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HOW TO SELECT THE APPROPRIATE TYPE OF CONTROL CHART IN METROLOGY
Author: Chen-Yun Hung, Gwo-Sheng Peng and Paul Kam-Wa Lui
Center for Measurement Standards (CMS) / Industrial Technology Research Institute (ITRI)
Abstract
According to Section 5.9 of ISO/IEC 17025:2005 [1], laboratories shall have quality control procedures for monitoring the validity of tests and
calibrations undertaken. The resulting data shall be recorded in such a way that trends are detectable and, where practicable, statistical techniques shall be
applied to the review of the results. In order to meet the requirements given above, control charts are commonly used to monitor the stability of the
measurement systems. However, the inappropriate selection of process parameters or control charts may result in a failure to detect changes in the stability
of measurement systems. For this reason, this paper focuses on how to select the appropriate types of control charts in metrology, such as for process
parameters characterized by trend or lower resolution. The accuracy of the measurement results could be continuously ensured through the correct use of
control charts.
Method (B) - When the data ci of process parameter drift linearly with time ti, assuming ci = 0 + 1ti + i, the probability distribution of ci will be a normal
distribution with an unfixed mean. Therefore, it is necessary to modify the process parameter if the control chart is constructed with a general model.
Because the data drift linearly with time, at a fixed check interval, if the process parameter is modified to be the difference between the two successive data
(xi = ci – ci-1), the probability distribution of xi will be a normal distribution with a fixed mean (1Δt). However, if the difference is used as a process
parameter, each set will have only one measurement, unlike the 𝒙 – R and 𝒙 – s control chart which can use the estimators of variation within groups to
estimate the variation between groups. Thus, it is suggested to apply the x – MR control chart to construct the control chart for individual measurements
and use moving range to replace range or standard deviation [2]. Assuming that xi is the i-th set of data, i = 1, ..., m, 𝑴𝑹𝒊 = |𝒙𝒊 − 𝒙𝒊−𝟏 | is the moving range
for two successive data, the control limits of control chart for individual measurements are calculated by Equation (10) to (12):
UCL = 𝒙 +
CL = 𝒙
Principle of Control Chart
The principle of a control chart was proposed by Dr. Walter A. Shewhart in 1924 [2], which is based on the upper and lower control limits established by
the confidence interval with a set of data. Thus, the control charts derived from this principle are called Shewhart control charts. Assuming that the process
parameter is xi, the mean is μx, and the standard deviation is σx, a general model can be expressed as Equation (1) to (3):
UCL = μx + kσx
CL = μx
LCL = μx – kσx
Upper Control Limit (UCL):
Centerline (CL):
Lower Control Limit (LCL):
(1)
(2)
(3)
When k = 3, a typical control limit established at three times the standard deviation, the probability of the measured value falling within the upper and
lower control limits is 99.73 %.
Determination of Process Parameters
LCL = 𝒙 –
𝒎
𝒊=𝟏 𝒙𝒊
(10)
(11)
𝑴𝑹
3
𝒅𝟐
(12)
𝒎
𝒊=𝟐 𝑴𝑹𝒊
where 𝒙 =
is the average of xi; 𝑴𝑹 =
is the average of MRi; and d2 varies from the moving range. When the moving range is two successive
𝒎
𝒎−𝟏
data points, d2 is 1.128.
Example - Table 1 shows the data ci of voltage measured by check interval. Since the trend of data is not applicable to the general control chart mentioned in
the previous section, Method (A) with regression analysis will be used to construct the control chart. In Figure 1, it can be observed that the control limits
for this type of control chart are not fixed, but vary with time in the form of linear model. For comparison, we modify the data ci to be xi = ci – ci-1. Figure 2
shows the x – MR control chart after the process parameter is modified. The control limits are conventional horizontal lines. Both types of control charts can
be applied to process parameters with drift characteristics. However, Method (B) control chart seems easier in calculation and construction as well as the
interpretation of patterns.
Table 1. Measured value ci once per three months.
The determination of process parameter is critical to the effectiveness of a control chart in monitoring whether the state of measurement system is incontrol or out-of-control. In general, the process parameters should be stable and sensitive enough to detect the signals of measurement systems. They are
usually related to measurement procedures and measurement equations. From the statistical point of view, if the process parameter data are subject to a
normal distribution, the interpretation of the control chart will be reasonable.
In practice, the determination of process parameters commonly includes utilizing check standards and reference standards to obtain a single measured
value, the difference between two measured values, or a ratio, etc. For example, in the caliper calibration system of length, the process parameter is defined
as the measured value of the check standard (caliper) which measures the reference standard (caliper checker or gauge block); in the accelerometer
calibration system of vibration, the process parameter is defined as the ratio of output voltages from the check standard and the reference standard. Many
more types of process parameters are described in [3]. It is recommended that as much time as possible is spent in collecting sufficient data when first
determining process parameters. This will ensure that the data are stable and subject to a normal distribution. Moreover, it is better for the process
parameters having high sensitivity to be able to detect the signals of measurement systems.
𝑴𝑹
3
𝒅𝟐
Date
Data (mV)
Dec. 2008 0.0050
Mar. 2009 0.0046
Jun. 2009 0.0045
Sep. 2009 0.0042
Dec. 2009 0.0040
Mar. 2010 0.0035
Jun. 2010 0.0034
Sep. 2010 0.0032
Dec. 2010 0.0030
Mar. 2011 0.0027
Jun. 2011 0.0026
Date
Data (mV)
Sep. 2011 0.0024
Dec. 2011 0.0019
Mar. 2012 0.0018
Jun. 2012 0.0016
Sep. 2012 0.0012
Dec. 2012 0.0010
Mar. 2013 0.0007
Jun. 2013 0.0006
Sep. 2013 0.0004
Dec. 2013 0.0000
Data (mV)
Method (A) control chart
0.0055
Data (mV)
Data
UCL
CL
LCL
0.0050
0.0045
0.0040
Method (B) control chart
0.0004
Data
UCL
CL
LCL
0.0002
0.0000
0.0035
-0.0002
0.0030
0.0025
-0.0004
0.0020
0.0015
-0.0006
0.0010
-0.0008
0.0005
0.0000
-0.0010
Date (month-year)
Date (month-year)
Figure 1. Method (A) control chart with ci.
Figure 2. Method (B) control chart with xi.
Appropriate Types of Control Charts
General Process Parameter
Most process parameters for metrology control charts have a normal distribution. Thus, Equation (1) to (3) can be applied to establish the upper and
lower limits of the control charts. At present, most laboratories use the general model provided by NBS SP 676-II [3]. When the data of process parameter
are c1, c2, …, cm, the control limits are expressed as Equation (4) to (6):
UCL = Ac + 3sc
CL = Ac
LCL = Ac – 3sc
where 𝑨𝐜 =
𝒎
𝒊=𝟏 𝒄𝒊
𝒎
is the average of ci; and 𝒔𝐜 =
𝟏
𝒎
𝟐
𝒊=𝟏(𝒄𝒊 −𝑨𝐜 ) 𝟐
𝒎−𝟏
(4)
(5)
(6)
is the standard deviation of ci.
Process Parameter with Drift Characteristic
The process parameters for some measurement systems drift over time, especially in the electrical metrology [3]. If this type of process parameters still
uses Equation (4) to (6) to calculate the centerline and the upper and lower control limits, the common interpretation criteria for control charts will not
effectively detect whether the measurement system is in control. For this type of process parameters, two types of control charts are presented as references
for laboratories.
Method (A) - When the data ci of process parameter drifts linearly with time ti, assume that the regression model is a simple linear regression model ci = 0 +
1ti + i, where 0 is the intercept, 1 is the slope, and i is the i-th random error [3]. Since 0 and 1 are unknown parameters, they will be estimated by b0
and b1 using the least squares method. The regression model with estimators of parameters is expressed as 𝒄𝒊 = 𝒃𝟎 + 𝒃𝟏 𝒕𝒊 . For this type of control chart, the
control limits are calculated by Equation (7) to (9).
UCL = 𝒄𝒊 + 3sc
CL = 𝒄𝒊
LCL = 𝒄𝒊 – 3sc
where 𝒔𝐜 =
𝒎 (𝒄
𝒊=𝟏 𝒊
−𝒄𝒊 )𝟐
𝒎−𝟐
is the standard deviation of the regression model.
(7)
(8)
(9)
Process Parameter with Lower Resolution
When the process parameter is defined as the measured value of the meter having lower resolution, on condition that the standard is extremely stable, it
will likely result in zero standard deviation and be unable to establish the upper and lower control limits of the control chart with a general model.
Otherwise, it is also possible that the control limits are smaller than the resolution due to low standard deviation, and the signals of measurement system are
detected by shifting only one resolution even though the measurement system is actually in control. If the above situation occurs, it is suggested that
technical staff with expertise appropriately adjust the upper and lower control limits. For example, if the measurement system is regarded as normal by
shifting one resolution from practical experience, the upper and lower limits can be set up by 1.5 times the resolution, as shown in Figure 3.
Conclusions
In order to construct a control chart monitoring the stability of the measurement system
effectively, the determination of process parameter is the primary factor. The second factor is the
selection of the appropriate type of control chart. This paper only provides several appropriate types
of control charts summarized by practical experience of the author. However, as the wide scope of
metrology, the types of control charts mentioned herein may not be able to meet the needs by each
laboratory. Laboratories should still cautiously consider the characteristics of measurement systems
or process parameters to select the appropriate types of control charts. If there are international
standards available, they should be followed first.
References
[1]ISO/IEC, “General requirements for the competence of testing and calibration laboratories,”
ISO/IEC 17025, 2005.
[2]R. DeVor, T. Chang, and J. Sutherland, Statistical Quality Design and Control, Prentice-Hall, 1992.
[3]C. Croarkin, , “Measurement Assurance Programs Part II: Development and Implementation, ”
National Bureau of Standards Special Publication 676-II, April 1985.
Figure 3. Lower-resolution control chart.
(resolution = 0.1 mm)
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