AcademyProbability
Academy
Experimental Errors
Level 1 - Math II (Physics) topic page in Probability.
Principle
A measurement can be modelled as a fixed true value plus random error. Random error varies from trial to trial, while systematic error shifts measurements in a consistent direction.
Notation
\(X\)
measured value random variable
\(\mu\)
fixed true value of the quantity being measured
\(\varepsilon\)
random error in the measurement
\(X=\mu+\varepsilon\)
measurement model
\(E[\varepsilon]=0\)
unbiased random error
\(\sigma^2\)
variance of the random error
Method
Step 1: Separate true value from error
Treat \(\mu\) as fixed. The randomness in \(X\) comes from \(\varepsilon\).
Measurement model
\[X=\mu+\varepsilon\]
Expectation
\[E[X]=E[\mu+\varepsilon]\]
Fixed true value
\[E[X]=\mu+E[\varepsilon]\]
Unbiased error
\[E[X]=\mu+0=\mu\]
Step 2: Derive the variance
A fixed shift changes the centre but not the spread.
Start with variance
\[\operatorname{Var}(X)=\operatorname{Var}(\mu+\varepsilon)\]
Fixed shift has no variance
\[\operatorname{Var}(X)=\operatorname{Var}(\varepsilon)\]
Error variance
\[\operatorname{Var}(X)=\sigma^2\]
Rules
Measurement model
\[X=\mu+\varepsilon\]
Unbiased random error
\[E[\varepsilon]=0\]
Measured mean
\[E[X]=\mu\]
Measured variance
\[\operatorname{Var}(X)=\sigma^2\]
Biased calibration model
\[X=\mu+b+\varepsilon\]
Examples
Question
A bolt has true diameter \(\mu\). A calliper reading is
\[X=\mu+\varepsilon\]
with \[E[\varepsilon]=0\]
What is \[E[X]\]
?Answer
Since \(\mu\) is fixed,
\[E[X]=E[\mu+\varepsilon]=\mu+E[\varepsilon]=\mu\]
The measurement is unbiased.Checks
- Random error changes from measurement to measurement.
- Systematic error is a consistent bias such as a calibration offset.
- If \(E[\varepsilon]=0\), then \(X\) is unbiased for the true value \(\mu\).
- After a measurement, the observed value of \(X\) is known, but \(\mu\) and \(\varepsilon\) may remain unknown.
- A fixed true value affects the mean of \(X\), not its variance.