Cp

Cp, usually referred as Capability, is a statistic that determines whether a process confirms to the specifications.

The following formula is used to calculate Cp:

Where:

In general, this statistic assumes that the process is normally distributed and in a strate of statistical control. The results are not reliable when the process is not in a state of statistical control and abnormally distributed. The general practice to determine the standard deviation estimate is based on a control chart.

Cpk

Cpk, referred as Capability Index, is a statistic that indicates both the centering and ability of the process to conform to specifications.

The following formula is used to calculate the Cpk:

Cpk = Min(Cpu, Cpl)

where:

A process is capble only when the Cpk value is 1.33 or greater. For a one-sided tolerance, the Cpk is equivalent to either Cpu or Cpl depending on which tolerance limit is specified.

Cpl

Cpl, Lower Capability Index, is a statistic that indicates both the centering and ability of the process to conform to specifications.

The following formula is used to calculate the Cpl:

Where:

Cpu

Cpu, Upper Capability Index, is a statistic that indicates both the centering and ability of the process to conform to specifications.

The following formula is used to calculate the Cpu:

Where:

Cr

Cr, Capability Ratio, is the inverse of Cp.

The following formula is used to calculate the Cr:

EstCp

The EstCp is similar to Cp. The difference is EstCp uses an estimate of standard deviation based on individuals, whereas Cp uses the estimate based on a control chart or individuals.

The following formula is used to calculate the EstCp:

Where:

The following formula is used to calculate the estimate of standard deviation for EstCp:

EstCpk

EstCpk is a statistic that indicates both the centering and ability of the process to conform to specifications.

The following formula is used to calculate the Cpk:

Where:

EstCpk and Cpk are equivalent when the estimate of standard deviation used to calculate Cpu and Cpl is based on individual measurements.

EstCpl

EstCpl is a statistic that indicates both the centering and ability of the process to conform to specifications.

The following formula is used to calculate the EstCpl:

Where:

Cpl and EstCpl are equivalent when the standard deviation estimate used is based on individuals. However, the standard deviation estimate used to calculate Cpl is based on a control chart, the values of these two statistics are slightly different. EstCpl always uses the standard deviation estimate based on individuals whereas Cpl use the standard deviation estimate based on individuals or an estimate from a control chart.

EstCpu

EstCpu is a statistic that indicates both the centering and ability of the process to conform to specifications.

The following formula is used to calculate the EstCpu:

Where:

Cpu and EstCpu are equivalent when the standard deviation estimate is based on individuals. The values are slightly different when the standard deviation estimate for Cpu is based on a control chart. EstCpu always uses the standard deviation estimate based on individuals whereas Cpu uses the standard deviation estimate based on individuals or an estimated from the control chart.

The following formula is used to calculate s_est:

EstCr

The EstCr is the estimated capability ratio and is the inverse of EstCp.

The following formula is used to calculate the EstCr:

EstPercentNCHigh

The EstPercentNCHigh statistic represents the total percentage of USL given a normal distribution with the same standard deviation to estimate from the sample data.

EstPercentNCLow

The EstPercentNCHigh statistic represents the total percentage of LSL given a normal distribution with the same standard deviation to estimate from the sample data.

EstPercentNCPPM

The EstPercentNCPPM statistic represents the parts per million (ppm) percent non-conforming based on the assumption that the data is from a normal distribution. That is, this quantity represents the ppm you would expect given a normal distribution with the same standard deviation as that estimated from the sample data.

EstPercentNCTotal

The EstPercentNCTotal statistic represents the total percentage of samples outside of the specifications given a normal distribution with the same standard deviation as that estimated from the sample data.

EstStdDev

The EstStdDev is a measure of the spread or variation in a distribution and is calculated based on an average range, average moving range, or average standard deviation from a control chart. The formula that is used to calculate the EstStdDev depends on the type of control chart.

For control charts based on a range or moving range, the following formula is used:

For control charts based on standard deviation, the following formula is used:

where

and d₂ and c₄ are constants based on the subgroup size of the sample.

EstZLSL

The following formula is used to calculate EstZLSL:

Where:

EstZUSL

The following formula is used to calculate EstZUSL:

Where:

Exponentially Weighted Moving Average (EWMA) Chart

The calculation of control limits for the EWMA chart depends on the type of dispersion chart that is being used.

For EWMA charts based on a range or moving range, the limits are calculated as follows:

For EWMA charts based on standard deviation charts, the control limits are calculated as follows:

Kurtosis

The kurtosis of a distribution describes the relative peakedness or flatness of the distribution relative to a normal distribution. A positive kurtosis indicates a peaked distribution while negative kurtosis indicates a flat distribution.

The following formula is used to calculate the kurtosis:

Max

The max is the largest value in a set of samples.

Mean

The mean of a distribution is the average of all samples. The following formula is used to calculate the mean:

Median

The median of a distribution is the middle value, that is, half of the numbers in the sample fall above the median and half below.

To calculate the median, sort the numbers from lowest to highest. If the number of samples is odd, pick the number in the middle of the sorted list to be the median. If the number of samples is even, pick the two values in the middle and average them to obtain the median.

Min

The min is the smallest value in a set of samples.

PercentNCHigh

The PercentNCHigh statistic represents the percentage of samples that are above the upper specification limit (USL).

The following formula is used to calculate this statistic:

PercentNCLow

The PercentNCHigh statistic represents the percentage of samples that are above the upper specification limit (USL). The following formula is used to calculate this statistic:

PercentNCPPM

The PercentNCPPM statistic represents the parts per million percent non-conforming. The following formula is used to calculate this statistic:

PercentNCTotal

The PercentNCTotal statistic represents the percentage of samples that are outside the specification limits. The following formula is used to calculate this statistic:

Pp

Pp and EstCp are equivalent.

Ppk

Ppk and EstCpk are equivalent.

Ppm

The ppm is the parts per million non-conforming. The following formula is used to calculate ppm:

Pr

The Pr is the estimated capability ratio and the inverse of EstCp. The following formula is used to calculate the EstCr:

Pr and EstCr are equivalent.

Range

The range is the difference between the max and min. The formula that is used to calculate the range is:

Shapiro-Wilk Test

Studies show that the Shapiro-Wilk (W—) test is the most powerful omnibus test of normality against a wide range of alternative distributions. In our test, the p-value of W is obtained through simulation. Hence, it is not a bug if the p-values obtained from two simulations (i.e., two successive runs of the test) are slightly different. Using the default simulation size of 1000, the absolute maximum error of the p-value is less than 3.3% at a 95% confidence level. If you get a marginal p-value, you should consider increasing the simulation size and performing the test again. The absolute maximum error is less than 1/sqrt(N) where N is the simulation size.

Although the W—test is the most powerful test of normality, its actual power is still low when the sample size is small.

Skewness

The skewness describes the asymmetry of a distribution about its mean. A positive value for skewness indicates the distribution is skewed to the positive side. A negative skewness value indicates an asymmetric tail toward the negative side. The following formula is used to calculate the skewness:

Standard Deviation

The standard deviation is a measure of the spread or variation in a distribution. The following formula is used to calculate the standard deviation:

Sum

The sum adds all samples. The following formula is used to calculate the sum:

TotalAboveUSL

The TotalAboveUSL is the count of the number of samples that are above the USL.

TotalBelowLSL

The TotalBelowLSL is the count of the number of samples that are below the LSL.

TotalNonConformities

The TotalNonConformities is the count of the number of samples that are outside the specification limits.

Variance

The variance of a distribution is the square of the standard deviation and an indication of the spread or dispersion of the distribution. The following formula is used to calculate the variance:

ZLSL

The following formula is used to calculate ZLSL:

Where:

ZMIN

The following formula is used to calculate ZMIN:

ZUSL

The following formula is used to calculate ZUSL:

Where:

Variables Control Chart Calculations

Control charts based on continuous data are called variables control charts. Typical control chart types include the following:

XBar and Range Chart Formulas

XBar chart control limits are calculated as follows:

Range chart control limits are calculated as follows:

Individuals and Moving Range Formulas

The moving range is defined as the absolute difference between consecutive samples. Individuals chart control limits are calculated as follows:

Moving range control limits are calculated as follows:

XBar and Standard Deviation Chart Formulas

The standard deviation for a subgroup is defined as follows:

Where n is the subgroup size.

The control limits for the XBar chart are calculated as follows:

is defined as follows and n is the number of subgroups.

The control limits for the standard deviation chart are calculated as follows:

XBar and Range Chart

The XBar and range chart is a control chart that is used to monitor processes when the data are included in a subgroup. The Western Electric (WECO) rules can be applied to the XBar and range chart. For more information on the WECO rules see the section on Western Electric Rules.

An example of an XBar and range chart is shown below:

The top chart plots the average of a subgroup of data while the bottom chart plots the range of the subgroup. The range is the difference between the largest and smallest values within a subgroup. The table below illustrates how subgroup averages and ranges are calculated:

The first subgroup consists of the three sample values 99.64, 100.98 and 99.37. The subgroup average is the sum of these three values divided by the number of samples in the subgroup. Sum = 99.64 + 100.99 + 99.37 = 300 Average = 299.99 / 3 = 100.00 The range is the difference between the largest and smallest values. Largest value = 100.99 Smallest value = 99.37 Range = Largest – Smallest = 100.99 – 99.37 = 1.61

XBar and Range Formulas

The control limits for the XBar chart are calculated using the following formulas:

Where:

The control limits for the range chart are calculated using the following formulas:

Where:

Box and Whisker Chart

The Box and Whisker Chart is ideal for displaying both the center and variability of a process on the same chart. The figure below displays and explains the components of a typical Box and Whisker Chart.

Fences are used to identify outliers.

Exponentially Weighted Moving Average (EWMA) Chart

The EWMA Chart is a weighted moving average control chart where the weight decreases as the samples get older. The figure below shows a standard control chart with EWMA values and control limits.

The green dots on the chart represent the exponentially weighted moving averages. These quantities are calculated as follows: For the first point we use the following formula:

For the remaining points, the following formula is used:

Control Limit Formulas

The calculation of control limits for the EWMA chart depends on the type of dispersion chart. For EWMA charts based on a range or moving range, the limits are calculated as follows:

For EWMA charts based on standard deviation charts, the control limits are calculated as follows:

Individuals and Moving Range Chart

The Individuals and Moving Range (IMR) chart is a control chart used to monitor processes where sampling is done on an individual basis rather than using subgrouped data. The WECO rules can be applied to an IMR chart.

Below is an example of an IMR chart.

The top chart plots each individual measurement while the bottom chart plots the moving range between consecutive samples. The moving range is the absolute difference between two consecutive samples. For example, in the chart above the first two sample values along with moving range are listed in the following table:

Individuals and Moving Range Formulas

Individuals chart control limits are calculated as follows:

Moving range control limits are calculated as follows:

In the formulas above, the quantity is defined as follows:

The quantity is defined as follows:

Where m is the number of values in the subgroup

The difference between data point xand its predecessor (moving range) is calculated as:

Next, the arithmetic mean (average) moving range of these values is calculated as:

The centerline, upper and lower control limits are calculated as:

Individual Charts

The average of individual values is calculated as follows:

Where m is the number of values in the subgroup

The difference between data point and its predecessor (moving range) is calculated as:

Next, the arithmetic mean (average) moving range of these values is calculated as:

The centerline, upper and lower control limits are calculated as:

Individuals Moving Range (MR) Chart

The chart calculations are:

starting with i = 2 (MR₁ = 0).

The arithmetic mean (average) moving range of these values is calculated as:

where m is the number of measurement values on the chart

Moving Range Charts

The range chart plots the moving range for each subgroup. The moving range is the absolute difference between the average for the subgroup minus the average of the previous subgroup. For each subgroup, starting with the second one, the moving range is calculated and plotted. The chart calculations are as follows:

The mean (average) of the values in each subgroup is calculated as:

where m is the subgroup size.

The moving range for each subgroup is calculated as:

starting with i=2 (MR₁ = 0).

The arithmetic mean (average moving range) of these values is calculated as:

where m is the number of measurement values

Western Electric Rules

A control chart is designed to help operators and managers determine if a process is producing consistent results with respect to a quality characteristic. One of the common techniques used to help make this determination is a set of rules commonly referred to as the Western Electric (WECO) rules.

The above figure has three colored zones. These zones are the regions between zero and three standard deviations from the mean broken into one standard deviation increments.

Our system implements the nine most common WECO rules.

Rule 1: A single point outside of the control limits

In this sample chart a single point is below the lower control limit and is marked as a rule violation with a red diamond marker.

Rule 2: Run above or below the centerline

The above figure has seven consecutive points above the centerline. The points marked with red diamond markers represent a violation of WECO Rule #2. The number of points required to trigger a failure is configurable.

Rule 3: Trend

WECO Rule 3 looks for either an upward or downward trend.

The above chart shows an steadily increasing trend marked with red diamond markers. This trend represents a violation of rule 3. The number of points required to trigger a failure is configurable.

Rule 4 – Cyclic pattern

A consecutive pattern of up and down points as illustrated above causes a violation of Rule 4. Again, the number of consecutive points that trigger the violation is configurable.

Rule 5 – Two out of three points in Zone A or beyond

Recall from above that Zone A is the region between two and three standard deviations from the centerline.

The two points marked with red diamonds are both in Zone A. Since the rule looks for two out of three consecutive points in Zone A or above, the condition marked on the chart represents a violation of Rule 5.

Rule 6 – Four out of five points in Zone B or beyond

Recall that Zone B is the region between one and two standard deviations from the centerline.

Since there are four out of five points in Zone B, there is a violation of Rule 6.

Rule 7 – Fifteen consecutive points in Zone C

Recall that Zone C is the region between zero and one standard deviation from the centerline.

The points marked with red diamonds represent a violation of Rule 7 because they all fall within Zone C and the number of consecutive points within Zone C is fifteen or greater. The number of points that are required to trigger a violation is configurable.

Rule 8 – Eight consecutive points none of which fall within Zone C

Here we see nine consecutive points none of which fall in Zone C. This is an example of a violation of Rule 8. The number of points required to trigger a violation is configurable.

Rule 9 – One point outside of the specifications

Some implementations of the WECO rules include this rule which is not a statistical rule but a rule based on process specifications.

Here we have two violations of Rule 9. One violation is due to a point above the upper specification limit while the other is due to a point below the lower specification limit.

Additional Western Electric Rule Notes

Since the WECO rules are statistical in nature, a violation could occur when there is no change in the process. However, a violation may not occur when there is a change in the process. There is a delicate balance between these two types of errors and it is directly related to the number of rules that you choose to use on your charts.

In general, the more rules you use, the more likely you are to find a false alarm. On the other hand, the fewer rules you use, the more likely you are to miss a change in the process when one has occurred. Thus, as with all statistical analysis, you must determine your objectives and use a common sense approach to implementing the WECO rules on your charts.