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, x true , 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: () true Absolute error x x x () Re () true Absolute error x lative 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+xwith 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 (ε r (x) or u r (x)), as the quotient of absolute uncertainty to the measured value: () () () r r x ux x u x x x The relative values are mainly useful to compare measurements between them. ------------------------------------------------------------------------------------------------------------------ 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 5 1 01 10 50 () , % 6 r 5 5 2 10 0 0001 50 10 The second measurement is, obviously, much more accurate than the first one. ------------------------------------------------------------------------------------------------------------------ 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:
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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.