Determining Measurement Uncertainty for Dimensional Measurements
The purpose of any measurement activity is to determine or quantify the size, location oramount of an object, substance or physical parameter of a feature we are measuring. We may betrying to determine the diameter of a wrist pin for an automobile engine (size), the X, Y and Z
coordinates of a dowel pin hole relative to a reference surface in a die component (location) or thepercentages of pigments in a specific shade of red paint (amount).
Following is an example of the equation of a dimensional measurement:Equation #1: Y = X + å Ci
WhereY = corrected value
X = measured value
Ci = corrections to the measured value
If we knew or could measure exactly all the elements that affect or alter the measured
value we could add or subtract the appropriate corrections Ci and exactly adjust the measured
value X to obtain the corrected value Y. However all of these elements have a certain degree ofpotential error or uncertainty associated with them and as a result the combined affect on thecorrected value is that we can only express it as being a certain value within a certain range to acertain level of confidence. This resulting “range” is the Expanded Uncertainty.
The individual performing a measurement, the equipment used to take the measurement
and the environment in which the measurement is taken are all potential sources of error (whichproduces uncertainty) in the measurement values obtained. Training of the individuals performing
the measurement activity, careful selection of the equipment and specification of the procedures
used to obtain measurements and control of the environment in which the measurements are taken
enable us to reduce to a minimum the magnitude of measurement error.
The formal determination of a value for uncertainty of a measurement requires that weanalyze and understand the interaction of the individual, the equipment and the environment todetermine the manner in which they contribute to the measurement error and the expectedmagnitude of their contributions.
Expanded uncertainty is the extent to which the measured values we obtain can be
expected to deviate from the actual value of the feature we are measuring. A stated value of
expanded uncertainty is typically expressed at a 95% confidence level, which means that in 95
cases out of 100, the measurement error will not exceed the stated amount. In the example, which
follows, we will calculate the Expanded Uncertainty of a 1­inch micrometer and although thisexample deals with dimensional measurement, the process of determination and the analyticalmethods involved are consistent for all measurement activities.
Begin the process of determination of uncertainty by establishing an uncertainty budget.
This is done by identifying all significant elements of the measurement process and determining
their individual contributions to the total uncertainty of the measurement. Reasoned judgments
must be made as to which elements are considered significant and must therefore be included andwhich ones will result in contributions so small as to be insignificant and can safely be excluded.
The micrometer which is the subject of our determination has the following characteristics:

Range of measurement: 0.0 to 1.0 in / 0.0 to 25.4 mm
Resolution: 0.0001 in / 0.00254 mm
Type of measurement display: Digital
It is equipped with a “ratchet” or “clutch” to minimize inconsistency in applied measuring force.
The gage block which will serve as the “master” is made of a special grade of steel, whichis capable of being hardened, and which will retain a high degree of dimensional stability.
For this specific micrometer, the following elements are identified as contributing to theoverall measurement uncertainty and therefore requiring a correction to the measured value toaccount for their contribution.
1.) C1 = Uncertainty of setting master2.) C2 = Uncertainty of repeatability
3.) C2 = Uncertainty of resolution
4.) C3= Uncertainty of thermometer5.) C4 = Uncertainty in CTE (Coefficient of Thermal Expansion)6.) C5 = Uncertainty due to temperature differential between micrometer and standard
Since the micrometer and the gage block are both made of steel which has a highcompressive strength and the micrometer has a ratchet on the thimble to minimize applied
measurement force, dimensional distortion of the micrometer frame and the gage block is
considered insignificant and therefore no correction is deemed necessary. This statement is an
example of reasoned judgment in determining what contributors are worthy of analysis and which
ones can be excluded without impacting the final determination.
The uncertainty budget used has 9 columns for entering data (see attached spreadsheet).
They are listed below with a brief explanation of what they represent.
Column 1.) The identifier for the correction being determined. C1, C2, C3 etc.
Column 2.) The source of the uncertainty associated with the correction identifier.
Column 3.) The statement or estimate of the uncertainty associated with the source. This is
typically an expanded uncertainty for Type B’s and may be an expanded or
standard uncertainty for Type A’s. It is necessary to determine which uncertainty
Is given for Type A’s. The units are typically micro inches (µin) or micrometers
Column 4.) Degrees of Freedom. This is an indicator of how much information was used in thestatistical determination of an uncertainty estimate. It is equal to the number ofmeasurements taken (n) – 1. If 30 measurements were taken then the degrees offreedom would be (n – 1) or 30 – 1 = 29. For Type B uncertainties the degrees of
freedom are by convention considered to be ( ¥ ) infinity.
Column 5.) Type of uncertainty. Type A uncertainties are those that are statistically determined.
Type B uncertainties are those that are determined by any other means. These means
may include previous measurement data, experience with or general knowledge ofthe relevant materials or instruments, data provided in calibration and other reports,
manufacturer’s specifications and uncertainties assigned to reference data taken fromhandbooks.
Column 6.) Type of distribution. Each contributor to the overall uncertainty has an underlying

distribution associated with the various values that might be obtained for it. There
are numerous types of distributions these values might take but for dimensional
uncertainties we will consider the 4 most common types as listed below.
Normal distribution:
This distribution results from processes that produce a population in which 68.26% of thepopulation is located within +/­1 Σ of the mean, 95.45% is located within +/­2 Σ of the mean and99.73% is located within +/­3 Σ from the mean.
Rectangular distribution:
This distribution results from processes that produce a population in which all values areequally likely to occur. 95% of the population is contained within +/­ 3 Σ.
Triangular distribution:
This distribution results from processes that produce a population in which most values areconcentrated near the center. 95% of the population is contained within +/­ 6 Σ.
“U” distribution:
This distribution results from processes that produce a population in which most values areconcentrated near the boundaries. 95% of the population is contained within +/­ 2 Σ.
Column 7.) Divisor. This is the number that divides the Expanded Uncertainty to obtain theStandard Uncertainty. It is the coefficient of S in the above descriptions.
Column 8.) Standard Uncertainty. If the stated or estimated uncertainty in column 3 is an
Expanded uncertainty then divide it by the divisor in column 7 to obtain theStandard uncertainty and enter it in this column. If the uncertainty in column 3
is a standard uncertainty then transfer it directly to this column. The units should bethe same as column 3.
Column 9.) Variance. The value in this column is the square of the Standard uncertainty found incolumn 8. (Variance = Standard Uncertainty2). The units should be µin2.
Below column 3 is a cell titled Sum of Variances. Place in this cell the sum of all thevariances in column 9.
Below the cell referenced above is a cell titled Combined Standard Uncertainty (Uc). The
Combined Standard Uncertainty is the square root of the Sum of the Variances. Obtain this value
and place it in this cell.
Below this cell is a cell titled Expanded Uncertainty (U). U = kUc where k is a coveragefactor determined as follows.
Rules for selecting values of the coverage factor k: (This list is not intended to be all inclusive)1.) If one or two contributors are dominant and their distribution is rectangular use k=1.65
for a result at a 95% confidence level.
2.) If three or more contributors are dominant and their distribution is rectangular use k =2.00 for a result at a 95% confidence level.

3.) If one contributor is dominant and its distribution is normal use k = 2.00 for a result at a95% confidence level. If the contributor is Type A and the degrees of freedom are small (less than30) obtain the value of k from a table of values for Student’s t­Distribution using the appropriate
degrees of freedom at a 95% confidence level.
4.) If a combination of contributors with normal and rectangular distributions is dominant
use k = 2.00 for a result at a 95% confidence level.
Data Collection
Uncertainty of Master: The master is a 1.00 in grade 2 gage block. The calibration
certificate for the gage block indicates that it has an expanded uncertainty of 3.00 µin. Since thestandard uncertainty is not determined statistically, there are no degrees of freedom. For the same
reason it is a Type B uncertainty. Gage blocks are sorted for size and categorized into grades basedon accuracy. The most accurate are grade 1, the next level of accuracy is grade 2 etc. This process
tends to produce (in all classes except the most accurate and the least accurate) a “U” distribution
where the majority of items will be near the boundaries and few toward the center. The divisor fora “U” distribution is 2 . The standard uncertainty is 2.12. The variance is 4.49
Determine the error due to repeatability and the error due to resolution. Use whichever is greaterin the final determination of Expanded Uncertainty.
Uncertainty of Repeatability: 30 measurements of the gage block were taken with themicrometer and the standard deviation (standard error) of these values was determined to be 38µin. Since this value is determined statistically, it has degrees of freedom equal to n­1 or 29. Also
since it is statistically determined it is a Type A uncertainty. The measurements produce a“Normal” distribution and since the standard error was determined the divisor is 1.00. Thevariance is 1,444.00
Uncertainty of Resolution: The micrometer reads to the nearest 50 µin. The expanded
uncertainty is one half of the resolution, 25 µin. This is a Type B uncertainty with no degrees offreedom since it is not determined statistically. The distribution is “Rectangular” since the actual
reading that is rounded off to determine the displayed value is equal to the displayed value +/­2.5µin and all 5 digits have an equal likelihood of occurrence. The divisor for a “Rectangular”
distribution is 3 . The standard uncertainty is 14.43. The variance is 208.33.
Next we need to examine thermal conditions in the lab and thermal characteristics of thematerials involved in the uncertainty determination. The laboratory is maintained at 68°F +/­1°F.
The heating and cooling system in the laboratory is controlled by a thermostat that turns on theheat when the temperature approaches 67°F and turns on the Air­conditioning when thetemperature approaches 69°F
Uncertainty of Thermometer: As stated, the laboratory temperature is controlled to +/­1°F however the thermometer is not exact and we need to determine the uncertainty associatedwith it. Expect that the thermometer is accurate to within +/­.5°F. The laboratory temperature
range is 68°F +/­1.5°F. The micrometer and the gage block are made of steel and the commonlystated value for CTE is 6 µin per inch of length per degree Fahrenheit. We will use the following
equation to calculate the resulting uncertainty associated with the inaccuracy of the thermometer.

Equation #2: DL = L * DT * a
Where…
DL = the error in length
L = the nominal length of the gage block
DT = the expected error of the thermometer in degrees Fahrenheit
a = CTE (Coefficient of Thermal Expansion)
Substituting actual values:
DL = 1.00*.5*6.00 µin =3.00 µin
The expanded uncertainty is 3.00 µin. It is a Type B uncertainty therefore there are nodegrees of freedom. It is unlikely that the worst case as calculated will always happen so consider
this as a “Rectangular” distribution for which the divisor is 3 . The standard uncertainty is 1.73.
The variance is 3.00.
Uncertainty of CTE (Coefficient of Thermal Expansion): The micrometer and the gage
block are made of steel and the commonly stated value for CTE is 6 µin per inch of length per
degree Fahrenheit. Although this value is satisfactory for most engineering calculations, it is not
an exact value and when considering precise measurements it is necessary to consider theuncertainty associated with the value for CTE. It is reasonable to expect that this value might varyby as much as 15% or .9 µin per inch of length per degree Fahrenheit. To calculate the resulting
uncertainty associated with the CTE we will use Equation #3 below.
Equation #3: DL = L * DT * Da
Where…
DL = the error in length
L = the nominal length of the gage block
DT = the deviation from standard temperature in the lab
Da = the difference in CTE of the gage block and the micrometer
Substituting actual values:
DL = 1.00*1.5*.9 µin = 1.35 µin
The expanded uncertainty is 1.35 µin. It is a Type B uncertainty therefore there are no
degrees of freedom. Again it is unlikely that the worst case as calculated will always happen soconsider this as a “Rectangular” distribution for which the divisor is 3 . The standard uncertainty
is 0.78. The variance is 0.61.
Uncertainty due to temperature differential: If the temperature of the micrometer andthe gage block are not the same they will experience unequal thermal expansion or contraction andthis will produce an error in the measurement. Laboratory procedures must address the need forboth items to be at the same temperature. Perhaps a policy states that items for test must be placedin the controlled environment of the laboratory a minimum of 24 hours prior to measurement sothat thermal equilibrium may be attained. Small fluctuations are still possible as the temperature
control system of the laboratory makes slight adjustments to maintain the stated temperature of68°F +/­1°F. It is reasonable to expect that under laboratory conditions the temperature differential
does not exceed 0.1°F. We will use Equation #2 below to calculate this error as follows.
Equation #2: DL = L * DT * aWhere…
DL = the error in lengthL = the nominal length of the gage block
DT = the temperature differential between the micrometer and the gage blocka = CTE (Coefficient of Thermal Expansion)
Substituting actual values:
DL = 1.00*0.1*6 µin = 0.60 µin

The expanded uncertainty is 0.60 µin. It is a Type B uncertainty therefore there are nodegrees of freedom. Again it is unlikely that the worst case as calculated will always happen soconsider this as a “Rectangular” distribution for which the divisor is 3 . The standard uncertainty
is 0.35. The variance is 0.12.
Summing all of the variances:
Variance of Master:
4.49(Larger of repeatability vs. resolution)
Variance of Repeatability:
1,440.00
Variance of uncertainty of thermometer:
3.00
Variance of uncertainty in CTE:
.61
Variance uncertainty due to temperature differential: .12å Variances = 1,448.22
Combined standard uncertainty: 1448.22 = 38.05
Referring to the rules for selecting a coverage factor, use k=2.00 per rule number 3 since
the one contributor is obviously dominant and it is normally distributed.
Expanded uncertainty (Note: k=2.00): 76.11 µin

Document Published by:Henry Alexander (PJLA Technical Committee Member)
Henry Alexander Engineering, Inc.
1367 Snyder Road
Norwalk, OH 44857
Approved by PJLA