The Size of an Error Depends on Context
Suppose a measured length is
\[ 99\text{ cm} \]when the accepted value is
\[ 100\text{ cm}. \]The measurement differs by
\[ 1\text{ cm}. \]Now suppose another measurement is \(1\) cm away from a reference value of only \(2\) cm.
Both absolute errors are \(1\) cm, but the second error is much larger relative to the quantity being measured.
\[ \boxed{ \text{absolute error asks: how far?} } \] \[ \boxed{ \text{relative error asks: how large compared with the reference?} } \]Reference and Measured Values
To calculate error, we need two values:
- a measured, estimated, observed, or approximated value,
- a reference value used for comparison.
Let
\[ x_m \]represent the measured or approximate value, and let
\[ x_r \]represent the reference or accepted value.
The reference might be:
- a calibrated standard,
- a known exact value,
- an accepted experimental value,
- a benchmark used for evaluation.
Signed Error
The signed error is
\[ \boxed{ e=x_m-x_r } \]The sign tells the direction of the discrepancy.
Positive Error
If
\[ e>0, \]the measured value is above the reference.
Negative Error
If
\[ e<0, \]the measured value is below the reference.
Zero Error
\[ e=0 \]means the measured and reference values agree exactly at the reported level.
Absolute Error
Absolute error ignores direction and measures only the size of the difference:
\[ \boxed{ E_{\text{abs}} = |x_m-x_r| } \]Absolute error has the same unit and dimension as the original quantity.
Example
Measured:
\[ 48.6\text{ g}. \]Reference:
\[ 50.0\text{ g}. \]Then
\[ E_{\text{abs}} = |48.6-50.0| = \boxed{1.4\text{ g}}. \]Absolute Error as Distance on a Number Line
Relative Error
Relative error compares the absolute error with the size of the reference value:
\[ \boxed{ E_{\text{rel}} = \frac{|x_m-x_r|}{|x_r|} } \]when
\[ x_r\neq0. \]Because the numerator and denominator have the same unit, the units cancel.
Relative error is therefore dimensionless.
Example
\[ \frac{1.4\text{ g}}{50.0\text{ g}} = 0.028. \]So the relative error is
\[ \boxed{0.028}. \]Percentage Error
Percentage error expresses relative error as a percentage:
\[ \boxed{ E_{\%} = \frac{|x_m-x_r|}{|x_r|} \times100\% } \]Using the previous example,
\[ 0.028\times100\% = \boxed{2.8\%}. \]Percentage error answers the question:
How large is the error as a percentage of the reference value?
Units and Dimension
Signed Error
Same unit as the original quantity.
Absolute Error
Same unit as the original quantity.
Relative Error
Dimensionless.
Percentage Error
Dimensionless percentage.
For a length measured in meters:
\[ E_{\text{abs}} \]is measured in meters, while
\[ E_{\text{rel}} \]has no length unit because
\[ \frac{\text{m}}{\text{m}}=1. \]Why Scale Matters
Consider two measurements, each with absolute error
\[ 1\text{ cm}. \]Case A
Reference:
\[ 100\text{ cm}. \]Relative error:
\[ \frac{1}{100}=0.01=1\%. \]Case B
Reference:
\[ 2\text{ cm}. \]Relative error:
\[ \frac{1}{2}=0.5=50\%. \]The same absolute error has radically different importance at different scales.
\[ \boxed{ \text{absolute error gives size;} \quad \text{relative error gives scale-aware size} } \]Comparing Errors across Quantities
Relative or percentage error is often better than absolute error when comparing measurements whose reference values differ greatly in magnitude.
Example
Measurement A:
\[ 1000\pm5. \]Measurement B:
\[ 20\pm1. \]A has the larger absolute error:
\[ 5>1. \]But relative errors are
\[ \frac5{1000}=0.005=0.5\%, \]and
\[ \frac1{20}=0.05=5\%. \]So B has the larger error relative to its own scale.
When the Reference Value Is Zero
The usual relative-error formula
\[ \frac{|x_m-x_r|}{|x_r|} \]is undefined when
\[ x_r=0. \]Division by zero is not permitted.
Therefore ordinary relative and percentage error cannot be computed using zero as the denominator reference.
In such cases, report a meaningful alternative such as:
- absolute error,
- error relative to another nonzero scale,
- an application-specific normalized measure.
The alternative reference should be stated explicitly.
Error versus Uncertainty
Error compares a result with a known or accepted reference value:
\[ x_m-x_r. \]Uncertainty describes the range or degree of doubt associated with a measurement when the true value is not known exactly.
These ideas are related but not identical.
A measurement may be reported as
\[ 20.0\pm0.2\text{ cm} \]without knowing the exact true length.
The \(\pm0.2\) cm is an uncertainty statement, not necessarily the known measurement error.
\[ \boxed{ \text{error requires a reference;} \quad \text{uncertainty describes limited knowledge} } \]Rounding and Approximation Error
Error also appears when exact numbers are replaced by approximations.
Example
Approximate
\[ \pi \]by
\[ 3.14. \]The absolute approximation error is
\[ |\pi-3.14|. \]Using
\[ \pi\approx3.14159265, \]the error is approximately
\[ 0.00159265. \]The relative error is approximately
\[ \frac{0.00159265}{\pi} \approx 0.000507. \]The percentage error is about
\[ \boxed{0.0507\%}. \]Error Bounds
Sometimes the exact error is unknown, but we can place an upper bound on it.
If a value \(x_a\) approximates \(x\) and
\[ |x_a-x|\leq\varepsilon, \]then \(\varepsilon\) is an absolute error bound.
For example,
\[ |x_a-x|\leq0.05 \]means the approximation differs from the true value by no more than \(0.05\).
This can also be written as an interval:
\[ x_a-0.05 \leq x \leq x_a+0.05. \]Rounding Bound
If a value is rounded to the nearest tenth, the absolute rounding error is at most
\[ \boxed{0.05} \]in the same unit.
Practical Applications
Measurement
Absolute and percentage error compare instrument readings with calibration standards.
Engineering
Relative error helps judge whether a deviation is important compared with the size of the designed quantity.
Numerical Computation
Approximation algorithms are evaluated using absolute and relative error bounds.
Science
Experimental results may be compared with accepted theoretical or reference values.
Forecasting
Prediction errors can be evaluated in raw units or normalized relative to the observed quantity.
Worked Examples
Example 1: Signed Error
Measured:
\[ 12.4\text{ cm}. \]Reference:
\[ 12.0\text{ cm}. \] \[ e=12.4-12.0 = \boxed{+0.4\text{ cm}}. \]The positive sign shows the measured value is above the reference.
Example 2: Absolute Error
\[ E_{\text{abs}} = |12.4-12.0| = \boxed{0.4\text{ cm}}. \]Example 3: Relative Error
\[ E_{\text{rel}} = \frac{0.4}{12.0} = 0.0333\ldots \] \[ \boxed{ E_{\text{rel}}\approx0.0333 }. \]Example 4: Percentage Error
\[ 0.0333\ldots\times100\% \approx \boxed{3.33\%}. \]Example 5: Comparing Two Measurements
First measurement:
\[ 500\text{ g} \]with absolute error
\[ 5\text{ g}. \]Relative error:
\[ \frac5{500}=1\%. \]Second measurement:
\[ 20\text{ g} \]with absolute error
\[ 1\text{ g}. \]Relative error:
\[ \frac1{20}=5\%. \]Although \(5\) g is the larger absolute error, the \(1\)-g error is larger relative to its reference.
Example 6: Error Bound from Rounding
A length is reported as
\[ 8.3\text{ m} \]to the nearest tenth.
The true value ordinarily satisfies
\[ 8.25\leq L<8.35. \]Therefore the absolute rounding error is less than or equal to approximately
\[ \boxed{0.05\text{ m}}. \]Common Errors
1. Forgetting the Absolute Value in Absolute Error
Absolute error is nonnegative:
\[ |x_m-x_r|\geq0. \]2. Reversing the Reference and Measured Values in Signed Error
If signed direction matters, define the convention before calculating.
3. Dividing Relative Error by the Measured Value
The standard formula used here divides by the reference value:
\[ \frac{|x_m-x_r|}{|x_r|}. \]4. Leaving Units on Relative Error
The units cancel because numerator and denominator have the same dimension.
5. Calling Relative Error a Percentage without Multiplying by \(100\%\)
A relative error of
\[ 0.04 \]corresponds to
\[ 4\%. \]6. Computing Percentage Error with a Zero Reference
The usual formula is undefined when the reference value is zero.
7. Confusing Error with Uncertainty
Known discrepancy from a reference and estimated uncertainty in an unknown true value are different concepts.
Practice
Absolute Error
Measured:
\[ 19.7\text{ m}. \]Reference:
\[ 20.0\text{ m}. \]Check Answer
\[ |19.7-20.0| = \boxed{0.3\text{ m}}. \]Relative Error
Using the same values, find the relative error.
Check Answer
\[ \frac{0.3}{20.0} = \boxed{0.015}. \]Percentage Error
Convert the previous relative error to a percentage.
Check Answer
\[ 0.015(100\%) = \boxed{1.5\%}. \]Scale Comparison
Which is larger relatively: an error of \(2\) in a reference value of \(100\), or an error of \(1\) in a reference value of \(10\)?
Check Answer
\[ \frac2{100}=2\%, \] \[ \frac1{10}=10\%. \]So the second error is larger relatively.
Zero Reference
Can the standard percentage-error formula be used when the reference value is \(0\)?
Check Answer
No. The formula would require division by zero.
The Deeper Idea
Absolute and relative error answer different mathematical questions.
Absolute error measures distance in the original measurement space:
\[ |x_m-x_r|. \]Relative error normalizes that distance by a reference scale:
\[ \frac{|x_m-x_r|}{|x_r|}. \]So relative error connects error measurement to the broader idea of normalization.
\[ \boxed{ \text{relative error} = \frac{\text{absolute discrepancy}}{\text{reference scale}} } \]This is why relative error is especially useful when comparing quantities of different sizes.
The same pattern appears throughout mathematics:
- rates divide one quantity by another,
- percentages compare a part with a reference whole,
- normalized values compare with a baseline,
- relative error compares discrepancy with reference magnitude.
The denominator supplies context.
Summary
- Signed error is \[ e=x_m-x_r. \]
- Absolute error is \[ E_{\text{abs}}=|x_m-x_r|. \]
- Signed and absolute error have the same unit as the original quantity.
- Relative error is \[ E_{\text{rel}} = \frac{|x_m-x_r|}{|x_r|}, \] when \(x_r\neq0\).
- Relative error is dimensionless.
- Percentage error is \[ E_{\%} = E_{\text{rel}}\times100\%. \]
- Absolute error measures raw discrepancy; relative error measures scale-adjusted discrepancy.
- Relative error is useful for comparing errors across quantities of different sizes.
- The standard relative-error formula is undefined when the reference value is zero.
- Error compares with a reference; uncertainty describes limited knowledge about the true value.
- Rounding and approximation can introduce calculable error.
- Error bounds state a maximum possible discrepancy when the exact error is not known.
Absolute error tells how far a result is from a reference. Relative and percentage error tell how important that distance is compared with the scale of the reference itself. Together, these measures make error both measurable and interpretable.
Measurement, Scale, and Precision
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