Authors: Ahmed Fathy, Ryan Shillington, Mariane Thomé Pires

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Table of Contents

Purpose

Users of QbDVision can upload data for different record types in QbDVision and analyze the results. This document is intended to help users of QbDVision understand the mathematical formulas behind the Process Analytics reports displayed.

Scope

It is assumed that the reader has a basic understanding of mathematical concepts including charting, graphing and can understand mathematical formulas.

References

1. FDA Ora Laboratory Manual Volume III Section 4 - Basic Statistics and Data Presentation

Definitions

Above and beyond the normal definitions in QbDVision documentation, the following terms are unique to this part of the software:

• Measurement - A measurement is a single number/value. Measurements are also often known as samples. 

• Subgroup - A subgroup is a group of 1 or more measurements. It’s usually a single batch or lot.

• Requirement - A requirement is a variable or attribute that subgroups contain measurements on. The requirement contains important information such as the expected results of measurements, the units of measurement, a description of what the data represents, etc.

• Chart - A chart is a visual representation of multiple measurements, often that span multiple subgroups.

• Metrics - A metric is a single number that usually describes an important relationship between specifications and real world data.

Rounding and Significant Digits

QbDVision follows the FDA guidelines for rounding and significant digits as specified in section 3 of FDA Ora Laboratory Manual Volume III Section 4.

Standard Formulas

Mean

When the mean () of a set of measurements () is referred to, this is the formula used to calculate it:

Standard deviation

For computing the standard deviation (), there is occasionally a special formula for approximating it in the notes of that section. If no other formula is provided, then this is the formula used:

where:

• is the total number of measurements across all subgroups.

• is a single measurement.

• is the mean of all of the measurements across all subgroups.

Notes

Since a large amount of data is required to compute a proper standard deviation, the symbol for standard deviation in this documentation is often to represent that it is an approximation of the true .

Charts

The following sections will outline each possible chart and describe the equations used to compute each.

Capability Histogram

Example

Here is an example of what this chart looks like:

Open

Usage

This chart shows the frequency of all measurements across all chosen subgroups.

Formulas and Constants

Bars

Each vertical bar represents a uniform range of values (measurements). The height of each bar is given by the number of measurements inside of that range.

Center Line

The center line is a straight line controlled by the “Target” value set on the requirement record in the Acceptance Criteria section.

Limit Lines

LSL (Lower Specification Limit) and USL (Upper Specification Limit)

These lines are straight lines controlled by the value entered by the user on the requirement in the Acceptance Criteria section.

Notes

None.

Capability Plot

Example

Here is an example of what this chart looks like:

Open

Usage

This chart describes the range of the data to 3 standard deviations on either side of the mean and compares it to the user entered LSL, Target and USL. This chart tells the user whether the data collected is conforming to the specifications they expected.

Formulas and Constants

All three lines always only contain 3 points. 

Overall

The overall capability plot (“Overall”) is calculated using:

where:

• is the mean of all measurements in all subgroups

• is the constant 3

• is the standard deviation of all measurements in all subgroups.

Within

The within capability plot (“Within”) is calculated using:

where:

• is the mean of all measurements in all subgroups

• is the constant 3

• is defined as:

  • where:

    • is the number of subgroups

    • is the number of measurements in subgroup

    • is measurement in subgroup

    • is the mean of all measurements in all subgroups

    • is the number of degrees of freedom defined as:

    • is a constant looked up from the table in Appendix A

Specs

The specification capability plot (“Specs”) is calculated using:

[LSL, Target, USL]

where:

• LSL is the Lower Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

• Target is entered by the user on the requirement in the Acceptance Criteria section.

• USL is the Upper Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

Notes

None.

Cumulative %Defective

Example

Here is an example of what this chart looks like:

Open

Usage

This chart is only for records with “measure” set to “Defects (Pass/Fail)” (i.e. every measurement is either “pass” or “fail”). This chart shows the running total of all measurements as subgroups are recorded. 

Formulas and Constants

Plotted Points

For each subgroup , the cumulative percentage (%) defective is calculated by:

where:

• is the number of the subgroup with subgroups before it

•  is the total number of measurements (regardless of pass or fail) in subgroup

• is 1 if measurement in subgroup is “fail” or 0 otherwise.

Center Line

The center line is a straight line which is the mean of failing measurements divided by the total number of measurements, multiplied by 100:

where:

• is the total number of subgroups

•  is the total number of measurements (regardless of pass or fail) in subgroup

• is 1 if measurement in subgroup is “fail” or 0 otherwise.

Limit Lines

There are no limit lines on this chart.

Notes

None.

Distribution of %Defective

Example

Here is an example of what this chart looks like:

Open

Usage

This chart is only for records with “measure” set to “Defects (Pass/Fail)” (i.e. every measurement is either “pass” or “fail”). This chart shows the frequency of the percent of failed measurements across all chosen subgroups.

Formulas and Constants

Bars

For each subgroup , the percentage (%) defective is calculated by:

where:

• is the total number of measurements (regardless of pass or fail) in subgroup

• is 1 if measurement is “fail” or 0 otherwise.

Given subgroups, this leads to having values. Each vertical bar represents a range of values. The height of each bar is given by the number of % defective () values inside of that range.

Center Line

The center line is a straight line controlled by the “Target” value set on the requirement record in the Acceptance Criteria section.

Limit Lines

USL (Upper Specification Limit)

These lines are straight lines controlled by the value entered by the user on the requirement in the Acceptance Criteria section.

Notes

There is never an LSL (Lower Specification Unit) displayed on this chart since it would always be zero. Ideally nothing is defective.

I Chart

Example

Here is an example of what this chart looks like:

Open

Usage

This chart shows all individual (I) measurements charted in sequential order. It is used to visualize the raw data.

Formulas and Constants

Plotted Points

Every measurement in every subgroup is plotted separately in this graph. The X axis is a running count of the total number of measurements across all subgroups.

Center Line

The center line is the mean of all measurements.

Control Limit Lines

LCL (Lower Control Limit)

The lower control limit (LCL) is calculated using:

where:

• is the mean of all measurements across all subgroups.

• is the constant 3.

• is the moving standard deviation (defined in the MR Chart notes).

UCL (Upper Control Limit)

where:

• is the mean of all measurements across all subgroups.

• is the constant 3.

• is the moving standard deviation (defined in the MR Chart notes).

Notes

• None

Last Subgroups

Example

Here is an example of what this chart looks like:

Open

Usage

This chart shows the raw measurements in each subgroup. It is framed by the USL and LSL which are entered by the user on the requirement in the Acceptance Criteria section.

Formulas and Constants

Plotted Points

Each point is a measurement. They are vertically arranged for each chosen subgroup.

Center Line

The center line is the median of all of the values. It is calculated the same way as the center line for the XBar chart described later.

Limit Lines

LSL (Lower Specification Limit) and USL (Upper Specification Limit)

These lines are straight lines controlled by the value entered by the user on the requirement in the Acceptance Criteria section

Notes

None.

MR Chart

Example

Here is an example of what this chart looks like:

Open

Usage

This chart shows the volatility of how the values are moving between measurements.

Formulas and Constants

Plotted Points

Every point is plotted as:

where:

• is the th measurement of all measurements in all subgroups.

Center Line

The center line is the mean of all measurements.

Control Limit Lines

LCL (Lower Control Limit)

The lower control limit (LCL) is calculated using:

where:

• is a constant looked up from the table in Appendix A

• is the constant 3.

• is the constant 2

• is an estimate of the standard deviation using the “Average Moving Range” method described in the notes section below.

• is a constant looked up from the table in Appendix A
  
  

UCL (Upper Control Limit)

where:

• is a constant looked up from the table in Appendix A

• is the constant 3.

• is the constant 2

• is an estimate of the standard deviation using the “Average Moving Range” method described in the notes section below.

• is a constant looked up from the table in Appendix A

Notes

Estimating Sigma (Average Moving Range method) ()

The moving standard deviation, is calculated using:

where:

• is the total number of measurements across all subgroups.

• is the index of the measurement in the list of all measurements from all subgroups.

• is the th measurement

• is the number of measurements in the moving range, which is a constant: 2

• is a constant looked up from the table in Appendix A

Normal Probability Plot

Example

Here is an example of what this chart looks like:

Open

Usage

This chart shows whether the data conforms to a normal distribution. This is often known as an “Anderson-Darling Test”. If the data strays from the red normality line in the center, this indicates that the data might possibly be out of control (or at least following a different distribution).

Formulas and Constants

Plotted Points

All of the measurements from all of the selected subgroups are sorted and represented as . A given point is plotted using its value on the x axis and its corresponding z value ()  on the y axis:

where:

• is the index of the measurement in the sorted list.

• is the total number of measurements.

• is an inverse Cumulative standard normal Distribution Function (iCDF). The definition of iCDF can be found here. 

Center Line

The red center line is a regression line based on the plotted points using the least sum of squares.

Control Limit Lines

None.

Anderson-Darling value (AD)

At the top of the chart is both the AD and P value. Given all measurements from all subgroups (), the AD value is computed as:

where:

• is the total number of measurements across all subgroups.

• is the index of the measurement in the list of all measurements from all subgroups.

• is the Cumulative Distribution Function (CDF) which is: 

  • where is the standard deviation of all measurements described in the Standard Formulas section under Standard Deviation

P value (P)

At the top of the chart is both the AD and P value. The first step is to compute which is the Anderson Darling value adjusted for small sample sizes:

The P value is then calculated:

• If :

• If :

• If :

• If :

where:

• is the constant representing the base of the natural logarithm.

Notes

None.

P Chart

Example

Here is an example of what this chart looks like:

Open

Usage

This chart shows the raw data of the proportion of measurements that failed for each subgroup.

Formulas and Constants

Plotted Points

For each subgroup , the proportion defective is calculated by:

where:

• is the total number of measurements (regardless of pass or fail) in subgroup

• is 1 if measurement is “fail” or 0 otherwise.

Center Line

The center line is a straight line which is the mean of failing measurements divided by the total number of measurements, multiplied by 100:

where:

• is the total number of subgroups

• is the total number of measurements (regardless of pass or fail) in subgroup

• is 1 if measurement in subgroup is “fail” or 0 otherwise.

Control Limit Lines

LCL (Lower Control Limit)

The lower control limit (LCL) is calculated as:

where:

• is the average of all proportions (see the center line above).

• is the constant 3

• is the average number of measurements per subgroup, defined as
  

  • where:

    • is the total number of subgroups

    •  is the total number of measurements (regardless of pass or fail) in subgroup

UCL (Upper Control Limit)

The upper control limit (UCL) is calculated as:

where:

• is the average of all proportions (see the center line above).

• is the constant 3

• is the average number of measurements per subgroup, defined as
  

  • where:

    • is the total number of subgroups

    •  is the total number of measurements (regardless of pass or fail) in subgroup

Notes

None.

R Chart

Example

Here is an example of what this chart looks like:

Open

Usage

This chart plots the range of each subgroup (i.e. max - min) to show how varied the measurements are in each subgroup. This is compared to the Upper Control Limit (UCL), (the average of all of the ranges), and the Lower Control Limit (LCL).

Formulas and Constants

Plotted Points

The point plotted for each subgroup is:

where:

• is the highest/largest measurement in subgroup

• is the lowest/smallest measurement in subgroup

Center Line

The center line is computed as:

where:

• is the number of subgroups

• is the number of measurements in subgroup

• is a constant looked up from the table in Appendix A

• is the estimated standard deviation (defined in the notes section below)

Control Limit Lines

LCL (Lower Control Limit)

LCL is defined as:

or zero, if LCL is negative, where:

• is the number of subgroups

• is the number of measurements in subgroup

• is a constant looked up from the table in Appendix A

• is the estimated standard deviation (defined in the notes section below)

• is the constant 3

• is a constant looked up from the table in Appendix A

UCL (Upper Control Limit)

UCL is defined as:

where:

• is the number of subgroups

• is the number of measurements in subgroup

• is a constant looked up from the table in Appendix A

• is the estimated standard deviation (defined in the notes section below)

• is the constant 3

• is a constant looked up from the table in Appendix A

Notes

Estimating Sigma ( Method)

For the purpose of estimating the standard deviation, , the following formula is used:

where:

• is the number of subgroups

• is the number of measurements in subgroup

• is defined below

• is the difference between the maximum and minimum measurement in subgroup

• is a constant looked up from the table in Appendix A

Function

is defined as:

where:

• is the number of measurements in subgroup

• is a constant looked up from the table in Appendix A

• is a constant looked up from the table in Appendix A

Subgroups without measurements

In previous versions of QbDVision (circa 2019), a user could add a subgroup by entering the average and standard deviation manually (known as “subgroups without measurements”). For the computation of the center line, LCL and UCL, the number of measurements in a subgroup without measurements is estimated to be the average of the number of measurements in the chosen subgroups that contain measurements. If all subgroups are without measurements, then the number of measurements is assumed to be 10. 

Rate of Defectives

Example

Here is an example of what this chart looks like:

Open

Usage

This chart compares the percent (%) of defective measurements against the number of measurements for each chosen sample group. 

Formulas and Constants

Plotted Points

For each subgroup , the percentage (%) defective is calculated by:

where:

•  is the total number of measurements (regardless of pass or fail) in subgroup

• is 1 if measurement is “fail” or 0 otherwise.

• is the number of measurements in subgroup

Instead of plotting the subgroups on the X axis, the sample size (i.e.. number of measurements, ) is used instead.

Center Line

The center line is a straight line which is the mean of failing measurements divided by the total number of measurements, multiplied by 100:

where:

• is the total number of subgroups

•  is the total number of measurements (regardless of pass or fail) in subgroup

• is 1 if measurement in subgroup is “fail” or 0 otherwise.

Limit Lines

There are no limit lines on this chart.

Notes

None.

S Chart

Example

Here is an example of what this chart looks like:

Open

Usage

This chart plots the standard deviation of each subgroup. This is compared to the Upper Control Limit (UCL), (the average of all of the standard deviations), and the Lower Control Limit (LCL).

Formulas and Constants

Plotted Points

Each point is calculated for subgroup as follows:

where:

• is the number of measurements in the subgroup

• is a single measurement

• is the average of all of the measurements in the subgroup

Center Line

The centerline is calculated as:

where:

• is the number of subgroups

• is the number of measurements in subgroup

• is a constant looked up from the table in Appendix A

• is the estimated standard deviation (defined in the notes section below)

Control Limit Lines

LCL (Lower Control Limit)

LCL is defined as:

where:

• is the number of subgroups

• is the number of measurements in subgroup

• is a constant looked up from the table in Appendix A

• is the estimated standard deviation (defined in the notes section below)

• is the constant 3

UCL (Upper Control Limit)

UCL is defined as:

where:

• is the number of subgroups

• is the number of measurements in subgroup

• is a constant looked up from the table in Appendix A

• is the estimated standard deviation (defined in the notes section below)

• is the constant 3

Notes

Estimating Sigma ( Method)

For the purpose of estimating the standard deviation, , the following formula is used:

where:

• is the number of subgroups

• is the number of measurements in subgroup

• is defined below

• is the difference between the maximum and minimum measurement in subgroup

• is a constant looked up from the table in Appendix A

Function

is defined as:

where:

• is the number of measurements in subgroup

• is a constant looked up from the table in Appendix A

Subgroups without measurements

In previous versions of QbDVision (circa 2019), a user could add a subgroup by entering the average and standard deviation manually (known as “subgroups without measurements”). For the computation of the center line, LCL and UCL, the number of measurements in a subgroup without measurements is estimated to be the average of the number of measurements in the chosen subgroups that contain measurements. If all subgroups are without measurements, then the number of measurements is assumed to be 10. 

XBar Chart

Example

Here is an example of what this chart looks like:

Open

Usage

This chart describes the average of all of the measurements in each subgroup and plots them, along with the Upper Control Limits (UCL), average of all of the measurements in all of the subgroups () and the Lower Control Limits (LCL).

Formulas and Constants

Plotted Points

Each point is the mean of the measurements in subgroup :

where:

• is the number of measurements in subgroup

• is measurement in subgroup

Center Line

The center line is sum of all measurements in all subgroups divided by the total number of measurements:

where:

• is the number of measurements across all subgroups

• is measurement (across all measurements in all subgroups)

• is the number of measurements in subgroup

• is the total number of subgroups

Control Limit Lines

LCL (Lower Control Limit)

The line for the LCL is computed as follows:

where:

• is the center line constant described above (average of all measurements in all subgroups)

• is the constant 3

• is the standard deviation across all subgroups described in the Standard Formulas section under Standard Deviation

• is the number of measurements in a given subgroup

• is the number of subgroups

UCL (Upper Control Limit)

The line for the UCL is computed as follows:

where:

• is the center line constant described above (average of all measurements in all subgroups

• is the constant 3

• is the standard deviation across all subgroups

• is the number of measurements in a given subgroup

• is the number of subgroups

Notes

Subgroups without measurements

In previous versions of QbDVision (circa 2019), a user could add a subgroup by entering the average and standard deviation manually (known as “subgroups without measurements”). For the computation of the LCL and UCL, the number of measurements in a subgroup without measurements is estimated to be the average of the number of measurements in the chosen subgroups that contain measurements. If all subgroups are without measurements, then the number of measurements is assumed to be 10. 

Metrics

This section discusses the metrics, which are single numbers, as opposed to charts that show a range of numbers.

In general, all of the metrics that start with “C” (ex. Cp, Cpk, etc.) are analyzing the data in each subgroup separately. The metrics that start with “P” (ex. Pp, Ppk, etc.) are analyzing the data across the subgroups.

Examples

Metrics look like the following in QbDVision. Here is an example from the dashboard:

Open

Here is an example from the control charts:

Open

LSL

Usage

The Lower Specification Limit (LSL) is entered into the requirement form in QbDVision and is the lower bound for a well-functioning process.

Formulas

LSL is the Lower Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

Notes

None.

USL

Usage

The Upper Specification Limit (USL) is entered into the requirement form in QbDVision and is the upper bound for a well-functioning process.

Formulas

USL is the Upper Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

Notes

None.

Within Stdev (Estimating Sigma [ Method])

Usage

This metric describes the extent in which the data deviates from the mean within each subgroup.

Formulas

For the purpose of estimating the standard deviation, , the following formula is used:

where:

• is the number of subgroups

• is the number of measurements in subgroup

• is measurement in subgroup

• is the mean of all measurements over all subgroups

• is the degrees of freedom, defined as:
  

• is a constant looked up from the table in Appendix A

Notes

None.

Cp

Usage

This metric compares the expected limits (USL & LSL) against the limits suggested by the standard deviation of the measurements in the selected subgroups.

Formulas

Here is the formula for this metric:

where:

• USL is the Upper Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

• LSL is the Lower Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

• is the constant 3

• is the standard deviation within the subgroups defined here.

Notes

None.

Cpl

Usage

This metric compares the difference between the mean and the LSL against the limit suggested by the standard deviation of the measurements in the selected subgroups.  

Formulas

Here is the formula for this metric:

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where:

• is the mean of all measurements over all subgroups

• LSL is the Lower Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

• is the constant 3

• is the standard deviation within the subgroups defined here.

Notes

None.

Cpu

Usage

This metric compares the difference between the USL and the mean against the limit suggested by the standard deviation of the measurements in the selected subgroups.  

Formulas

Here is the formula for this metric:

where:

• USL is the Upper Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

• is the mean of all measurements over all subgroups

• is the constant 3

• is the standard deviation within the subgroups defined here.

Notes

None.

Cpk

Usage

This metric is the minimum between the Cpl (testing the LSL) and the Cpu (testing the USL).

Formulas

The Cpk is the minimum between the Cpl and the Cpu.

Notes

None.

Cpm

Usage

This metric compares the expected limits (USL & LSL) against the limits suggested by how far the measurements are off from the target. 

Formulas

Here is the formula for this metric:

where:

• USL is the Upper Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

• LSL is the Lower Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

• is the constant 3

• is the number of subgroups

• is the number of measurements in subgroup

• is measurement in subgroup

• is the Target entered by the user on the requirement in the Acceptance Criteria section.

Notes

None.

Overall Stdev (Estimating Sigma [ Method])

Usage

This metric describes the extent in which the data deviates from the mean over all the subgroups.

Formulas

For the purpose of estimating the standard deviation, , the formula is described in the Standard Formulas section under Standard Deviation but divided by the unbiasing constant, :

where:

• is the formula described in the Standard Formulas section under Standard Deviation

• is a constant looked up from the table in Appendix A

• is the total number of measurements across all subgroups

Notes

None.

Pp

Usage

This metric compares the expected limits (USL & LSL) against the limits suggested by the standard deviation of the measurements in the selected subgroups.

Formulas

Here is the formula for this metric:

where:

• USL is the Upper Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

• LSL is the Lower Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

• is the constant 3

• is the standard deviation over all subgroups defined here.

Notes

None.

Ppl

Usage

This metric compares the difference between the mean and the LSL against the limit suggested by the standard deviation of the measurements in the selected subgroups.  

Formulas

Here is the formula for this metric:

where:

• is the mean of all measurements over all subgroups

• LSL is the Lower Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

• is the constant 3

• is the standard deviation over all subgroups defined here.

Notes

None.

Ppu

Usage

This metric compares the difference between the USL and the mean against the limit suggested by the standard deviation of the measurements in the selected subgroups.  

Formulas

Here is the formula for this metric:

where:

• USL is the Upper Specification Limit entered by the user on the requirement in the Acceptance Criteria section.

• is the mean of all measurements over all subgroups

• is the constant 3

• is the standard deviation over all subgroups defined here.

Notes

None.

Ppk

Usage

This metric is the minimum between the Ppk (testing the LSL) and the Ppu (which tests the USL).

Formulas

The Ppk is the minimum between the Ppl and the Ppu.

Notes

None.

Appendix A - Mathematical Constants

Constants when n <= 25

Approximation when n > 25

is estimated using the following formula:

is estimated using the following formula:

is estimated using the following formula:

where:

• is the number of measurements or subgroups, depending on the chart

• is the gamma function.