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## Probability of Reject Rate

Hi all

I have what I consider to be a somewhat complicated question, at least to me it seems that way. I want to know if it is possible to use the population standard deviation and mean of Xi's to predict the reject rate of a packaged group of Xi's.

So for context, I work for a food manufacturer that makes candy bars. I have access to the population data of any production run, which will last anywhere from 1 - 6 days or roughly 4.8 million bars. There two groups of packaged  Xi's, a 4 count and a 72 count.

Here are the Target units of conversion and lower spec limits.

1 Bar = 40 Grams (LSL = 36.4)

1 4 count carton = 160 Grams (LSL = 149.2)

1 12 count carton = 480 Grams (LSL = 460)

An example observed population Mean and Standard Deviation are 40.5 and 1.96 respectively.

The reason I want to determine this information is we use X-Ray technology to blow off lightweight Xi's before they go into a carton. However, i've recently seen a trend where the X-Ray will fall out of calibration and add an additional 3.5 grams of weight to the reading. This in turn takes bars that would otherwise be considered lightweight and allows them to pass and make their way into a carton. From there if too many light bars make it into a carton,the mean weight will drop below the LSL and blow off causing carton waste. This is significant because 1 carton is more expensive than even 2 or 3 wrappers.

Do I have all the proper data to acheive my goal or am I missing something? If not, could someone help me figure this out? Thanks!

1 ACCEPTED SOLUTION

Accepted Solutions

## Re: Probability of Reject Rate

Taking your figures at face value (and assuming I didn't make a mistake) I made some random data that I think might help you. Generally such simulations can be a nice way to sidestep some theory. To see the result, do 'File > New > New Script' and copy and paste the code below into the resulting window. Then do 'Edit > Run Script' (may take a second or so), and inspect the results.

``````NamesDefaultToHere(1);

n12 = 10000;	// Number of boxes of 12 bars made
n = 12*n12;		// Number of bars made

// Generate data assuming an in-control process
dt = New Table( "Candy Bar Production",
New Column( "Weight",
Numeric,
"Continuous",
Format( "Best", 12 ),
Formula( Random Normal( 40.5, 1.96 ) ),
Set Property( "Spec Limits", {LSL( 36.4 ), Target( 40 ), Show Limits( 0 )} )
),
New Column( "4 Pack",
Numeric,
"Nominal",
Format( "Best", 12 ),
Formula( Modulo( Row(), Floor(n/4) ) )
),
New Column( "12 Pack",
Numeric,
"Nominal",
Format( "Best", 12 ),
Formula( Modulo( Row(), Floor(n/12) ) )
)
);

// Make data for the 4 pack weights
dt4 = dt << Summary(
Group( :Name( "4 Pack" ) ),
Sum( :Weight ),
statistics column name format( "column" )
);
Column(dt4, "Weight") << Set Property( "Spec Limits", {LSL( 149.2 ), Target( 160 ), Show Limits( 0 )} );

// Make data for the 12 pack weights
dt12 = dt << Summary(
Group( :Name( "12 Pack" ) ),
Sum( :Weight ),
statistics column name format( "column" )
);
Column(dt12, "Weight") << Set Property( "Spec Limits", {LSL( 460 ), Target( 480 ), Show Limits( 0 )} );

// Use Control Chart Builder
dt << Control Chart Builder(Variables( Y( :Weight ) ));
dt4 << Control Chart Builder(Variables( Y( :Weight ) ));
dt12 << Control Chart Builder(Variables( Y( :Weight ) ));
``````

How useful this is will depend on the details of your situation. I'm very cautious when I see the word 'distribution' in the context of Statistical Process Control: As noted in the code, this data was generated assuming a perfectly 'in control' process (independent, random samples from a fixed Gaussian distribution), and if that was really true you wouldn't need SPC.

3 REPLIES 3

## Re: Probability of Reject Rate

Taking your figures at face value (and assuming I didn't make a mistake) I made some random data that I think might help you. Generally such simulations can be a nice way to sidestep some theory. To see the result, do 'File > New > New Script' and copy and paste the code below into the resulting window. Then do 'Edit > Run Script' (may take a second or so), and inspect the results.

``````NamesDefaultToHere(1);

n12 = 10000;	// Number of boxes of 12 bars made
n = 12*n12;		// Number of bars made

// Generate data assuming an in-control process
dt = New Table( "Candy Bar Production",
New Column( "Weight",
Numeric,
"Continuous",
Format( "Best", 12 ),
Formula( Random Normal( 40.5, 1.96 ) ),
Set Property( "Spec Limits", {LSL( 36.4 ), Target( 40 ), Show Limits( 0 )} )
),
New Column( "4 Pack",
Numeric,
"Nominal",
Format( "Best", 12 ),
Formula( Modulo( Row(), Floor(n/4) ) )
),
New Column( "12 Pack",
Numeric,
"Nominal",
Format( "Best", 12 ),
Formula( Modulo( Row(), Floor(n/12) ) )
)
);

// Make data for the 4 pack weights
dt4 = dt << Summary(
Group( :Name( "4 Pack" ) ),
Sum( :Weight ),
statistics column name format( "column" )
);
Column(dt4, "Weight") << Set Property( "Spec Limits", {LSL( 149.2 ), Target( 160 ), Show Limits( 0 )} );

// Make data for the 12 pack weights
dt12 = dt << Summary(
Group( :Name( "12 Pack" ) ),
Sum( :Weight ),
statistics column name format( "column" )
);
Column(dt12, "Weight") << Set Property( "Spec Limits", {LSL( 460 ), Target( 480 ), Show Limits( 0 )} );

// Use Control Chart Builder
dt << Control Chart Builder(Variables( Y( :Weight ) ));
dt4 << Control Chart Builder(Variables( Y( :Weight ) ));
dt12 << Control Chart Builder(Variables( Y( :Weight ) ));
``````

How useful this is will depend on the details of your situation. I'm very cautious when I see the word 'distribution' in the context of Statistical Process Control: As noted in the code, this data was generated assuming a perfectly 'in control' process (independent, random samples from a fixed Gaussian distribution), and if that was really true you wouldn't need SPC.

Highlighted

## Re: Probability of Reject Rate

Does your formula for creating the column sub groups only work for random data? Every time I try to make a colum in such manner so I can summarize it using that column I get error messages.

Highlighted

## Re: Probability of Reject Rate

The formulas for the columns '4 Pack' and '12 Pack' don't depemd on the rest of the table, random entries or otherwise. But they were also a 'quick and dirty' way to get what I wanted, and there are certainly more robust ways to define such 'grouping columns'.

This seems a little tangential to the original question, but if you can give further details of the code you are using and the error you see, I'm sure someone can help.

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