2. Measurement Process Characterization
2.3. Calibration
2.3.5. Control of artifact calibration
2.3.5.2. Control of bias and long-term variability

## Example of Shewhart control chart for mass calibrations

Example of a control chart for mass calibrations at the kilogram level Mass calibrations usually start with the comparison of four kilogram standards using a high precision balance as a comparator. Many of the measurements at the kilogram level that were made at NIST between 1975 and 1989 were made on balance #12 using a 1,1,1,1 calibration design. The restraint for this design is the known average of two kilogram reference standards. The redundancy in the calibration design produces individual estimates for the two test kilograms and the two reference standards.
Check standard There is no slot in the 1,1,1,1 design for an artifact check standard when the first two kilograms are reference standards; the third kilogram is a test weight; and the fourth is a summation of smaller weights that act as the restraint in the next series. Therefore, the check standard is a computed difference between the values of the two reference standards as estimated from the design. The convention with mass calibrations is to report the correction to nominal, in this case the correction to 1000 g, as shown in the control charts below.
Need for monitoring The kilogram check standard is monitored to check for:
1. Long-term degradation in the calibration process
2. Anomalous behavior at specific times
Monitoring technique for check standard values Check standard values over time and many calibrations are tracked and monitored using a Shewhart control chart. The database and control limits are updated when needed and check standard values for each calibration run in the next cycle are compared with the control limits. In this case, the values from 117 calibrations between 1975 and 1985 were averaged to obtain a baseline and process standard deviation with v = 116 degrees of freedom. Control limits are computed with a factor of k = 3 to identify truly anomalous data points.
Control chart of kilogram check standard measurements showing a change in the process after 1985
Interpretation of the control chart The control chart shows only two violations of the control limits. For those occasions, the calibrations were discarded and repeated. The configuration of points is unacceptable if many points are close to a control limit and there is an unequal distribution of data points on the two sides of the control chart -- indicating a change in either:

• process average which may be related to a change in the reference standards
or
• variability which may be caused by a change in the instrument precision or may be the result of other factors on the measurement process.
Small changes only become obvious over time Unfortunately, it takes time for the patterns in the data to emerge because individual violations of the control limits do not necessarily point to a permanent shift in the process. The Shewhart control chart is not powerful for detecting small changes, say of the order of at most one standard deviation, which appears to be approximately the case in this application. This level of change might seem insignificant, but the calculation of uncertainties for the calibration process depends on the control limits.
Re-establishing the limits based on recent data and EWMA option If the limits for the control chart are re-calculated based on the data after 1985, the extent of the change is obvious. Because the exponentially weighted moving average (EWMA) control chart is capable of detecting small changes, it may be a better choice for a high precision process that is producing many control values.
Revised control chart based on check standard measurements after 1985
Sample code The original and revised Shewhart control charts can be generated using both Dataplot code and R code. The reader can download the data as a text file.