Summary Physics Laboratory
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TimonBeeftink
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Chapter 1: Introduction
1.1 Why error analysis?
It is impossible to indicate the exact value of a physical quantity when it is determined experimentally; due to
equipment, persons doing the experiment etc. There will always be an error in the final quantity. Error analysis
describes how errors in separate measurements propagate and influence the final result.
1.2 Sources of error
Possible errors during an experiment are:
reading error: when reading a value from a scale, a reading error always occurs.
adjustment error: when adjusting a certain quantity in an experiment, an error is made.
methodical error: this arises if a measurement instrument influences the measured quantity.
instrumentation error: due to the calibration accuracy of an apparatus an error occurs.
errors in the experimental setup: occur if the theory is imperfect or things are neglected.
1.3 Error classification
Errors can be divided into two different classes:
random errors: if its sign and magnitude vary unpredictably when a measurement is repeated.
systematic errors: if its sign and magnitude remain the same the whole experiment.
1.4 Representation of the random error
There are two ways in which a random error can be represented:
assigned error: if one single measurement (or a few) of a quantity is performed, an estimate has to
made of the uncertainty in the result. The error ∆𝑥 is then assigned in such a way, that the interval 𝑥 −
∆𝑥 to 𝑥 + ∆𝑥 is approximately 68% of the distribution of values that could be attributed to the value.
statistical error: a statistical error is derived from the spread in the results of a large number of repeated
measurements of a single quantity. The statistical error 𝑠𝑥 is then defined such that it covers
approximately 68% of 𝑥 − 𝑠𝑥 ≤ 𝑋 ≤ 𝑥 + 𝑠𝑥 .
The value of a quantity 𝑋 is commonly written as 𝑋 = 𝑥 ± ∆𝑥, where 𝑥 is the best estimate for 𝑋, based on
measurement results, and ∆𝑥 the error as defined above. All errors can be given as either an absolute error or a
relative error. The absolute error has the same dimension as the quantity it belongs to, the relative error is a
fraction and is dimensionless. This yields the following table:
assigned error statistical error
absolute error ∆𝑥 𝑠𝑥
relative error (fraction) ∆𝑥 𝑠𝑥
|𝑥| |𝑥|
relative error (percentage) ∆𝑥 𝑠𝑥
100 100
|𝑥| |𝑥|
1.5 Rounding off and notation of measurement results
Since the accuracy of an experiment is limited, not all digits of the calculated result are meaningful. Meaningful
digits are called significant digits. The general rule is that an error is always rounded off to 1 significant digit and
that it is always rounded off upwards. A measurement result is rounded off in such a way that its last digit is at
the same decimal place as the least significant digit of the error. Some examples:
𝑣 = 2,71828 𝑚𝑠 −1 ± 2 𝑚𝑚𝑠 −1 → 𝑣 = 2,718 ± 0,002 𝑚𝑠 −1
𝐶 = 4722 𝜇𝐹 ± 0,42 𝑚𝐹 → 𝐶 = 4,7 ± 0,5 𝑚𝐹 = (4,7 ± 0,5) ∙ 103 𝜇𝐹
𝑅 = 68 ± 22 𝑘𝛺 → 𝑅 = 68,00 ± 0,03 𝑀𝛺
o Remark: the relatively small error ∆𝑅 warrants more significant digits in 𝑅 then given.
1.6 Accuracy versus precision
In measurement theory, there is a distinction between accuracy and precision. An accurate measurement is one
which has a small error, which means that the final result is very close to the true value of the quantity. A precise
measurement is one with many significant digits. But a precise measurement can be inaccurate.
1
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Chapter 2: Propagation of errors
2.1 Quantities dependent on a single variable
Errors in the measurement lead to an error in the final quantity: so-called propagation of errors. When you are
dealing with one variable, you can apply the following general formula for a function 𝑦 = 𝑦(𝑥):
𝑑𝑦
∆𝑦 = ∆𝑥
𝑑𝑥
2.2 Quantities dependent on more than one variable
For quantities dependent on more than one (independent) variable, there are error propagation formulae:
Relation between 𝑍 and 𝐴, 𝐵 Relation between standard errors
𝑍 =𝐴+𝑐 ∆𝑍 = ∆𝐴
𝑍 = 𝑐𝐴 ∆𝑍 = 𝑐∆𝐴
𝑍 =𝐴±𝐵 (∆𝑍)2 = (∆𝐴)2 + (∆𝐵)2
𝑍 = 𝐴𝐵 or 𝑍 =
𝐴
∆𝑍 2 ∆𝐴 2 ∆𝐵 2
𝐵 ( ) =( ) +( )
𝑍 𝐴 𝐵
𝑍 = 𝐴𝑛 ∆𝑍 ∆𝐴
=𝑛
𝑍 𝐴
𝑍 = ln 𝐴 ∆𝐴
∆𝑍 =
𝐴
𝑍 = 𝑒𝐴 ∆𝑍
= ∆𝐴
𝑍
Calculating errors can most easily be done by using partial derivatives (derivative with respect to one specific
variable, while keeping the others constant). ∆𝑄𝑥 , ∆𝑄𝑦 , ∆𝑄𝑧 , … can be calculated for any general function given
by: 𝑄 = 𝑓(𝑥, 𝑦, 𝑧, … ). This yields:
𝜕𝑓
∆𝑄 = ∆𝑥
𝜕𝑥
This results in:
𝜕𝑓 2 𝜕𝑓 2 𝜕𝑓 2
∆𝑄 = √( ∆𝑥) + ( ∆𝑦) + ( ∆𝑧)
𝜕𝑥 𝜕𝑦 𝜕𝑧
Example
Exercise
A resistor with resistance 𝑅 carries a current 𝐼. The power 𝑃 dissipated as heat by the resistor is given by
𝑃 = 𝐼 2 𝑅. A resistor with 𝑅 = 330 Ω is used, the accuracy of 𝑅 is listed by the factory as 5%. The current is
measured: 𝐼 = 0,28 ± 0,01 𝐴.
Calculate the error in the power 𝑃 and write the final result in the correct notation.
Solution
Using the method of partial derivatives yields:
𝑃 = 𝐼 2 𝑅 = 25,872 𝑊
𝜕𝑃 2 𝜕𝑃 2
∆𝑃 = √( ∆𝐼) + ( ∆𝑅) = √(2𝐼𝑅∆𝐼)2 + (𝐼 2 ∆𝑅)2 = 2,2558 𝑊
𝜕𝐼 𝜕𝑅
So: 𝑃 = 𝑃 ± ∆𝑃 = 26 ± 3 𝑊
2.3 Dependent errors
When errors depend on each other, you cannot sum up the errors quadratically. The exact procedure to calculate
the final error depends on how the measured results interact with each other.
2
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written by
TimonBeeftink
www.stuvia.com
Downloaded by: svenlay |
Distribution of this document is illegal
, Stuvia.com - The Marketplace to Buy and Sell your Study Material
Chapter 1: Introduction
1.1 Why error analysis?
It is impossible to indicate the exact value of a physical quantity when it is determined experimentally; due to
equipment, persons doing the experiment etc. There will always be an error in the final quantity. Error analysis
describes how errors in separate measurements propagate and influence the final result.
1.2 Sources of error
Possible errors during an experiment are:
reading error: when reading a value from a scale, a reading error always occurs.
adjustment error: when adjusting a certain quantity in an experiment, an error is made.
methodical error: this arises if a measurement instrument influences the measured quantity.
instrumentation error: due to the calibration accuracy of an apparatus an error occurs.
errors in the experimental setup: occur if the theory is imperfect or things are neglected.
1.3 Error classification
Errors can be divided into two different classes:
random errors: if its sign and magnitude vary unpredictably when a measurement is repeated.
systematic errors: if its sign and magnitude remain the same the whole experiment.
1.4 Representation of the random error
There are two ways in which a random error can be represented:
assigned error: if one single measurement (or a few) of a quantity is performed, an estimate has to
made of the uncertainty in the result. The error ∆𝑥 is then assigned in such a way, that the interval 𝑥 −
∆𝑥 to 𝑥 + ∆𝑥 is approximately 68% of the distribution of values that could be attributed to the value.
statistical error: a statistical error is derived from the spread in the results of a large number of repeated
measurements of a single quantity. The statistical error 𝑠𝑥 is then defined such that it covers
approximately 68% of 𝑥 − 𝑠𝑥 ≤ 𝑋 ≤ 𝑥 + 𝑠𝑥 .
The value of a quantity 𝑋 is commonly written as 𝑋 = 𝑥 ± ∆𝑥, where 𝑥 is the best estimate for 𝑋, based on
measurement results, and ∆𝑥 the error as defined above. All errors can be given as either an absolute error or a
relative error. The absolute error has the same dimension as the quantity it belongs to, the relative error is a
fraction and is dimensionless. This yields the following table:
assigned error statistical error
absolute error ∆𝑥 𝑠𝑥
relative error (fraction) ∆𝑥 𝑠𝑥
|𝑥| |𝑥|
relative error (percentage) ∆𝑥 𝑠𝑥
100 100
|𝑥| |𝑥|
1.5 Rounding off and notation of measurement results
Since the accuracy of an experiment is limited, not all digits of the calculated result are meaningful. Meaningful
digits are called significant digits. The general rule is that an error is always rounded off to 1 significant digit and
that it is always rounded off upwards. A measurement result is rounded off in such a way that its last digit is at
the same decimal place as the least significant digit of the error. Some examples:
𝑣 = 2,71828 𝑚𝑠 −1 ± 2 𝑚𝑚𝑠 −1 → 𝑣 = 2,718 ± 0,002 𝑚𝑠 −1
𝐶 = 4722 𝜇𝐹 ± 0,42 𝑚𝐹 → 𝐶 = 4,7 ± 0,5 𝑚𝐹 = (4,7 ± 0,5) ∙ 103 𝜇𝐹
𝑅 = 68 ± 22 𝑘𝛺 → 𝑅 = 68,00 ± 0,03 𝑀𝛺
o Remark: the relatively small error ∆𝑅 warrants more significant digits in 𝑅 then given.
1.6 Accuracy versus precision
In measurement theory, there is a distinction between accuracy and precision. An accurate measurement is one
which has a small error, which means that the final result is very close to the true value of the quantity. A precise
measurement is one with many significant digits. But a precise measurement can be inaccurate.
1
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Chapter 2: Propagation of errors
2.1 Quantities dependent on a single variable
Errors in the measurement lead to an error in the final quantity: so-called propagation of errors. When you are
dealing with one variable, you can apply the following general formula for a function 𝑦 = 𝑦(𝑥):
𝑑𝑦
∆𝑦 = ∆𝑥
𝑑𝑥
2.2 Quantities dependent on more than one variable
For quantities dependent on more than one (independent) variable, there are error propagation formulae:
Relation between 𝑍 and 𝐴, 𝐵 Relation between standard errors
𝑍 =𝐴+𝑐 ∆𝑍 = ∆𝐴
𝑍 = 𝑐𝐴 ∆𝑍 = 𝑐∆𝐴
𝑍 =𝐴±𝐵 (∆𝑍)2 = (∆𝐴)2 + (∆𝐵)2
𝑍 = 𝐴𝐵 or 𝑍 =
𝐴
∆𝑍 2 ∆𝐴 2 ∆𝐵 2
𝐵 ( ) =( ) +( )
𝑍 𝐴 𝐵
𝑍 = 𝐴𝑛 ∆𝑍 ∆𝐴
=𝑛
𝑍 𝐴
𝑍 = ln 𝐴 ∆𝐴
∆𝑍 =
𝐴
𝑍 = 𝑒𝐴 ∆𝑍
= ∆𝐴
𝑍
Calculating errors can most easily be done by using partial derivatives (derivative with respect to one specific
variable, while keeping the others constant). ∆𝑄𝑥 , ∆𝑄𝑦 , ∆𝑄𝑧 , … can be calculated for any general function given
by: 𝑄 = 𝑓(𝑥, 𝑦, 𝑧, … ). This yields:
𝜕𝑓
∆𝑄 = ∆𝑥
𝜕𝑥
This results in:
𝜕𝑓 2 𝜕𝑓 2 𝜕𝑓 2
∆𝑄 = √( ∆𝑥) + ( ∆𝑦) + ( ∆𝑧)
𝜕𝑥 𝜕𝑦 𝜕𝑧
Example
Exercise
A resistor with resistance 𝑅 carries a current 𝐼. The power 𝑃 dissipated as heat by the resistor is given by
𝑃 = 𝐼 2 𝑅. A resistor with 𝑅 = 330 Ω is used, the accuracy of 𝑅 is listed by the factory as 5%. The current is
measured: 𝐼 = 0,28 ± 0,01 𝐴.
Calculate the error in the power 𝑃 and write the final result in the correct notation.
Solution
Using the method of partial derivatives yields:
𝑃 = 𝐼 2 𝑅 = 25,872 𝑊
𝜕𝑃 2 𝜕𝑃 2
∆𝑃 = √( ∆𝐼) + ( ∆𝑅) = √(2𝐼𝑅∆𝐼)2 + (𝐼 2 ∆𝑅)2 = 2,2558 𝑊
𝜕𝐼 𝜕𝑅
So: 𝑃 = 𝑃 ± ∆𝑃 = 26 ± 3 𝑊
2.3 Dependent errors
When errors depend on each other, you cannot sum up the errors quadratically. The exact procedure to calculate
the final error depends on how the measured results interact with each other.
2
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