HOW TO SELECT THE APPROPRIATE TYPE OF CONTROL CHART IN METROLOGY

2014 NCSL International Workshop and Symposium

Author: Chen-Yun Hung, Gwo-Sheng Peng and Paul Kam-Wa LuiCenter for Measurement Standards (CMS) / Industrial Technology Research Institute (ITRI)

AbstractAccording 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.

Principle of Control ChartThe 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):

Upper Control Limit (UCL): UCL = μx + kσx (1)Centerline (CL): CL = μx (2)Lower Control Limit (LCL): LCL = μx – kσx (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 ParametersThe determination of process parameter is critical to the effectiveness of a control chart in monitoring whether the state of measurement system is in-

control 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.

Appropriate Types of Control ChartsGeneral 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 (4)CL = Ac (5)LCL = Ac – 3sc (6)

where is the average of ci; and = is the standard deviation of ci.

Process Parameter with Drift CharacteristicThe 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 . For this type of control chart, the control limits are calculated by Equation (7) to (9).

UCL = + 3sc (7)CL = (8)LCL = – 3sc (9)

where is the standard deviation of the regression model.

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 – R and – 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, | 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 = + 3 (10)CL = (11)LCL = – 3 (12)

where is the average of xi; is the average of MRi; and d2 varies from the moving range. When the moving range is two successive 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.

Date Data (mV) Date Data (mV)Dec. 2008 0.0050 Sep. 2011 0.0024Mar. 2009 0.0046 Dec. 2011 0.0019Jun. 2009 0.0045 Mar. 2012 0.0018Sep. 2009 0.0042 Jun. 2012 0.0016Dec. 2009 0.0040 Sep. 2012 0.0012Mar. 2010 0.0035 Dec. 2012 0.0010Jun. 2010 0.0034 Mar. 2013 0.0007Sep. 2010 0.0032 Jun. 2013 0.0006Dec. 2010 0.0030 Sep. 2013 0.0004Mar. 2011 0.0027 Dec. 2013 0.0000Jun. 2011 0.0026  

Process Parameter with Lower ResolutionWhen 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.

ConclusionsIn 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.

Table 1. Measured value ci once per three months.

Figure 1. Method (A) control chart with ci. Figure 2. Method (B) control chart with xi.

Figure 3. Lower-resolution control chart. (resolution = 0.1 mm)

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

0.0055

Data (mV)

Date (month-year)

Method (A) control chart

DataUCLCLLCL

-0.0010

-0.0008

-0.0006

-0.0004

-0.0002

0.0000

0.0002

0.0004

Data (mV)

Date (month-year)

Method (B) control chart

DataUCLCLLCL