A Target with Room Around It
Suppose a manufactured shaft is specified as
\[ 20.00\pm0.05\text{ mm}. \]The nominal target is
\[ 20.00\text{ mm}. \]The tolerance allows values up to
\[ 0.05\text{ mm} \]below or above that target.
So acceptable values satisfy
\[ 19.95\leq d\leq20.05. \]A tolerance therefore converts a target value into an interval.
\[ \boxed{ \text{acceptable range} = \text{nominal value} \pm \text{tolerance} } \]Nominal Values
A nominal value is the stated target, designated value, or reference value around which acceptable variation is allowed.
For example, in
\[ 50.0\pm0.2\text{ mm}, \]the nominal value is
\[ \boxed{50.0\text{ mm}}. \]The nominal value does not mean every actual object must measure exactly \(50.0\) mm.
Instead, it provides the center or reference for the tolerance specification.
What Is Tolerance?
A tolerance specifies how much deviation from a nominal value is permitted.
For a symmetric tolerance,
\[ x_0\pm t, \]where:
- \(x_0\) is the nominal value,
- \(t\) is the allowed deviation.
The acceptable interval is
\[ \boxed{ x_0-t\leq x\leq x_0+t } \]Lower and Upper Bounds
A lower bound is the smallest permitted value.
An upper bound is the largest permitted value.
For
\[ x_0\pm t, \]the bounds are:
\[ \boxed{ L=x_0-t } \]and
\[ \boxed{ U=x_0+t }. \]Example
For
\[ 12.0\pm0.3, \]we have
\[ L=11.7, \qquad U=12.3. \]Tolerance as an Interval
Interval Representation
Tolerance ranges can be written in several equivalent forms.
Suppose
\[ x=8.0\pm0.2. \]Inequality Form
\[ 7.8\leq x\leq8.2. \]Interval Notation
\[ \boxed{ x\in[7.8,8.2] } \]Absolute-Value Form
\[ \boxed{ |x-8.0|\leq0.2 } \]These are different representations of the same tolerance condition.
Symmetric Tolerance
A symmetric tolerance allows the same deviation above and below the nominal value.
For
\[ 100\pm3, \]the lower and upper bounds are
\[ 97 \]and
\[ 103. \]The nominal value lies exactly halfway between the bounds.
Asymmetric Tolerance
Not all tolerances allow equal variation in both directions.
A specification may be written as
\[ 50.0^{+0.4}_{-0.1}\text{ mm}. \]This means:
\[ 50.0-0.1 \leq x \leq 50.0+0.4. \]Therefore,
\[ \boxed{ 49.9\leq x\leq50.4 } \]millimeters.
Asymmetric tolerances are useful when exceeding the target is more acceptable than falling below it, or vice versa.
Maximum Possible Deviation
For a symmetric tolerance
\[ x_0\pm t, \]the maximum permitted absolute deviation from the nominal value is
\[ \boxed{ |x-x_0|\leq t } \]for every accepted value.
Example
If
\[ 25.0\pm0.5\text{ cm}, \]then every accepted value differs from the nominal value by at most
\[ \boxed{0.5\text{ cm}}. \]Bounds from Rounding
Rounding creates natural measurement or approximation bounds.
If a value is reported as
\[ 6.4 \]to the nearest tenth, then the underlying value ordinarily satisfies
\[ 6.35\leq x<6.45. \]The maximum rounding error is
\[ \boxed{0.05}. \]More generally, rounding to the nearest unit \(u\) creates a maximum rounding error of
\[ \boxed{ \frac{u}{2} }. \]Measurement Bounds
A measured value can also be represented by a lower and upper bound when the measurement uncertainty is known.
For example,
\[ 12.6\pm0.2\text{ cm} \]can be associated with the interval
\[ 12.4\leq L\leq12.8 \]centimeters.
The interval expresses a range supported by the measurement statement.
Its interpretation depends on how the uncertainty was obtained, so a measurement interval should not automatically be treated as a manufacturing tolerance.
Tolerance versus Uncertainty
Tolerance
A requirement: how far an actual value is allowed to deviate and still be acceptable.
Uncertainty
A statement about limited knowledge: how precisely a measurement result is known.
These can interact, but they answer different questions.
A part might have tolerance
\[ 10.00\pm0.05\text{ mm} \]while the measuring instrument has uncertainty
\[ \pm0.02\text{ mm}. \]The first is a design specification. The second concerns confidence in the measurement.
Acceptance Decisions
A simple acceptance rule is:
\[ \boxed{ L\leq x\leq U } \]where \(L\) and \(U\) are the specified lower and upper bounds.
Example
Specification:
\[ 15.00\pm0.10\text{ mm}. \]Acceptable interval:
\[ 14.90\leq x\leq15.10. \]A measured value of
\[ 15.07\text{ mm} \]falls inside the range.
\[ \boxed{\text{accepted}} \]A measured value of
\[ 15.14\text{ mm} \]falls outside the range.
\[ \boxed{\text{rejected under this simple rule}} \]Combining Tolerances
When several bounded quantities are combined, their possible deviations can accumulate.
Simple Worst-Case Addition
Suppose
\[ A=10.0\pm0.2 \]and
\[ B=5.0\pm0.1. \]For
\[ S=A+B, \]the nominal sum is
\[ 15.0. \]In a simple worst-case bound, the tolerances add:
\[ 0.2+0.1=0.3. \]So
\[ \boxed{ S=15.0\pm0.3 } \]is guaranteed to contain every possible sum if each input independently stays within its stated bound.
More advanced uncertainty analysis may combine independent random uncertainties differently. That is a separate statistical question.
Relative and Percentage Tolerance
Tolerance can also be compared with the nominal value.
For symmetric tolerance \(t\) around nominal value \(x_0\neq0\),
\[ \boxed{ \text{relative tolerance} = \frac{t}{|x_0|} } \]and
\[ \boxed{ \text{percentage tolerance} = \frac{t}{|x_0|}\times100\% } \]Example
For
\[ 200\pm4, \]the relative tolerance is
\[ \frac4{200}=0.02, \]so the percentage tolerance is
\[ \boxed{2\%}. \]Practical Applications
Manufacturing
Part dimensions are accepted only when they fall inside engineering tolerance limits.
Construction
Lengths, elevations, clearances, and alignments are often specified with allowable deviations.
Electronics
Components such as resistors are commonly manufactured with percentage tolerances around a nominal value.
Science
Measurement bounds express ranges supported by instruments and experimental procedures.
Numerical Computation
Error bounds specify how far an approximation may be from the exact value.
Worked Examples
Example 1: Find the Bounds
Given
\[ 30.0\pm0.4\text{ mm}, \]the lower bound is
\[ 30.0-0.4=29.6, \]and the upper bound is
\[ 30.0+0.4=30.4. \] \[ \boxed{ 29.6\leq x\leq30.4 } \]Example 2: Convert to Absolute-Value Form
The interval
\[ 7.8\leq x\leq8.2 \]has midpoint
\[ 8.0 \]and half-width
\[ 0.2. \]So
\[ \boxed{ |x-8.0|\leq0.2 }. \]Example 3: Asymmetric Tolerance
\[ 40.0^{+0.6}_{-0.2} \]gives
\[ 40.0-0.2=39.8 \]and
\[ 40.0+0.6=40.6. \] \[ \boxed{ 39.8\leq x\leq40.6 } \]Example 4: Rounding Bound
A measurement is reported as
\[ 18.7\text{ cm} \]to the nearest tenth.
Then
\[ \boxed{ 18.65\leq L<18.75 } \]and the maximum rounding error is \(0.05\) cm.
Example 5: Acceptance Test
Specification:
\[ 5.00\pm0.03\text{ cm}. \]Acceptable interval:
\[ 4.97\leq x\leq5.03. \]Measured value:
\[ 5.02\text{ cm}. \]Since
\[ 4.97\leq5.02\leq5.03, \]the part is
\[ \boxed{\text{within tolerance}}. \]Example 6: Combined Worst-Case Tolerance
Two lengths are
\[ 12.0\pm0.1\text{ cm} \]and
\[ 8.0\pm0.2\text{ cm}. \]The nominal total is
\[ 20.0\text{ cm}, \]and the worst-case tolerance is
\[ 0.1+0.2=0.3\text{ cm}. \] \[ \boxed{ 20.0\pm0.3\text{ cm} } \]Common Errors
1. Confusing the Nominal Value with a Guaranteed Actual Value
The nominal value is the target or designation, not necessarily the measured value of every object.
2. Forgetting to Apply Tolerance in Both Directions
For \(x_0\pm t\), subtract \(t\) for the lower bound and add \(t\) for the upper bound.
3. Assuming Every Tolerance Is Symmetric
Specifications may allow different positive and negative deviations.
4. Confusing Tolerance with Uncertainty
Tolerance defines acceptability; uncertainty describes measurement knowledge.
5. Treating a Rounded Value as Infinitely Exact
Rounding implies a range of underlying values.
6. Combining Bounds as Though Deviations Must Cancel
For guaranteed worst-case bounds, deviations may reinforce one another rather than cancel.
Practice
Find the Bounds
Convert
\[ 25.0\pm0.2 \]to an interval.
Check Answer
\[ \boxed{ 24.8\leq x\leq25.2 }. \]Absolute-Value Form
Write
\[ 9.5\leq x\leq10.5 \]as an absolute-value inequality.
Check Answer
The midpoint is \(10\) and the half-width is \(0.5\):
\[ \boxed{ |x-10|\leq0.5 }. \]Asymmetric Bounds
Find the interval for
\[ 12.0^{+0.3}_{-0.1}. \]Check Answer
\[ \boxed{ 11.9\leq x\leq12.3 }. \]Rounding
A value is reported as \(4.2\) to the nearest tenth. What is the ordinary rounding interval?
Check Answer
\[ \boxed{ 4.15\leq x<4.25 }. \]Percentage Tolerance
Find the percentage tolerance for
\[ 80\pm2. \]Check Answer
\[ \frac2{80}\times100\% = \boxed{2.5\%}. \]The Deeper Idea
Tolerance turns a single target value into a set of acceptable values.
\[ \boxed{ x_0 \longrightarrow [x_0-t,\;x_0+t] } \]That means tolerance is naturally connected to set and interval language.
The same condition can be written as:
\[ x\in[x_0-t,x_0+t], \]or
\[ |x-x_0|\leq t. \]So a practical manufacturing specification is also a mathematical statement about distance on the number line.
This connects several foundational ideas:
- intervals describe allowed regions,
- absolute value measures distance,
- error measures deviation from a reference,
- tolerance sets a maximum permitted deviation.
The mathematics is the same structure viewed from different applications.
Summary
- A nominal value is the target or designated reference value.
- A tolerance specifies how much deviation from the nominal value is allowed.
- For symmetric tolerance \[ x_0\pm t, \] the lower and upper bounds are \[ x_0-t \quad\text{and}\quad x_0+t. \]
- Tolerance ranges can be written as inequalities, intervals, or absolute-value conditions.
- Symmetric tolerances allow equal positive and negative deviation.
- Asymmetric tolerances allow different deviations above and below the nominal value.
- A tolerance places a maximum permitted deviation from the nominal value.
- Rounded values naturally imply measurement or approximation bounds.
- Measurement bounds describe ranges supported by measured data.
- Tolerance is an acceptance requirement; uncertainty describes limited measurement knowledge.
- Acceptance decisions compare a measured value with specified bounds.
- Worst-case bounds for sums can be formed by adding component tolerances.
- Relative and percentage tolerance compare permitted deviation with the nominal scale.
Tolerance and measurement bounds turn vague ideas such as “close enough” into precise mathematical intervals. Once the bounds are known, acceptance, comparison, and worst-case reasoning become explicit rather than subjective.
Measurement, Scale, and Precision
- Previous: Absolute, Relative, and Percentage Error
- Measurement, Scale, and Precision — return to the lesson directory.