

x, and the “true” value of such quantity, xtrue, is called absolute error (usually shortened as

error), that can also be expressed as relative error when we compare it with the “true” value

of such quantity:

( ) trueAbsolute error x x x ( )

Re ( )true

Absolute error xlative error x

x

Unfortunately, the “true” value of a quantity is usually unknown, its errors can be

quantified with difficulty, and we will have to “estimate” the true value through

measurements.

Anyway is sure that, when the measurement of a physical quantity is carried out, to only

give the value of the measurement (x) does not give idea about its validity, being necessary the

measurement goes with another value (x ó u(x)). This value is saying us that the “true” value

of that quantity will be in the interval x-x, x+x with some probability. Obviously, if x is

increased, it will be more likely to find the “true” value inside the interval, and vice versa. x

(also written as u(x)) is called absolute uncertainty of a measurement, being the correct way

to express it with its error: x±x or x(u(x)). We will use in this document x better than u(x).

And from the absolute uncertainty can also be calculated the relative uncertainty (εr(x) or

ur(x)), as the quotient of absolute uncertainty to the measured value:

( )( ) ( )r r

x u xx u x

x x

The relative values are mainly useful to compare measurements between them.

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Example: two measured lengths with an absolute uncertainty of 5 cm are not equally

accurate if the measured lengths are 50 cm and 50 Km, respectively. The relative uncertainties

in both cases are, in percentage:

( ) , %r

51 0 1 10

50 ( ) , %6

r 5

52 10 0 0001

50 10

The second measurement is, obviously, much more accurate than the first one.

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The words error and uncertainty, even though they refer to different concepts, are used

interchangeably when referring to the absolute uncertainty of a measurement. When we want

to refer to the relative values, then we have to add the word relative.

There are two standards or rules to correctly write a measurement with its error. These

rules must be obeyed always the result of a measurement is given:

a) The error of measurement will be rounded in order it has, as maximum, two

significant figures, but it can also be written with only one significant figure. This

choosing is, in some way, arbitrary, depending on the committed error. As an example,

if our error is 0,89 m and we round it to 0,9 m, the difference is slightly higher than 1 %,

value that could justify the rounding by giving the error in a more simple way. But if an

error of 1,4 m is rounded to 1 m, the difference is close to 40 %, something justifiably

with difficulty. Anyway, the person is measuring must decide how many significant

figures, trying to combine the rigour with the simplicity in the result.

Note: Significant figures are:

i. Any figure different than zero.

ii. Zeros placed between two figures different than zero.

iii. For any value >1, zeros placed on right of comma are significant figures.

b) The last significant figure of measurement shouldn’t have less decimal order than that

of the error. It is, the measurement can’t be written more precise then the error. We

have on first to round the error and after, to round the measurement according the

decimal order of the error.

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Examples:

Incorrect measurements Correct measurements

48,721 ± 0,32 V 48,72 ± 0,32 V = 48,72(0,32) V

4,6 ± 0,0182 V 4,6 ± 0,018 V = 4,6(0,018) V

563,1 ± 30 cm 563 ± 30 cm = 563(30) cm

872·10-6 ± 0,8656·10-4 N 8,72·10-4 ± 0,87·10-4 N = (8,72 ± 0,87) ·10-4 N

= 8,72·10-4(0,87·10-4) N = 8,72(0,87)·10-4 N

4,67825·10-8 ± 4,61·10-10 A 4,678·10-8 ± 0,046·10-8 A=(4,678 ± 0,046) ·10-8 A

= 4,678·10-8(0,046·10-8) A = 4,678(0,046)·10-8 A

0,23±3,0 ºC 0±3 ºC = 0(3) ºC

2 SOMETHING ABOUT STATISTICS

The relationship between uncertainty or error and probability can be better understood if

we introduce some idea about statistics.

Statistics demonstrates that those processes depending on lots of independent factors, as

it happens in the majority of measurements we carry out at the laboratory or in a lot of social,

demographic and economic processes, show a statistically “normal” behaviour. Such

behaviour is defined by a function of distribution of probabilities given by the so called Gauss’s

bell, symmetric curve whose equation is ( )

( )

2

2

x x

2

2

1f x e

2

. It means that the probability to

find a value of the studied variable between two values a and b equals the area enclosed by

the Gauss’s bell, the X axis and the values a y b:

This distribution is characterized by two parameters:

The mean value (or average value, x ). This

value is also the most likely, giving us an

idea about the position of the curve along

the X axis.

σ, the called standard deviation, whose

main meaning is that when we take the

interval ,x x , then the probability

to find randomly a given value of x inside

such interval is 68%. It gives us an idea

about how narrow or broad the distribution

is, and therefore if the more likely values are fare or close to the mean value.

If we increase the interval to ,x 2 x 2 , then the probability to randomly find a

given value of x inside this interval is 95%, and if we increase still more the interval up to

,x 3 x 3 , then it is almost sure than the value of x will be found inside the

interval (probability 99,7%).

3 ACCURACY AND PRECISION. SISTEMATIC AND RANDOM ERRORS

When we carry out a measurement, we are

looking for to obtain a measurement as close as

possible to the “true” value, and moreover, if we

repeat the measurement several times, the results

must show a little dispersion. It is, any measurement

system must have two features:

Accuracy or validity (Exactitud): is the feature

by that the mean value is very close to the

“true” value.

Precision or Reliability (Precisión o

Fiabilidad): is the feature by that when the

same quantity is measured several times in

the same conditions, it always leads us to

similar values.

( ) ( )b

a

p a x b f x dx

a b

x x x

Accuracy and precision are associated to two types of errors:

Random (or accidental) errors: These errors are randomly produced, both up and

down, and they highlight a lack of reliability. For example, the reading on a ruler when

we interpolate between two lines of a tape, or the reaction time of a person when he is

measuring with a chronometer. As these errors are randomly produced, both up and

down, the carrying out of several measurements of the same quantity and their

further statistical processing allows its reduction.

Systematic errors: These errors are

always produced in the same sense

(up or down), and they highlight a lack

of accuracy on measurement. They are

produced by a bad measurement

system, always producing an error in

the same sense, or because of a bad

calibration of measuring devices. They

are difficult to be detected, and only

by taking several measurements they

can’t be removed. Only thinking about

the method to measure, or a correct

calibration of measuring devices can

reduce them, even though some

techniques to detect and correct them

exist. On picture can be seen the difficulties to correct them: as the “true” value is not

known, everything happens as if the bullseyes would have been removed on the above

picture.

4 DIRECT AND INDIRECT MEASUREMENTS

A quantity can be basically measured in two different ways:

Direct measurements: They are measurements taken in a direct way, by using a

measuring device. Examples of direct measurements are the measurement of a length

with a tape, a time with a chronometer, or a resistance with an ohmmeter.

Indirect measurements: They are measurements obtained by applying a physical law

or an equation, from known quantities or quantities directly measured. An example of

this type of measurements would be the calculation of the volume of a sphere from its

radius. The radius is directly measured with a tape, and the volume is calculated with

the equation 34V R

3

5 ERRORS ON DIRECT MEASUREMENTS

When we are using a measuring device to take direct measurements of a quantity, we said

that the repetition of several measurements reduces the random errors, but we’ll see that

sometimes it will be enough only one measurement.

Random: Low Systematic ?

Random: Low Systematic ?

Random: High Systematic ?

Random: High Systematic ?

a) Repetition of several measurements

Ideally, we should take infinite measurements of the same quantity, and if we accept

that this quantity behaves normally, then its function of distribution of probabilities

would be defined, and therefore its mean value ( x ) and its standard deviation (σ). As a

consequence, by considering an interval with amplitude σ, 2σ, or 3σ around the mean

value, we would have probabilities of 68, 95 ó 99,7% respectively, that its value would

be inside the interval.

But to take infinite measurements is not possible, obviously, and instead to do that, we

will take only a few measurements (N, a sample of the whole population). From these

measurements, we will “estimate” a value for x and another one for σ. The number of

needed measurements (N) will be discussed further.

For x is taken the mean value of measurements (x1, x2,…, xN):

...i

i 1 N

x

xN

This value can be computed on Excel with the command PROMEDIO

And for σ is taken a value taking in account the error due to the dispersion of

measurements. In order to compute it, we must on first to calculate the standard

deviation of the sample (Sx):

...

( )

( )

2i

i 1 Nx

x x

SN 1

This value can be computed on Excel with the command DESVEST.M

The standard deviation is related to the dispersion of measurements, but avoiding that

some measurements cancel other measurements (term ( )2ix x ). Moreover, the term

N-1 in the denominator is saying us that it makes no sense to calculate this parameter

when only one measurement is carried out.

And with the theory of error propagation, that will be further explained, it can be

demonstrated that error of the mean of the N taken measurements (xi) is:

...

( )

( ) ( )( )

2i

x i 1 NA A

x xS

x u xN N 1N

This value is known as error (or uncertainty) type A, and it is not directly available on

Excel, but it can be calculated with .DESVEST M

N

b) Taking only one measurement.

Anyway, sometimes is impossible, difficult, or unnecessary to repeat the measurement

several times, being enough to take only one measurement to know the quantity we

are measuring. It happens when the own measurement can destroy the sample we are

measuring, when the manufacturer of the measuring device gives us information about

the error on measurement, or when the time taken on several measurements can’t be

justified.

The most confident way to calculate the error in this situation, the called error (or

uncertainty) type B, comes from the technical characteristic sheets delivered by the

manufacturer, if they exist. But if they are not available and we don’t have any other

confident source of information, we can calculate the error from the resolution of

measuring device (a). This parameter is defined as the minimum magnitude can read

the measuring device. It equals the shortest distance between two lines of the ruler on

an analog device, or one unit of the less significant figure shown on the display of a

digital device.

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Example: The resolution of the digital device shown on picture (ammeter) is a=0,001 mA.

The resolution of the analog device shown on picture (voltmeter) is a=0,5 V

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When the resolution (a) is known, statistically can be demonstrated that a realistic

magnitude for the type B error is:

( ) ( )B B

ax u x

2 3

Anyway, is very important to take in account that the type B error must be evaluated in

a subjective way by the user. Even though the measuring device gives us a clear and

unambiguous magnitude for B, is possible that the measuring system advices to

consider a higher value. That would be the case, as an example, when we have to

determine the position of a moving object, if its position can’t be determined with equal

accuracy than that of the tape. Another example would be the case when we measure

the radius of a sphere with a straight and rigid ruler. We will be able to measure the

radius only approximately, and therefore B will be, probably, higher than the value

given by the ruler. But when an arbitrary value is given for the type B error, it must be

justified.

mA

Digital (left) and analog (right) devices

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Example: We want to measure the radius of a sphere with a rigid ruler whose shortest

distance between two lines of the scale (resolution) is 1 mm. The type B error of the ruler

would be ,B

10 29 mm

2 3 .But to measure the radius, we note the difficulty to

make coincident the origin of the ruler and the centre of the sphere (undefined position),

and it is also difficult to determine which line of the scale is coincident with the surface of

the sphere. Therefore, we consider difficult to read the radius of the sphere with less of 4

divisions of the scale (4 mm), reason why we will take for the type B error:

, ,B

41 16 1 2 mm

2 3 .

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Moreover, when only one measurement is carried out, is very important to have

available some measurement or procedure to detect simple mistakes, such as to get

wrong when reading the measurement, a mistake when assembling the measurement

system, or a failure on any device.

By summarizing, if we carry out a set of several measurements, the error of the measured

magnitude (typical error) will be the type A error, but if we carry out only one measurement,

its error will be the type B error. An exception to this rule occurs when we take several

measurements all of them very close, and therefore with type A error null or very low. As the

error of any measurement is at least that of the measuring device (type B error), the typical

error when several measurements are carried out is the highest of type A and B errors.

If we take only one measurement: ( ) ( )Bx x

If we take several measurements: ( ) max( ( ), ( ))A Bx x x

But we have still an important detail to be clarified:

6 HOW MANY REPETITIONS OF A MEASUREMENT SHOULD WE TAKE?

The answer to this question is not trivial. We have just said that with several measurements

of a quantity we can minimize the random errors, but how many measurements are enough?

As a general rule, Statistics recommends to start with three measurements of that quantity

we want to know (x1, x2, x3). From these measurements we calculate the dispersion (D) in % of

set of measurements, defined as:

max min 100x x

Dx

According the result of the computed dispersion:

o If D < 2%, then the three taken measurements are enough. We compute their mean

value, their type A error, we round them and we give the result.

o If D > 2%, then we have to carry out three more measurements (6 as a whole) and

verify that the resulting dispersion of the set of six measurements is < 8%.

o If the dispersion of the before set of six measurements is > 8%, then we have to carry

out nine more measurements (15 as a whole), and verify that the dispersion is < 12 %.

o If the dispersion of the before set of 15 measurements is > 12 %, then we have to

increase up to a minimum of 50 measurements and a statistical data processing of

taken measurements.

7 GENERAL EXAMPLES

Example: We are measuring an intensity of current with a digital multi-meter acting as an

ammeter, and we don’t have the technical sheets of the multi-meter. The display of the multi-

meter shows us four figures. On first, we take three readings of the intensity, with results:

I1=1,452 mA I1=1,424 mA I1=1,444 mA

From these results, we on first calculate its average or mean value (Command PROMEDIO in

Excel):

ii 1 N

av

I1 451 1 424 1 444

I I 1 439666N 3

... , , ,

, ....

and the type A error, as the standard deviation over N (command DESVEST.M/ N in Excel):

...

( )( , , ) ( , , ) ( , , )

( ) ,( ) ( )

22 2 2i

i 1 NA

I I1 451 1 43966 1 424 1 43966 1 444 1 43966

I 0 00808N 1 N 3 1 3

We calculate the type B error, by taking in account that the resolution of the measuring

device is 0,001 mA:

,( ) ,B

0 001I 0 00029 mA

2 3

As A(I) > B(I), the typical error of the intensity is (I)= A(I)=0,0081 mA and the result:

I=1,4397±0,0081 mA

The dispersion of the measurements is

I I 1 451 1 424D 100 100 1 87

I 1 4397

max min , ,

, %,

As the dispersion is <2 %, the three measurements carried out are enough to validate the

result.

All these calculations can be summarized on a table, with the equations on the bottom and

correctly rounded:

I1 (mA)

I2 (mA)

I3 (mA)

Iav (mA)

A (mA)

B (mA)

(mA)

1,451 1,424 1,444 1,4397 0,0081 0,00029 0,0081

...i

i 1 N

I

IN

...

( )

( )

2i av

i 1 NA

I I

N 1 N

,B

0 001

2 3 max( , )A B

Result: I=1,4397±0,0081 mA

Example: We are measuring an intensity of current with the digital multi-meter of before

example and we only have time enough to carry out one measurement. But when we are

measuring, the reading is constantly changing. We note that the most repeated reading is

around 10,24 mA, but this reading is oscillating between a lowest value of 9,871 mA, and a

highest value of 10,73 mA.

With only one measurement, there is not type A error. If the reading of ammeter wouldn’t

oscillate, then its type B error would be (remind that the resolution is now 0,01 mA):

,( ) ,B

0 01I 0 0029 mA

2 3

But as there is an oscillation of the reading, the resolution is the difference between the

upper and lower limits of oscillation (10,73-9,871) and the type B error is:

, ,( ) , ,B

10 73 9 871I 0 2479 0 25 mA

2 3

Therefore, the measured intensity is:

I=10,24±0,25 mA

8 REFERENCES

“Evaluación de datos de medición — Guía para la expresión de la incertidumbre de medida”. Centro español de metrología, CEM, JCGM © 2008. www.cem.es (spanish), www.bipm.org. (english).

“Introducción al análisis de errores”, John R. Taylor, 2ª Ed. Ed. Reverté, Barcelona, 2014.

“Glosario básico de términos estadísticos”, Instituto Nacional de Estadística e Informática (INEI), Lima, 2006.