# Statistics In QW5

### QW5 Help Files

This section is intended to provide a definition of terms used in QW 5 as well as information on the calculations and rules followed by QW 5 in developing the statistics that appear when a graph of a variable is displayed.

These calculations are based on having a fixed USL, LSL and Target.

Only entered values (non-null) are included in calculations in QW 5.

The Following changes with the release of Quality Window Version 5.0.0.783

 Currently Reports as: Replaced by: Cpk – Process Capability Index Ppk – Process Performance Index Cp – Process Capability Rate Pp – Process Performance Rate

*Cpk and Cp are still reported.

### Definitions of Statistics used in QW 5

Average
Represented by Avg
The mathematical average of all data points in a field. Also known as the Arithmetic Mean. where n = number of points in the field

Calc % above TGT
Represented by C%>TGT
Calculated percent above target.

Calc % above UCL
represented by C%>UCL
Calculated percent above upper control limit.

Calc % above USL represented by C%>USL
Calculated percent above upper specification limit.

Calc % above UWL represented by C%>UWL
Calculated percent above upper warning limit.

Calc % below LCL represented by C%<LCL
Calculated percent below lower control limit.

Calc % below LSL represented by C%<LSL
Calculated percent below upper specification limit.

Calc % below LWL represented by C%<LWL
Calculated percent below lower warning limit.

Calc % below TGT represented by C%<TGT
Calculated percent below target.

Calc % OSL represented by C%OSL
This statistic uses a table to infer the theoretical %OSL.  The table assumes normality, control and the right number of samples so that the Average and SD are representative of the true population.  Unless these assumptions hold true, the number will be inaccurate.  When you have calculated quantities and no observed quantities, first suspect the assumptions above have been violated … particularly with large sample sizes.  It is calculated as the value of the percentage (or parts per million) of a normal distribution curve that will fall outside the specification limits (points in the white band of the chart) if a large number of samples are taken.  This value is theoretical and is accurate only if the variable is in statistical control and the data is normally distributed.

To calculate the C%OSL you need :

• the average (Avg)
• the standard deviation (Std Dev – pop) of the population
• the Upper Specification Limit (USL)
• the Lower Specification Limit (LSL)
• a Z-Table to find the proportion out (Pz). Calc N above TGT represented by Cn>TGT
Calculated number above target.

Calc N above UCL represented by Cn>UCL
Calculated number above upper control limit.

Calc N above USL represented by Cn>USL
Calculated number above upper specification limit.

Calc N above UWL represented by Cn>UWL
Calculated number above upper warning limit.

Calc N below LCL represented by Cn<LCL
Calculated number below upper control limit.

Calc N below LSL represented by Cn<LSL
Calculated number below lower specification limit.

Calc N below LWL represented by Cn<LWL
Calculated number below lower warning limit.

Calc N below TGT represented by Cn<TGT
Calculated number below target.

Calc N OSL represented by CnOSL

Calc ppm above TGT represented by Cppm>TGT
Calculated parts per million above target.

Calc ppm above UCL represented by Cppm>UCL
Calculated parts per million above upper control limit.

Calc ppm above USL represented by Cppm>USL
Calculated parts per million above upper specification limit.

Calc ppm above UWL represented by Cppm>UWL
Calculated parts per million above upper warning limit.

Calc ppm below LCL represented by Cppm<LCL
Calculated parts per million below lower control limit.

Calc ppm below LSL represented by Cppm<LSL
Calculated parts per million below lower specification limit.

Calc ppm below LWL represented by Cppm<LWL
Calculated parts per million below lower warning limit.

Calc ppm below TGT represented by Cppm<TGT
Calculated parts per million below target.

Calc ppm OSL represented by CppmOSL
Calculated parts per million out of specification limits.

Calc ppm UCI represented by CppmUCI
The Calculated parts per million Upper Confidence Interval is an adjustment to the defect estimate to account for sampling error using the normal distribution theory.

The UCI ppm values are defect estimates that represent the worst case scenario of the actual defect level since it takes into account the potential error associated with low sample size. The larger the sample size, the lower the UCI ppm value. For example, if random samples were repeatedly taken from the same population and used to determine the defect estimate in ppm, the 95% UCI ppm would represent the ppm number that 95% of the samples estimates would fall below and 5% would fall above.

Calc Lower Warning Limit represented by CLWL
Calculated lower warning limit.

Calc Upper Warning Limit represented by CUWL
Calculated upper warning limit.

Clearance see Ppk.

Co-efficient Variation represented by %CV
The absolute value of (the standard deviation divided by the average) multiplied by 100. Control Limits
Upper Control Limit represented by UCL
Lower Control Limit represented by LCL

Control limits reflect target +/- 3S based on the historical performance of each line in question. Choosing a best time (absence of special causes) for basing the historical performance on is preferred as long as special causes can be eliminated or corrected.  Control limits are based on the best capability of each line are preferably symmetric.  +/- 3S can be used for any chart with 98-100% confidence without assumptions as to normality and control.  Control Limits in QW 5.0 can be fixed, blank or continuously calculated values.  These values are used in the program to set the values where color changes occur in the Charts.  In QW 5.0 the control limits are relative to the target and specification limits and establish a “warning track” inside the specification limits.

Correlation Coefficient
All Null values within variable X are filled with extrapolated values
All Null values within variable Y are filled with extrapolated values
All leading and trailing pairs of X, Y data are removed if either X or Y are null.
Determine R (Correlation Coefficient) If R > 1 then R = 1
If R < -1 then R = -1

Determine T (Significance Test) Determine if Significant.

If | T | > 1.96 highlight the value (95% significant)

Display Correlation Coefficient as Current Average represented by Avg-current

Current Standard Deviation represented by S-current

f-calc
Tests whether the Variation of two populations are different.  Last Value represented by Last Value

Lower Capability Limit represented by Cpl
(AVG – LSL) / ) / (3*SD MR)

Lower Control Limit represented by LCL

Lower Performance Limit represented by Ppl
(AVG – LSL) / (3*SD Pop)

Lower Process Capability (-3S) represented by Avg-3S

Lower Process Capability (-4S) represented by Avg-4S

Lower Spec. Limit represented by LSL

Lower Warning Limit represented by LWL

Maximum Value represented by Max
The highest value found in the data selected.

Median represented by M
The median is a measure of control tendency derived from taking the middlemost or most central point in a set of sorted points.  Half of the points will lie above the median, and the other half will lie below the median.  If the number of points is odd the middle point is the median.  If there is an even number of points the median is the average of the middle points.

Minimum Value represented by Min Value
The lowest value found in the data selected.

Moving Range Average represented by The Moving Range Average is based on succesive differences between individual values. Moving Range Upper Control Limit represented by MRUCL
Is the Moving Range Upper Control Limit Number of Points represented by N
The number of non-null values found in the data selected.

Obs % above TGT represented by O%>TGT
Observed percent above target.

Obs % above UCL represented by O%>UCL
Observed percent above upper control limit.

Obs % above USL represented by O%>USL
Observed percent above upper specification limit.

Obs % above UWL represented by O%>UWL
Observed percent above upper warning limit.

Obs % below LCL represented by O%<LCL

Obs % below LSL represented by O%<LSL

Obs % below LWL represented by O%<LWL

Obs % below TGT represented by O%<TGT

Obs % on TGT represented by O%=TGT

Obs % OSL represented by  O%OSL

Obs N above TGT represented by On>TGT

Obs N above UCL represented by On>UCL

Obs N above USL represented by On>USL

Obs N above UWL represented by On>UWL

Obs N below LCL represented by On<LCL

Obs N below LSL represented by On<LSL
Obs N below LWL represented by On<LWL

Obs N below TGT represented by On<TGT

Obs N on TGT represented by On=TGT

Obs N OSL represented by OnOSL

Obs ppm above TGT represented by Oppm>TGT

Obs ppm above UCL represented by Oppm>UCL

Obs ppm above USL represented by Oppm>USL

Obs ppm above UWL represented by Oppm>UWL

Obs ppm below LCL represented by Oppm<LCL

Obs ppm below LSL represented by Oppm<LSL

Obs ppm below LWL represented by Oppm<LWL

Obs ppm below TGT represented by Oppm<TGT

Obs ppm on TGT represented by Oppm=TGT

Obs ppm OSL represented by OppmOSL

Obs ppm UCI represented by OppmUCI
The Observed parts per million Upper Confidence Interval is an adjustment to the defect estimate to account for sampling error using the binomial distribution theory.

The UCI ppm values are defect estimates that represent the worst case scenario of the actual defect level since it takes into account the potential error associated with low sample size. The larger the sample size, the lower the UCI ppm value. For example, if random samples were repeatedly taken from the same population and used to determine the defect estimate in ppm, the 95% UCI ppm would represent the ppm number that 95% of the samples estimates would fall below and 5% would fall above.
Point
The value of each cell in a data collection.

Process Capability Index represented by Cpk
Requires two separate calculations and then you select the smaller of the two results.

These equations describe the clearance between the process distribution curve and the specification limits in terms of standard deviation.  The concern would be on whichever end of the distribution curve is closest to the specification limit (could be equal if the process is perfectly centered within the specification limits). A value of 1.33 or greater is considered good.  A value of 1.33 would mean that the process average is four deviations away from the nearest specification limit.  Compare Cr with Cpk.  The process variation is in the numerator of Cr, and the denominator of Cpk (also Cp).   Low Cr and high Cpk numbers are good. Process Capability Rate represented by Cp
The inverse of Capability Ratio. Process Performance Index represented by Ppk
Requires two separate calculations and then you select the smaller of the two results.

These equations describe the clearance between the process distribution curve and the specification limits in terms of standard deviation.  The concern would be on whichever end of the distribution curve is closest to the specification limit (could be equal if the process is perfectly centered within the specification limits). A value of 1.33 or greater is considered good.  A value of 1.33 would mean that the process average is four deviations away from the nearest specification limit.  Compare Cr with Ppk. The process variation is in the numerator of Cr, and the denominator of Ppk (also Pp). Low Cr and high Ppk numbers are good. Process Performance Rate represented by Pp
The  inverse of Capability Ratio. Range AVG represented by R-AVG

Range LWL represented by R-LWL

Range UCL represented by R-UCL

Range UWL represented by R-UWL

Rule Violation represented by Rule

SD Low
SD Low is defined as the lowest standard deviation of a consecutive group of m points with a population of n points.

m is determined by multiplying n by 20%

• if the result is < 10 then m = 10
• if the result is > 100 then m = 100

Sigma represented by Sigma

Specification Limits
Upper Spec. Limit represented by USL
Lower Spec. Limit represented by LSL

Fixed specification limits must be outside fixed control limits and the target.  They reflect a shutdown scenario or a high probability that a real process or product problem exists that must be corrected immediately.  They are defined via product and process research.  All lines have the process capability to run within specifications with a margin that allows for control charts to provide their intended benefit of early warning.  A Cr of < .75 and Tz of < .5 is preferred to allow early warning. avoid overadjustment and reactive operation.  Specifications should not be calculated based on lead line capability.  This will be a dilution of the quality system in terms of real meaning to the business.

Std Dev – mr represented by S-mr
The standard deviation of the Moving Range Average. Std Dev – pop represented by S-pop
The measurement of variability around the average

Take each individual data point and calculate the deviation of that point from the average, square the deviation (multiply it by itself), add up all the results, divide the total by one less than the number of points, then take the square root of the result. Sum represented by Sum

t-calc
Tests whether the Average of two populations are different. Target represented by TGT
The target should reflect the optimum place to average the process relative to product and process research.  The process should have the capability to average a target.  If not, the 3, 5 and 7 point rules cannot be used without creating overadjustment.  Each lines’ capability may require a unique target in order to use 3, 5 and 7 point control rules for certain variables.  The target can be fixed, blank or continuously calculated.

Target Deviation represented by T-Dev

In the statistics of QW 5 it is the separation between the process average  and the Target value of the variable.  The value shown is in units of measure of the variable and can be positive or negative, depending on whether the process average is above or below target.

T-Dev = Avg – TGT
where TGT = the target value for the variable

Targeting represented by Tz
In QW 5 this is determined by dividing the Target Deviation by the Standard Deviation

This produces a number that measures how well the process is centered on target in standard deviations.  A good process should be less than 0.5 away from target (can be plus or minus).  A good Target Z does not necessarily mean the process is always in specification.

• If Cr .5 then Tz is calculated depending on the type of spec. limits and calculated Average.
• If TGT and both spec. limits are fixed the Avg is compared against Tgt +/- 4.2% of the spec. range.
• If TGT and only one spec. limits are fixed the Avg is compared against Tgt +/- 8.4% of Tgt to spec. range.
• If AVG is within these limits then Tz = 0.
• If AVG is greater than these limits then Tz = +1.
• If AVG is less than these limits then Tz = -1. Upper Capability Limit represented by Cpu
(USL – AVG) /) / (3*SD MR)

Upper Performance Limit represented by Ppu
(USL – AVG) /) / (>3*SD Pop)

Upper Process Capability (+3S) represented by Avg +3S

Upper Process Capability (+4S) represented by Avg +4S

Upper Spec Limit represented by USL

Upper Warning Limit represented by UWL
Variation represented by Cr

A term that describes the size of the process variability relative to the size of the specification range.  It does not consider where the process is centered so a good value for capability ratio does not mean that all the values are within specification.

When both specifications are fixed Quality Window calculates CR as 