# Margin of error

In practical measurement technology , the error limits are agreed or guaranteed maximum values ​​for positive or negative deviations of the display (output) of a measuring device from the correct value . Conceptually, error limits must be strictly distinguished from the actual measurement errors and from the measurement uncertainty .

When buying a measuring device, the actual deviations are generally not specified, but a reputable manufacturer usually guarantees their maximum values ​​under specified conditions. Error limits depend on the technical effort and on fundamental limits. The amount of random measurement errors is often negligibly small compared to the error limit; otherwise it should be taken into account when determining the error limit.

In a more recent metrological standard, the term limit deviation is used instead of the term error limit . Outside of measurement technology, the term error limit corresponds to the term deviation limit amount .

## Definitions

There is an upper and a lower limit of error. Usually both are the same size and are then referred to as symmetrical error limits . The error limits are always amounts and are therefore given without a sign. ${\ displaystyle G}$

It applies to the (absolute) deviation or the (absolute) error ${\ displaystyle F}$

${\ displaystyle | F | \ leq G}$ .

Accordingly, there is a relative error limit such that the relative deviation or the relative error applies ${\ displaystyle g}$${\ displaystyle f}$

${\ displaystyle | f | \ leq g}$ .

The reference value for the relative margin of error is like the relative error of the correct value  ; ${\ displaystyle x_ {r}}$

${\ displaystyle g = G / | x_ {r} |}$ .

## Notation

The displayed (output) value is then in a range ${\ displaystyle x_ {a}}$

${\ displaystyle x_ {r} -G \ leq x_ {a} \ leq x_ {r} + G}$ .

This is shortened to the spelling

${\ displaystyle x_ {a} = x_ {r} \ pm G}$ ,

which must by no means be interpreted as if it could only assume two values. ${\ displaystyle x_ {a}}$

If the relative error limit should appear in the result, this is possible by excluding: ${\ displaystyle x_ {r}}$

${\ displaystyle x_ {a} = x_ {r} \ cdot \ left (1 \ pm {\ frac {G} {x_ {r}}} \ right) = x_ {r} \ cdot (1 \ pm g)}$ .

In no way is it allowed to write, because then a value with the unit of the measured variable and a value with the unit one would have to be added. ${\ displaystyle x_ {r} \ pm g}$

## Quantitative information

When specifying uncertainties and error limits quantitatively, the quality of the information must be kept in mind.

• Example : A statement “5%” should contain an estimate and stand for “about 5%”; In this context, the “5” is never mathematically exact, so that any number of zeros can be appended to it after the comma. An indication of “4.8%” is hardly an indication of increased care.

No “fine” results can be derived from a “rough” starting position, because the rules for error propagation of error limits for mutually independent values ​​result (see below: Calculating with error limits):

The result can never be more precise than what is put into it. (An exception applies to random errors: Here, after repeated measurements, the mean value becomes more precise than the individual measured value).
• Example : 5% 15.6 V = 0.8 V and not 0.78 V,
unless 5.0% can state responsible.

This requirement corresponds to the requirement in DIN 1333 : Uncertainties are given with a significant digit, except for the numbers 1 or 2, in which case two significant digits are given.

• Example : 5% 35.6 V = 1.8 V and not 2 V.

A leading zero is not significant.

• Example : The specification 0.8 V contains only one significant digit.

It is part of the concept of the limit value that it may only be rounded up and not down; The same applies to the uncertainty according to DIN 1333. Actually, an error limit 5% · 6.2 V = 0.31 V would be rounded up to 0.4 V and not rounded down to 0.3 V; but one should keep a certain eye here, because already 4.8% · 6.2 V <0.3 V.

It is not wrong to calculate more precisely in intermediate steps so that rounding errors do not build up, and only to consider the error limits when the result is reached, see also significant digits .

Information and examples on measuring device error limits can be found

## Calculating with error limits

If a measurement result can only be calculated from several independent measured values , then, mathematically speaking, it is a function of several independent variables${\ displaystyle y}$${\ displaystyle x_ {i}}$${\ displaystyle y}$${\ displaystyle x_ {i}}$

${\ displaystyle y = y (x_ {1}, \ x_ {2}, \ \ cdots)}$

Changes in the independent variable by a small amount are transferred with the function and lead to a change in the dependent variable by a value , according to the rules of mathematics ${\ displaystyle \ Delta x_ {i}}$${\ displaystyle \ Delta y}$

${\ displaystyle \ Delta y \ approx {\ frac {\ partial y} {\ partial x_ {1}}} \ Delta x_ {1} + {\ frac {\ partial y} {\ partial x_ {2}}} \ Delta x_ {2} + \ cdots}$ .

If one does not know the changes (measurement errors or measurement deviations) themselves, but only their limit values ​​(error limits) , then only the error limit of the result can be specified; in the sense of the limit value, the most unfavorable sign combination of the summands is to be used as a basis ${\ displaystyle G_ {i}}$ ${\ displaystyle G_ {y}}$

${\ displaystyle G_ {y} = \ left | {\ frac {\ partial y} {\ partial x_ {1}}} \ right | G_ {1} + \ left | {\ frac {\ partial y} {\ partial x_ {2}}} \ right | G_ {2} + \ cdots}$ .

This formula is simplified to easily memorable rules for the four basic arithmetic operations

• with addition and subtraction ,${\ displaystyle \ quad G_ {y} = G_ {1} + G_ {2} + \ cdots}$
i.e. the sum of the absolute error limits,

and using the relative error limits ${\ displaystyle g_ {i} = G_ {i} / | x_ {i} | \ {;} \ quad g_ {y} = G_ {y} / | y |}$

• for multiplication and division ,${\ displaystyle \ quad g_ {y} = g_ {1} + g_ {2} + \ cdots}$
thus the sum of the relative error limits.

Example : Ohm's law should be used to determine from and . ${\ displaystyle U = I \ cdot R}$${\ displaystyle U}$${\ displaystyle I}$${\ displaystyle R}$

If = 2 mA · (1 ± 2%) and = 12 kΩ · (1 ± 5%), then = 24 V · (1 ± 7%).${\ displaystyle I}$${\ displaystyle R}$${\ displaystyle U}$