Measurement Process Characterization_8 docx

measurements the solution for the design for n=4 angle blocks is as follows:The solution for the reference standard is shown under the first column of thesolution matrix; for the check s

Block sizes Angle blocks normally come in sets of

1, 3, 5, 20, and 30 seconds

1, 3, 5, 20, 30 minutes

1, 3, 5, 15, 30, 45 degreesand blocks of the same nominal size from 4, 5 or 6 different sets can becalibrated simultaneously using one of the designs shown in this catalog

Design for 4 angle blocks

Restraint The solution to the calibration design depends on the known value of a

reference block, which is compared with the test blocks The reference block

is designated as block 1 for the purpose of this discussion

Check

standard

It is suggested that block 2 be reserved for a check standard that is maintained

in the laboratory for quality control purposes

Series 1: 2-3-2-1-2-4-2Series 2: 5-2-5-1-5-3-5Series 3: 4-5-4-1-4-2-4Series 4: 3-4-3-1-3-5-3

Measurements

for 6 angle

blocks

Series 1: 2-3-2-1-2-4-2Series 2: 6-2-6-1-6-3-6Series 3: 5-6-5-1-5-2-5Series 4: 4-5-4-1-4-6-4Series 5: 3-4-3-1-3-5-3

The equations explaining the seven measurements for the first series in terms

of the errors in the measurement system are:

with B a bias associated with the instrument, d is a linear drift factor, X is the

value of the angle block to be determined; and the error terms relate torandom errors of measurement

The check block, C, is measured before and after each test block, and the

difference measurements (which are not the same as the differencemeasurements for calibrations of mass weights, gage blocks, etc.) areconstructed to take advantage of this situation Thus, the 7 readings arereduced to 3 difference measurements for the first series as follows:

For all series, there are 3(n - 1) difference measurements, with the first

subscript in the equations above referring to the series number The differencemeasurements are free of drift and instrument bias

Design matrix As an example, the design matrix for n = 4 angle blocks is shown below.

1 1 1 1

0 1 -1 0 -1 1 0 0

0 1 0 -1

0 -1 0 1 -1 0 0 1

0 0 -1 1

0 0 1 -1 -1 0 1 0

0 -1 1 0 The design matrix is shown with the solution matrix for identificationpurposes only because the least-squares solution is weighted (Reeve) toaccount for the fact that test blocks are measured twice as many times as thereference block The weight matrix is not shown

measurements the solution for the design for n=4 angle blocks is as follows:

The solution for the reference standard is shown under the first column of thesolution matrix; for the check standard under the second column; for the firsttest block under the third column; and for the second test block under thefourth column Notice that the estimate for the reference block is guaranteed

to be R*, regardless of the measurement results, because of the restraint that

is imposed on the design Specifically,

Solutions are correct only for the restraint as shown

For n blocks, the differences between the values for the blocks measured in

the top ( denoted by "t") and bottom (denoted by "b") positions are denotedby:

The standard deviation of the average (for each block) is calculated fromthese differences to be:

s test = K1s1where K 1 is shown under "Factors for computing repeatability standarddeviations" for each design and is the repeatability standard deviation asestimated from the design Because this standard deviation may seriouslyunderestimate the uncertainty, a better approach is to estimate the standarddeviation from the data on the check standard over time An expandeduncertainty is computed according to the ISO guidelines

2 Measurement Process Characterization

2.3 Calibration

2.3.4 Catalog of calibration designs

2.3.4.5 Designs for angle blocks

2.3.4.5.1 Design for 4 angle blocks

Y(21) 0 -5.0516438 -1.2206578 2.2723000

Y(22) 0 7.3239479 7.3239479 9.3521166

Y(23) 0 -1.2206578 -5.0516438 2.2723000

Y(31) 0 -1.2206578 2.2723000 -5.0516438

Y(32) 0 7.3239479 9.3521166 7.3239479

Y(33) 0 -5.0516438 2.2723000 -1.2206578

R* 1 1 1 1 R* = VALUE OF REFERENCE ANGLE BLOCK

FACTORS FOR REPEATABILITY STANDARD DEVIATIONS

2 Measurement Process Characterization

2.3 Calibration

2.3.4 Catalog of calibration designs

2.3.4.5 Designs for angle blocks

2.3.4.5.2 Design for 5 angle blocks

Y(13) 0.00000 2.48697 -0.89818 -4.80276 -0.78603 Y(21) 0.00000 -5.48893 -0.21200 -1.56370 3.26463 Y(22) 0.00000 5.38908 5.93802 4.71618 7.95672 Y(23) 0.00000 -0.89818 -4.80276 -0.78603 2.48697 Y(31) 0.00000 -0.21200 -1.56370 3.26463 -5.48893 Y(32) 0.00000 5.93802 4.71618 7.95672 5.38908 Y(33) 0.00000 -4.80276 -0.78603 2.48697 -0.89818 Y(41) 0.00000 -1.56370 3.26463 -5.48893 -0.21200 Y(42) 0.00000 4.71618 7.95672 5.38908 5.93802 Y(43) 0.00000 -0.78603 2.48697 -0.89818 -4.80276 R* 1 1 1 1 1.

R* = VALUE OF REFERENCE ANGLE BLOCK

FACTORS FOR REPEATABILITY STANDARD DEVIATIONS

2 Measurement Process Characterization

2.3 Calibration

2.3.4 Catalog of calibration designs

2.3.4.5 Designs for angle blocks

2.3.4.5.3 Design for 6 angle blocks

Y(11) 0.0000 3.2929 -5.2312 -0.7507 -0.6445 -0.6666

Y(12) 0.0000 6.9974 4.6324 4.6495 3.8668 3.8540

Y(13) 0.0000 3.2687 -0.7721 -5.2098 -0.6202 -0.6666

Y(21) 0.0000 -5.2312 -0.7507 -0.6445 -0.6666 3.2929

Y(22) 0.0000 4.6324 4.6495 3.8668 3.8540 6.9974

Y(23) 0.0000 -0.7721 -5.2098 -0.6202 -0.6666 3.2687

Y(31) 0.0000 -0.7507 -0.6445 -0.6666 3.2929 -5.2312

Y(32) 0.0000 4.6495 3.8668 3.8540 6.9974 4.6324

Y(33) 0.0000 -5.2098 -0.6202 -0.6666 3.2687 -0.7721

Y(41) 0.0000 -0.6445 -0.6666 3.2929 -5.2312 -0.7507

Y(42) 0.0000 3.8668 3.8540 6.9974 4.6324 4.6495

Y(43) 0.0000 -0.6202 -0.6666 3.2687 -0.7721 -5.2098

Y(51) 0.0000 -0.6666 3.2929 -5.2312 -0.7507 -0.6445

Y(52) 0.0000 3.8540 6.9974 4.6324 4.6495 3.8668

Y(53) 0.0000 -0.6666 3.2687 -0.7721 -5.2098 -0.6202

R* 1 1 1 1 1 1

R* = VALUE OF REFERENCE ANGLE BLOCK

FACTORS FOR REPEATABILITY STANDARD DEVIATIONS

SIZE K1

2 Measurement Process Characterization

temperature of the bath with a standard resistance thermometer at the

beginning, middle and end of each run of K test thermometers The test

thermometers themselves are measured twice during the run in thefollowing time sequence:

where R1, R2, R3 represent the measurements on the standard resistance

thermometer and T1, T2, , T K and T'1, T'2, , T' K represent the pair

of measurements on the K test thermometers.

Assumptions

regarding

temperature

The assumptions for the analysis are that:

Equal time intervals are maintained between measurements onthe test items

deviations

The residual variance is given by

.The standard deviation of the indication assigned to the ith testthermometer is

and the standard deviation for the estimates of shift and drift are

respectively

2 Measurement Process Characterization

2.3 Calibration

2.3.4 Catalog of calibration designs

2.3.4.7 Humidity standards

Humidity standards The calibration of humidity standards

usually involves the comparison ofreference weights with cylinderscontaining moisture The designs shown

in this catalog are drift-eliminating andmay be suitable for artifacts other thanhumidity cylinders

List of designs

2 reference weights and 3 cylinders

●

2 Measurement Process Characterization

2.3 Calibration

2.3.4 Catalog of calibration designs

2.3.4.7 Humidity standards

2.3.4.7.1 Drift-elimination design for 2

reference weights and 3 cylinders

OBSERVATIONS 1 1 1 1 1

Y(1) +

Y(2) +

Y(3) +

Y(4) +

Y(5) - +

Y(6) - +

Y(7) +

Y(8) +

Y(9) - +

Y(10) +

RESTRAINT + +

CHECK STANDARD + DEGREES OF FREEDOM = 6 SOLUTION MATRIX DIVISOR = 10 OBSERVATIONS 1 1 1 1 1

Y(1) 2 -2 0 0 0

Y(2) 0 0 0 2 -2

Y(3) 0 0 2 -2 0

Y(4) -1 1 -3 -1 -1

Y(5) -1 1 1 1 3

Y(6) -1 1 1 3 1

Y(7) 0 0 2 0 -2

Y(8) -1 1 -1 -3 -1

Y(9) 1 -1 1 1 3

Y(10) 1 -1 -3 -1 -1

R* 5 5 5 5 5

R* = average value of the two reference weights FACTORS FOR REPEATABILITY STANDARD DEVIATIONS WT K1 1 1 1 1 1 1 0.5477 +

1 0.5477 +

1 0.5477 +

2 0.8944 + +

3 1.2247 + + +

0 0.6325 + -

Explanation of notation and interpretation of tables

2 Measurement Process Characterization

2.3 Calibration

2.3.5 Control of artifact calibration

Purpose The purpose of statistical control in the calibration process is to

guarantee the 'goodness' of calibration results within predictable limitsand to validate the statement of uncertainty of the result Two types ofcontrol can be imposed on a calibration process that makes use ofstatistical designs:

Control of instrument precision or short-term variability

1

Control of bias and long-term variability

Example of a Shewhart control chart

estimates the basic precision of the instrument Designs should bechosen to have enough measurements so that the standard deviationfrom the design has at least 3 degrees of freedom where the degrees of

freedom are (n - m + 1) with

n = number of difference measurements

The check standard should be of the same type and geometry as itemsthat are measured in the designs These artifacts must be stable andavailable to the calibration process on a continuing basis There should

be a check standard at each critical level of measurement For example,for mass calibrations there should be check standards at the 1 kg; 100 g,

10 g, 1 g, 0.1 g levels, etc For gage blocks, there should be check

standards at all nominal lengths.

A check standard can also be a mathematical construction, such as thecomputed difference between the calibrated values of two referencestandards in a design

Database of

check

standard

values

The creation and maintenance of the database of check standard values

is an important aspect of the control process The results from eachcalibration run are recorded in the database The best way to record thisinformation is in one file with one line (row in a spreadsheet) of

information in fixed fields for each calibration run A list of typicalentries follows:

2 Measurement Process Characterization

separate control chart is required for each instrument except whereinstruments are of the same type with the same basic precision, in whichcase they can be treated as one

The baseline is the process standard deviation that is pooled from k = 1, , K individual repeatability standard deviations, , in the database,each having degrees of freedom The pooled repeatability standarddeviation is

with degrees of freedom

UCL Usually, only the upper control limit is of interest because we are

primarily interested in detecting degradation in the instrument'sprecision A possible complication is that the control limit is dependent

on the degrees of freedom in the new standard deviation and iscomputed as follows:

The quantity under the radical is the upper percentage point from the

F table where is chosen small to be, say, 05 The other two termsrefer to the degrees of freedom in the new standard deviation and thedegrees of freedom in the process standard deviation

Limitation

of graphical

method

The graphical method of plotting every new estimate of repeatability on

a control chart does not work well when the UCL can change with each

calibration design, depending on the degrees of freedom The algebraicequivalent is to test if the new standard deviation exceeds its controllimit, in which case the short-term precision is judged to be out ofcontrol and the current calibration run is rejected For more guidance,see Remedies and strategies for dealing with out-of-control signals

As long as the repeatability standard deviations are in control, there isreason for confidence that the precision of the instrument has notdegraded

2 Measurement Process Characterization

Need for

monitoring

precision

The precision of the balance is monitored to check for:

Slow degradation in the balance

The standard deviations over time and many calibrations are tracked and monitored using a

control chart for standard deviations The database and control limits are updated on a yearly or bi-yearly basis and standard deviations for each calibration run in the next cycle are compared with the control limits In this case, the standard deviations from 117

calibrations between 1975 and 1985 were pooled to obtain a repeatability standard

deviation with v = 3*117 = 351 degrees of freedom, and the control limits were computed

at the 1% significance level.

read mass.dat t id y bal s ds let n = size s

y1label MICROGRAMS x1label TIME IN YEARS xlimits 75 90

x2label STANDARD DEVIATIONS ON BALANCE 12 characters * blank blank blank

lines blank solid dotted dotted let ss=s*s

let sp=mean ss let sp=sqrt(sp) let scc=sp for i = 1 1 n let f = fppf(.99,3,351)

let f=sqrt(f) let sul=f*scc plot s scc sul vs t

Further

interpretation

of the control

chart

However, it is also clear from the pattern of standard deviations over time that the precision

of the balance was gradually degrading and more and more points were approaching the control limits This finding led to a decision to replace this balance for high accuracy calibrations.

2 Measurement Process Characterization

2.3 Calibration

2.3.5 Control of artifact calibration

2.3.5.2 Control of bias and long-term

The check standard values are denoted by

The baseline is the process average which is computed from the checkstandard values as

The process standard deviation is

with (K - 1) degrees of freedom.

with denoting the upper critical value from the

t-table with v = (K - 1) degrees of freedom.

let v1 = k-1let t = tppf(alphau, v1)return the following value:

THE COMPUTED VALUE OF THE CONSTANT T =0.2570583E+01

C > UCL

to see if the failure was a chance occurrence Check standard valuesthat remain in control, especially over a period of time, provideconfidence that no new biases have been introduced into themeasurement process and that the long-term variability of the processhas not changed.

Out-of-control

signals that

recur require

investigation

Out-of-control signals, particularly if they recur, can be symptomatic

of one of the following conditions:

Change or damage to the reference standard(s)