Computing from standard values
- Last UpdatedNov 04, 2024
- 9 minute read
When the cl_source for a characteristic or for a QM specification that overrides the characteristic is configured to use the given standard mean and standard deviation (cl_source = 1), the given standard mean (std_avg) and the standard deviation (std_deviation) are used to calculate the control limits to validate the measurements recorded for a characteristic.
If the standard mean (std_avg) or the standard deviation (std_deviation) for a characteristic is not given, the mean or the standard deviation (estimate or population) is calculated from the data. The standard deviation is calculated based on the following:
-
The sigma estimate (sigma_est) configured for a characteristic.
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The default chart type linked to the characteristic.
If the standard deviation is given, it is adjusted based on the chart type.
The mean and standard deviation are only used to calculate control limits for a characteristic. These values are not used for calculating statistical values such as Cp and CpK. The target value configured for a specification, that is associated with the characteristic is used as the standard average in the following conditions:
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The given standard average (std_avg) contains a non null value.
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The std_avg_is_target is set to True for the characteristic.
The following table describes how the given standard deviation, CL, LCL and UCL are adjusted based on the chart type. The examples given below have the following:
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Standard mean set to 0.6 for variable-type characteristics
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Standard deviation set to 0.8 for variable-type characteristics
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Standard proportion set to 0.6 for binary-type characteristics
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Standard defects per unit set to 10 for counted-type characteristics
The examples in the table below are only applicable to the final chart point that can be plotted in a chart for a characteristic. The control limits for the other chart points can be calculated using the formula for a chart type.
|
Chart Type |
Formula |
Comments |
Example |
|---|---|---|---|
|
ix |
CL = Given Standard Mean LCL = CL – (3 * Given Std. Dev.) UCL = CL + (3 * Given Std. Dev.) |
Since individual data is considered for this chart type, the sample size is always 1. |
CL = 0.6 The given standard mean is 0.6, and the given standard deviation is 0.8. LCL = 0.6 – (3 * 0.8) = –1.80 UCL = 0.6 + (3 * 0.8) = 3.00 |
|
mr (ix + mr) |
CL = Given Std. Dev. * d2 [2] LCL = CL – (3 * (Given Std. Dev. * d3 [2])) UCL = CL + (3 * (Given Std. Dev. * d3 [2])) if LCL < 0, then 0; otherwise LCL. |
Since this is a moving range chart, the range is always between 2 measurements, hence the d2 factor is always for sample_size=2. |
CL = 0.8 * 1.12838 = 0.90270 The given standard deviation is 0.8 and the d2 factor value for a sample size of 2 is 1.12838. LCL = 0.90270 – (3 * (0.8 * 0.85250)) = – 1.14330 Since LCL is less than 0, LCL = 0 UCL = 0.90270 + (3 * (0.8 * 0.85250)) = 2.94871 The d3 factor value for a sample size of 2 is 0.85250. |
|
mr |
CL = Given Std. Dev. * d2 [Moving Average Span] LCL = CL – (3 * (Given Std. Dev. * d3 [Moving Average Span])) UCL = CL + (3 * (Given Std. Dev. * d3 [Moving Average Span])) if LCL < 0, then 0; otherwise LCL. |
The moving average span configured for a characteristic is used to adjust the given standard deviation for calculating the control limits. |
CL = 0.8 * 1.69257 = 1.35406 The given standard deviation is 0.8 and the d2 factor value for a sample size of 3 is 1.69257. LCL = 1.35406 – (3 * (0.8 * 0.88837)) = – 0.77803 Since LCL is less than 0, LCL = 0 UCL = 1.35406 + (3 * (0.8 * 0.88837)) = 3.48614 The d3 factor value for a sample size of 3 is 0.88837. |
|
xbar |
CL = Given Standard Mean ASTD =
UCL = Mean + (3 * ASTD) where ASTD is the adjusted standard deviation from the given standard deviation. |
The sample size from a sample is used to adjust the given standard deviation for calculating the control limits (center line). |
ASTD = The given standard deviation is 0.8 and the sample size obtained from this sample is 4. CL =0.6 (Mean) LCL = 0.6 – (3 * 0.4) = –0.600 UCL = 0.6 + (3 * 0.4) = 1.800 |
|
range |
CL = Given Std. Dev. * d2 [Sample Size] LCL = CL – (3 * (Given Std. Dev. * d3 [Sample Size])) UCL = CL + (3 * (Given Std. Dev. * d3 [Sample Size])) if LCL < 0, then 0; otherwise LCL. |
The sample size from a sample is used to adjust the given standard deviation for calculating the control limits (center line). |
CL = 0.8 * 2.05875 CL = 1.64700 The given standard deviation is 0.8 and the d2 factor value for a sample size of 4 is 2.05875. LCL = 1.64700 – (3*(0.8*0.87981)) = –0.46454 Since LCL is less than 0, LCL = 0 UCL = 1.64700+(3*(0.8*0.87981)) = 3.75854 The d3 factor value for a sample size of 4 is 0.87981. |
|
sigma |
Sample Size = Sample Size of a sample CL = Given Std. Dev. * C4 [Sample Size] LCL = CL – (3 * (Given Std. Dev. * UCL = CL + (3 * (Given Std. Dev. * if LCL < 0, then 0; otherwise LCL. |
The sample size from a sample is used when adjusting the center line, LCL and UCL. |
CL = 0.8 * 0.92132 = 0.73705 The given standard deviation is 0.8 and the C4 factor value for a sample size of 4, obtained from this sample is 0.92132. LCL = 0.73705 – (3 * (0.8 * )) Since LCL is less than 0, LCL = 0 UCL = 0.73705 + (3 * (0.8 * |
|
ma |
CL = Given Standard Mean ASTD =
UCL = Mean + (3 * ASTD) The adjusted standard deviation from the given standard deviation is ASTD. |
The moving average span configured for a characteristic is used to adjust the given standard deviation for calculating the control limits (center line). |
ASTD = The given standard deviation is 0.8 and the moving average span configured for a characteristic is 3. CL =0.6 (Mean) LCL = 0.6 – (3 * 0.46188) = – 0.78564 UCL = 0.6 + (3 * 0.46188) = 1.98564 |
|
sigma |
CL = Given Std. Dev. * C4 [Moving Average Span] LCL = CL – (3 * (Given Std. Dev. * ))
|
The C4 value is based on the moving average span configured for a characteristic. |
CL = 0.8 * 0.88623 = 0.70898 The given standard deviation is 0.8 and the C4 factor value for a moving average span of 3 configured for this characteristic is 0.88623. LCL = 0.70898 – (3 * (0.8 * )) Since LCL is less than 0, LCL = 0 UCL = 0.70898 + (3 * (0.8 * |
|
p |
CL = Std. Mean (Given) LCL = CL – (3 * UCL = CL + (3 * where n is the actual sample size from a sample, and Std. Mean (Given) is the given mean value. If LCL < 0 then LCL = 0; otherwise LCL. If UCL > 1 then UCL = 1; otherwise UCL. |
The given mean value is used as a center line for all the chart points. However, the LCL and UCL are calculated for each chart point. |
Sample Size: 15 CL = 0.6 (Mean/proportion) where 0.6 is the given mean value. LCL = 0.6 – (3 * UCL = 0.6 + (3 * |
|
np |
CL = Std. Mean (Given) * n LCL = CL – (3 * )
If LCL < 0 then LCL = 0; otherwise LCL. |
The CL, LCL and UCL are calculated for each chart point. |
Sample Size: 15 CL = 0.6 * 15 = 9.0 CL = 9.0 LCL = 9.0 – (3 * ) = 3.30790
UCL = 14.69210 |
|
c |
CL = Given Avg. No. of Defects per Unit * n LCL = CL – (3 * )
If LCL < 0, then LCL = 0; otherwise LCL. |
The CL, LCL and UCL are calculated for each chart point. |
Sample Size: 15 CL = 10 * 15 = 150 CL = 150 LCL = 150 – (3 * UCL = 150 + (3 * |
|
u |
CL = Given Avg. No. of Defects per Unit LCL = CL – (3 * )
If LCL < 0, then LCL = 0; otherwise LCL. |
The LCL and UCL are calculated for each chart point. |
Sample Size: 15 CL = 10.0 LCL = 10 – (3 * = 7.55051 UCL = 10 + (3 * = 12.44949 |
|
DPMO |
where n is the actual sample size from the sample, AvgNoDPU is the given average number of defects per unit for this characteristic (std_avg), and NoDefectsOpps is the number of defects opportunities (num_defects_opp) configured for this characteristic. If LCL < 0, then LCL = 0; otherwise LCL. |
The LCL and UCL are calculated for each chart point. |
Sample Size: 15
|
LCL = Mean – (3 * ASTD)
= 0.4
))
= – 0.19609
LCL = Mean – (3 * ASTD)
= 0.46188
UCL = CL + (3 * (Given Std. Dev. * ))
= – 0.40282
)
) = 0.22053
UCL = CL + (3 * )
where 
UCL = 9.0 + (3 * ) = 
UCL = CL + (3 * )
) = 113.25765
UCL = CL + (3 * )
)
