The Central Idea
A measured value should be interpreted together with information about how precisely it was obtained or reported.
If a quantity is reported as
\[ 12.4\text{ cm}, \]the value usually represents an approximation rather than an infinitely exact length.
Measurement mathematics therefore asks:
- what values could the true quantity reasonably have?
- how precise is the measurement?
- how large is the possible or observed error?
Exact and Approximate Values
An exact value is known without measurement uncertainty.
Examples include:
\[ 12\text{ objects}, \]when the objects were counted exactly, or
\[ \pi \]when used symbolically as an exact mathematical constant.
An approximate value represents a quantity only to limited precision.
Measured Values Are Usually Approximate
Physical measurements depend on instruments and procedures.
A ruler, scale, stopwatch, sensor, or measuring tape has finite resolution.
Therefore, a reported measurement such as
\[ 8.7\text{ cm} \]normally represents a range of possible true values.
Precision
Precision describes the fineness or resolution with which a quantity is measured or reported.
Reporting
\[ 4.2\text{ m} \]gives less numerical precision than reporting
\[ 4.237\text{ m}. \]More digits do not automatically guarantee greater accuracy, but they indicate a more finely reported numerical value.
Accuracy and Precision Are Different
Accuracy concerns closeness to the true or accepted value.
Precision concerns the resolution or repeatability of the measurement.
A measurement can be precise but inaccurate if it is consistently offset from the true value.
Resolution
The resolution of an instrument is the smallest change it can reliably display or distinguish.
If a ruler is marked every millimeter, its smallest marked interval is
\[ 1\text{ mm}. \]A measurement reported to the nearest millimeter reflects that scale of resolution.
Rounding Creates an Interval
Suppose a length is reported as
\[ 12.4\text{ cm} \]to the nearest tenth of a centimeter.
Values that round to \(12.4\) lie from \(12.35\) up to but not including \(12.45\).
Thus,
\[ \boxed{ 12.35\le L<12.45 }. \]Lower and Upper Bounds
A lower bound is a value at or below all allowed values in the interval.
An upper bound is a value at or above all allowed values.
For a measurement reported as \(12.4\) to the nearest tenth:
\[ \boxed{\text{lower bound}=12.35} \]and
\[ \boxed{\text{upper bound}=12.45}. \]Why the Upper Endpoint Is Usually Excluded
Under ordinary rounding to the nearest tenth,
\[ 12.45 \]rounds to \(12.5\), not \(12.4\).
Therefore the interval is typically written
\[ [12.35,12.45). \]General Bound from Rounding Unit
Suppose a value \(m\) is rounded to the nearest unit of size \(u\).
Then the true value \(x\) lies approximately within half a unit:
\[ \boxed{ m-\frac{u}{2} \le x < m+\frac{u}{2} }. \]Example: Nearest Whole Number
A length is reported as
\[ 27\text{ m} \]to the nearest meter.
The rounding unit is
\[ u=1\text{ m}. \]Therefore,
\[ \boxed{ 26.5\text{ m}\le L<27.5\text{ m} }. \]Example: Nearest Hundredth
A measurement is reported as
\[ 3.47\text{ kg} \]to the nearest hundredth.
Since
\[ u=0.01, \]half the rounding unit is
\[ 0.005. \]Therefore,
\[ \boxed{ 3.465\le m<3.475 }. \]Error
Error measures the difference between an approximate value and a true or accepted value.
If the true value is \(T\) and the approximation is \(A\), the signed error may be written
\[ \boxed{E=A-T}. \]A positive error means the approximation is too large; a negative error means it is too small.
Absolute Error
The absolute error ignores the sign and measures the size of the discrepancy:
\[ \boxed{ |E|=|A-T| }. \]Absolute error has the same units as the measured quantity.
Example: Absolute Error
Suppose the accepted length is
\[ 10.00\text{ cm} \]and a measurement gives
\[ 9.92\text{ cm}. \]Then
\[ |E| = |9.92-10.00| = 0.08. \] \[ \boxed{|E|=0.08\text{ cm}}. \]Relative Error
Absolute error does not tell us how large the error is compared with the size of the quantity being measured.
The relative error is
\[ \boxed{ \frac{|A-T|}{|T|} }. \]Relative error is dimensionless.
Percent Error
Percent error expresses relative error as a percentage:
\[ \boxed{ \text{percent error} = \frac{|A-T|}{|T|}\times100\% }. \]Example: Percent Error
Suppose the accepted value is
\[ 50.0\text{ m} \]and the measured value is
\[ 49.4\text{ m}. \]The absolute error is
\[ 0.6\text{ m}. \]Relative error:
\[ \frac{0.6}{50.0}=0.012. \]Therefore,
\[ \boxed{\text{percent error}=1.2\%}. \]Same Absolute Error, Different Significance
An absolute error of
\[ 1\text{ cm} \]is large when measuring a \(5\text{ cm}\) object but small when measuring a \(1000\text{ cm}\) object.
Relative error captures this difference because it compares error with the scale of the quantity.
Maximum Error from Rounding
If a quantity is rounded to the nearest unit \(u\), the maximum rounding error is half that unit:
\[ \boxed{ |E|<\frac{u}{2} }. \]In many practical contexts this is summarized as an uncertainty of about
\[ \pm\frac{u}{2}. \]Example: Maximum Rounding Error
A mass is reported as
\[ 8.3\text{ kg} \]to the nearest tenth.
The maximum rounding error is
\[ \frac{0.1}{2}=0.05. \]So the measurement may be summarized as approximately
\[ \boxed{8.3\pm0.05\text{ kg}}. \]Uncertainty Notation
A measured value is often written in the form
\[ \boxed{x\pm\Delta x}. \]This means the reported central value is \(x\) and the stated uncertainty is \(\Delta x\).
The corresponding interval is
\[ x-\Delta x\le X\le x+\Delta x. \]Example: Interpreting Uncertainty
Suppose a length is reported as
\[ 15.0\pm0.2\text{ cm}. \]Then the stated interval is
\[ 14.8\text{ cm} \le L \le 15.2\text{ cm}. \]Significant Figures
Significant figures communicate the precision of a numerical value by identifying digits that meaningfully contribute to the reported measurement.
For example,
\[ 4.20 \]has three significant figures.
The final zero indicates that the value has been reported to the hundredths place.
Leading Zeros
Leading zeros are generally not significant because they only locate the decimal point.
For example,
\[ 0.0042 \]has two significant figures:
\[ 4,\qquad2. \]Zeros between Nonzero Digits
Zeros between nonzero digits are significant.
For example,
\[ 5.007 \]has four significant figures.
Trailing Zeros after a Decimal Point
Trailing zeros to the right of a decimal point are significant when they are intentionally reported.
Thus,
\[ 2.500 \]communicates more reported precision than
\[ 2.5. \]Ambiguity in Whole Numbers
A number such as
\[ 1500 \]may be ambiguous if the intended number of significant figures is not otherwise stated.
Scientific notation removes the ambiguity:
\[ 1.5\times10^3, \] \[ 1.50\times10^3, \]and
\[ 1.500\times10^3 \]clearly indicate different levels of reported precision.
Decimal Places versus Significant Figures
Decimal places count positions to the right of the decimal point.
Significant figures count meaningful reported digits regardless of decimal location.
For example,
\[ 0.00450 \]has five decimal places but three significant figures.
Rounding after Calculation
Intermediate values should usually retain more precision than the final answer.
Rounding too early can accumulate error.
A reliable habit is:
- keep exact forms when possible;
- retain extra calculator digits during intermediate work;
- round once at the end to the required precision.
Example: Avoiding Early Rounding
Suppose
\[ x=\sqrt{2}. \]Using
\[ x\approx1.4 \]too early produces
\[ x^2\approx1.96, \]while the exact value is
\[ x^2=2. \]The discrepancy comes entirely from premature rounding.
Bounds for Sums
Suppose
\[ a_L\le a\le a_U \]and
\[ b_L\le b\le b_U. \]Then
\[ \boxed{ a_L+b_L \le a+b \le a_U+b_U }. \]Bounds for Differences
If
\[ a_L\le a\le a_U \]and
\[ b_L\le b\le b_U, \]then
\[ \boxed{ a_L-b_U \le a-b \le a_U-b_L }. \]Why Difference Bounds Reverse One Pair
To make
\[ a-b \]as small as possible, choose the smallest \(a\) and the largest \(b\).
To make it as large as possible, choose the largest \(a\) and the smallest \(b\).
Bound calculations follow the structure of the operation.
Bounds for Products of Positive Quantities
If all relevant bounds are nonnegative, then
\[ a_L\le a\le a_U, \qquad b_L\le b\le b_U \]implies
\[ \boxed{ a_Lb_L \le ab \le a_Ub_U }. \]When negative values are possible, all endpoint products must be considered carefully.
Bounds for Quotients of Positive Quantities
Suppose
\[ 0Bounds in Geometric Measurement
If side lengths are approximate, computed perimeters, areas, and volumes are also uncertain.
Bounds let us determine a guaranteed interval for the resulting geometric quantity.
Example: Perimeter Bounds
A rectangle is measured as
\[ 8.2\text{ cm} \]by
\[ 5.4\text{ cm}, \]each to the nearest tenth.
Therefore,
\[ 8.15\le l<8.25 \]and
\[ 5.35\le w<5.45. \]Since
\[ P=2l+2w, \]the lower perimeter bound is
\[ 2(8.15)+2(5.35)=27.0, \]and the upper bound is
\[ 2(8.25)+2(5.45)=27.4. \]Thus,
\[ \boxed{ 27.0\text{ cm}\le P<27.4\text{ cm} }. \]Example: Area Bounds
Using the same rectangle,
\[ 8.15\le l<8.25 \]and
\[ 5.35\le w<5.45. \]Since all values are positive,
\[ A_{\min}=8.15(5.35)=43.6025, \] \[ A_{\max}=8.25(5.45)=44.9625. \]Therefore,
\[ \boxed{ 43.6025\text{ cm}^2 \le A < 44.9625\text{ cm}^2 }. \]Relative Uncertainty
If a measured quantity is reported as
\[ x\pm\Delta x, \]its relative uncertainty may be expressed as
\[ \boxed{ \frac{\Delta x}{|x|} }. \]Multiplying by \(100\%\) gives percent uncertainty.
Example: Percent Uncertainty
Suppose
\[ x=20.0\pm0.2\text{ cm}. \]The relative uncertainty is
\[ \frac{0.2}{20.0}=0.01. \]Therefore,
\[ \boxed{1\%} \]is the stated percent uncertainty.
Error Propagation in Addition
For a simple worst-case bound, if
\[ z=x+y \]and the absolute uncertainties are \(\Delta x\) and \(\Delta y\), then a conservative bound is
\[ \boxed{ \Delta z\le\Delta x+\Delta y }. \]The same principle applies to subtraction.
Example: Adding Measured Lengths
Suppose
\[ x=5.0\pm0.1\text{ cm} \]and
\[ y=8.0\pm0.2\text{ cm}. \]Then
\[ z=x+y=13.0\text{ cm} \]with conservative uncertainty
\[ \Delta z\le0.1+0.2=0.3. \]Thus,
\[ \boxed{z\approx13.0\pm0.3\text{ cm}}. \]Relative Error in Products
For small uncertainties, products and quotients are often analyzed using relative rather than absolute error.
If
\[ z=xy, \]then a common first-order approximation is
\[ \boxed{ \frac{\Delta z}{|z|} \approx \frac{\Delta x}{|x|} + \frac{\Delta y}{|y|} }. \]This is an approximation, not an exact worst-case identity in every setting.
Relative Error in Powers
If
\[ y=x^n, \]then for small relative uncertainty,
\[ \boxed{ \frac{\Delta y}{|y|} \approx |n| \frac{\Delta x}{|x|} }. \]The exponent amplifies relative uncertainty.
Why Area Can Be More Sensitive than Length
For a square,
\[ A=s^2. \]A small relative uncertainty in \(s\) produces approximately twice that relative uncertainty in \(A\):
\[ \frac{\Delta A}{A} \approx 2\frac{\Delta s}{s}. \]Why Volume Can Be Even More Sensitive
For a cube,
\[ V=s^3. \]Therefore,
\[ \frac{\Delta V}{V} \approx 3\frac{\Delta s}{s}. \]This is another consequence of dimensional powers.
Measurement Error and Scaling
If a measured length is used to compute area or volume, uncertainty can grow because the length appears with a power.
Dimensional scaling affects not only the quantity itself but also the sensitivity of the result to input error.
Systematic Error
A systematic error consistently shifts measurements in a similar direction.
Examples include:
- a miscalibrated scale;
- a ruler with an incorrect zero point;
- a sensor with a consistent bias.
Repeated measurements do not necessarily remove systematic error.
Random Error
Random error causes measurements to vary unpredictably around a central value.
Repeated measurements may help estimate this variability and reduce the effect of random fluctuations when values are averaged.
Repeated Measurements
If several measurements are taken,
\[ x_1,x_2,\ldots,x_n, \]their arithmetic mean is
\[ \boxed{ \bar{x} = \frac{x_1+x_2+\cdots+x_n}{n} }. \]The mean can provide a useful central estimate when random variation is present.
Range as a Simple Measure of Spread
A basic description of measurement spread is
\[ \boxed{ \text{range} = x_{\max}-x_{\min} }. \]More advanced statistics provide stronger tools for quantifying variability, but the range gives a simple first indication of consistency.
Precision from Repeated Measurements
Measurements clustered tightly together indicate high repeatability or precision.
Measurements spread widely apart indicate lower precision.
Precision alone does not reveal whether the cluster is centered on the true value.
Calibration
Calibration compares an instrument against a known reference to detect or correct systematic offsets.
Good measurement practice therefore involves both precision and calibration.
Tolerance
A tolerance specifies an allowed departure from a target value.
For example,
\[ 10.00\pm0.05\text{ mm} \]specifies an acceptable interval from
\[ 9.95\text{ mm} \]to
\[ 10.05\text{ mm}. \]Tolerance versus Measurement Uncertainty
Tolerance and uncertainty are related but different.
A tolerance describes what values are acceptable.
Measurement uncertainty describes how confidently the actual value is known.
A part can be measured precisely yet still fall outside tolerance.
Bounds and Decision Making
Bounds are useful when a decision depends on whether a quantity definitely lies above or below a threshold.
If the entire measurement interval lies below a maximum allowed value, the decision is clear.
If the interval crosses the threshold, more precise measurement may be required.
Example: Threshold Decision
A component must be no longer than
\[ 10.00\text{ cm}. \]A measurement is reported as
\[ 9.96\pm0.03\text{ cm}. \]The possible interval is
\[ 9.93\le L\le9.99. \]Since the upper bound is below \(10.00\text{ cm}\), the component satisfies the stated maximum within this uncertainty model.
Common Error: Treating a Rounded Value as Exact
A value reported as
\[ 6.2\text{ cm} \]to the nearest tenth should not automatically be treated as exactly
\[ 6.200000\ldots\text{ cm}. \]The reported precision implies a range of possible true values.
Common Error: Confusing Accuracy with Precision
Repeated values such as
\[ 10.52,\quad10.51,\quad10.52 \]are precise because they are tightly clustered.
They are not necessarily accurate unless the true value is near that cluster.
Common Error: Using Signed Error When Magnitude Is Requested
Signed error can be positive or negative:
\[ A-T. \]Absolute error is always nonnegative:
\[ |A-T|. \]Read the wording carefully.
Common Error: Dividing Percent Error by the Approximation
The standard percent-error formula compares the absolute error with the true or accepted value:
\[ \frac{|A-T|}{|T|}\times100\%. \]Using the approximation in the denominator changes the quantity being reported.
Common Error: Rounding Too Early
Premature rounding introduces additional numerical error that may compound through later calculations.
Keep extra digits or exact symbolic values until the final reporting step.
Common Error: Overreporting Precision
A calculation should not imply more measurement certainty than the input data justify.
Reporting
\[ 12.4387621\text{ cm} \]from measurements known only to the nearest tenth gives a misleading appearance of precision.
Common Error: Using Only Central Values in a Bounds Problem
If the task asks for maximum or minimum possible area, volume, or rate, calculate with the appropriate endpoint bounds rather than only the rounded central measurements.
Common Error: Applying Positive-Quantity Bound Rules across Zero
Product and quotient shortcuts such as
\[ a_Lb_L\le ab\le a_Ub_U \]assume appropriate sign conditions.
If intervals include negative values or zero, endpoint combinations must be examined more carefully.
A Reliable Procedure for Finding Rounding Bounds
- Identify the place value or rounding unit.
- Express that unit as \(u\).
- Compute half the unit, \(\frac{u}{2}\).
- Subtract it from the reported value for the lower bound.
- Add it to the reported value for the upper bound.
- Use the correct endpoint convention for the rounding rule.
A Reliable Procedure for Error Calculations
- Identify the approximate value \(A\).
- Identify the true or accepted value \(T\).
- Compute signed error \(A-T\) if direction matters.
- Compute absolute error \(|A-T|\) if magnitude matters.
- Divide by \(|T|\) for relative error.
- Multiply by \(100\%\) for percent error.
- Attach units only to absolute error, not to relative or percent error.
A Reliable Procedure for Bounds in Formulas
- Write a lower and upper bound for every measured input.
- Identify whether the formula increases or decreases with each input in the relevant range.
- Choose endpoint combinations that produce the minimum and maximum outputs.
- Check sign conditions before using product or quotient shortcuts.
- Carry enough precision through the calculation.
- Report the resulting interval with correct units.
Example: Full Error Workflow
A measured distance is
\[ 24.6\text{ m}, \]while the accepted distance is
\[ 25.0\text{ m}. \]Signed error:
\[ E=24.6-25.0=-0.4\text{ m}. \]Absolute error:
\[ |E|=0.4\text{ m}. \]Relative error:
\[ \frac{0.4}{25.0}=0.016. \]Percent error:
\[ 0.016(100\%)=1.6\%. \]Therefore,
\[ \boxed{ E=-0.4\text{ m},\qquad |E|=0.4\text{ m},\qquad \text{percent error}=1.6\% }. \]Example: Full Bounds Workflow
A square side is reported as
\[ 6.4\text{ cm} \]to the nearest tenth.
Therefore,
\[ 6.35\le s<6.45. \]Since
\[ A=s^2 \]and all values are positive,
\[ 6.35^2\le A<6.45^2. \]Thus,
\[ 40.3225\le A<41.6025. \] \[ \boxed{ 40.3225\text{ cm}^2 \le A < 41.6025\text{ cm}^2 }. \]Practice 1: Rounding Bounds
A length is reported as
\[ 18.7\text{ cm} \]to the nearest tenth. State its lower and upper bounds.
Practice 2: Nearest Whole Unit
A distance is reported as
\[ 42\text{ m} \]to the nearest meter. State the interval of possible true values.
Practice 3: Absolute Error
The accepted value is \(12.0\text{ cm}\) and the measured value is \(11.7\text{ cm}\). Find the absolute error.
Practice 4: Percent Error
The accepted value is \(80\) and the approximation is \(78\). Find the percent error.
Practice 5: Maximum Rounding Error
A value is reported to the nearest hundredth. What is the maximum rounding error?
Practice 6: Significant Figures
How many significant figures are in
\[ 0.00640? \]Practice 7: Area Bounds
A square side is reported as \(5.0\text{ cm}\) to the nearest tenth. Find the lower and upper bounds for its area.
Practice 8: Percent Uncertainty
A measurement is reported as
\[ 40.0\pm0.5\text{ m}. \]Find the stated percent uncertainty.
Practice 9: Sum of Uncertainties
Suppose
\[ x=7.0\pm0.1 \]and
\[ y=9.0\pm0.2. \]Using the conservative addition rule, report \(x+y\) with uncertainty.
Practice 10: Power Sensitivity
A cube edge has approximately \(2\%\) relative uncertainty. Using the first-order power rule, estimate the relative uncertainty in its volume.
Practice Answers
Practice 1
\[ \boxed{ 18.65\text{ cm}\le L<18.75\text{ cm} }. \]Practice 2
\[ \boxed{ 41.5\text{ m}\le d<42.5\text{ m} }. \]Practice 3
\[ |11.7-12.0|=0.3. \] \[ \boxed{0.3\text{ cm}}. \]Practice 4
\[ \frac{|78-80|}{80}\times100\% = \frac{2}{80}\times100\% = 2.5\%. \] \[ \boxed{2.5\%}. \]Practice 5
The rounding unit is \(0.01\), so half is
\[ 0.005. \] \[ \boxed{0.005}. \]Practice 6
The leading zeros are not significant, while the trailing zero after the decimal is significant.
\[ \boxed{3\text{ significant figures}}. \]Practice 7
The side satisfies
\[ 4.95\le s<5.05. \]Therefore,
\[ 4.95^2\le A<5.05^2. \] \[ \boxed{ 24.5025\text{ cm}^2 \le A < 25.5025\text{ cm}^2 }. \]Practice 8
\[ \frac{0.5}{40.0}\times100\% = 1.25\%. \] \[ \boxed{1.25\%}. \]Practice 9
\[ x+y=16.0, \]with uncertainty
\[ 0.1+0.2=0.3. \] \[ \boxed{16.0\pm0.3}. \]Practice 10
For
\[ V=s^3, \]the relative uncertainty is approximately
\[ 3(2\%)=6\%. \] \[ \boxed{6\%\text{ approximately}}. \]What to Remember
- Measured values are generally approximate rather than infinitely exact.
- Precision describes measurement resolution or reported fineness.
- Accuracy describes closeness to the true or accepted value.
- Rounding to a unit \(u\) usually implies bounds within \(\frac{u}{2}\).
- Absolute error is \(|A-T|\).
- Relative error compares absolute error with the magnitude of the true value.
- Percent error is relative error multiplied by \(100\%\).
- Significant figures communicate reported numerical precision.
- Bounds can be propagated through formulas to estimate possible output ranges.
- Rounding should usually be postponed until the final reporting step.
The central principle is: a measured number is most meaningful when its precision, uncertainty, and possible range are understood along with its numerical value.
Where We Go Next
Bounds and measurement error describe the reliability of spatial quantities. The next lesson, Decomposition, Accumulation, and Dimensional Analysis, brings the section together by showing how complex measurements can be built from simpler pieces, accumulated across regions, and checked through dimensional structure.
That lesson will connect perimeter, area, volume, integration-like reasoning, composite figures, units, and model verification into a common measurement framework.