Sample Size for Attribute Binary Data

optomist1

A Sea of Statistics
Trusted Information Resource
#1
Hi To All Covers,

I seem to be experiencing a brain cramp of sorts.....I have conducted eight separate observational studies of a process that is measured in a binary manner; a go or no-go outcome.

Using minitab, I have constructed a p - bar chart; sample sizes have varied from 35 to 121 pieces or units. The p-bar over the eight separate observational studies is 8%.

Question: What sample size do I need to "ensure" that all subsequent observational studies & sample sizes are correct; will be large enough to detect a defect if one is present.

Regards,
optomist1
 
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Bev D

Heretical Statistician
Staff member
Super Moderator
#2
Re: Sample Size Attribute Binary Data

Will You be doing SPC? Monitoring the process to see if the defect rate changes
Or will you be inspecting lots to see if they meet some acceptable defect level?
 

optomist1

A Sea of Statistics
Trusted Information Resource
#3
Re: Sample Size Attribute Binary Data

Hi Bev,

Strictly SPC, monitoring the process.....

Regards,
Marty
 

optomist1

A Sea of Statistics
Trusted Information Resource
#4
Re: Sample Size Attribute Binary Data

Hi Bev,

A little more detail, I conducted 11 observational studies, the most recent p-bar is 8%.

Thank you for your assistance.

Regards,
Marty
 

Steve Prevette

Deming Disciple
Staff member
Super Moderator
#5
Hi To All Covers,

I seem to be experiencing a brain cramp of sorts.....I have conducted eight separate observational studies of a process that is measured in a binary manner; a go or no-go outcome.

Using minitab, I have constructed a p - bar chart; sample sizes have varied from 35 to 121 pieces or units. The p-bar over the eight separate observational studies is 8%.

Question: What sample size do I need to "ensure" that all subsequent observational studies & sample sizes are correct; will be large enough to detect a defect if one is present.

Regards,
optomist1
Going back to the original question - which appears to be - if the average failure rate is 8%, how many do I need to sample to "ensure" that I detect that there is a failure.

If I replace "ensure" with 95% confidence (a standard threshold out there), then my sample size should be 36

That is - =BINOMDIST(0,36,0.08,FALSE) = 0.049

If the failure rate is 8% and I sample 36 units, there is less than 5% chance I will not find a defect. THIS IS ASSUMING a number of things - such as each unit has an identical probability of failure, and all units are independent from each other.

So, when I do the p-chart, I'd suggest making sure each subset is at least 36 in size.
 

Statistical Steven

Statistician
Staff member
Super Moderator
#6
Re: Sample Size Attribute Binary Data

Hi Bev,

A little more detail, I conducted 11 observational studies, the most recent p-bar is 8%.

Thank you for your assistance.

Regards,
Marty
Marty

If you are doing SPC, then the sample size is a function of more than just ensuring that all future observational studies are correct. Since you quote an 8% average, the upper and lower limits can be calculated for any sample size. How much "sensitivity" to you require to a shift from 8%?
 

Statistical Steven

Statistician
Staff member
Super Moderator
#7
Going back to the original question - which appears to be - if the average failure rate is 8%, how many do I need to sample to "ensure" that I detect that there is a failure.

If I replace "ensure" with 95% confidence (a standard threshold out there), then my sample size should be 36

That is - =BINOMDIST(0,36,0.08,FALSE) = 0.049

If the failure rate is 8% and I sample 36 units, there is less than 5% chance I will not find a defect. THIS IS ASSUMING a number of things - such as each unit has an identical probability of failure, and all units are independent from each other.

So, when I do the p-chart, I'd suggest making sure each subset is at least 36 in size.
Steve, a sample size of 36 still gives a lower control limit that is less than 0. How do we reconcile that a sample of 36 with no defects is within the common cause variability of a process with 8% defective rate?
 

Steve Prevette

Deming Disciple
Staff member
Super Moderator
#8
Steve, a sample size of 36 still gives a lower control limit that is less than 0. How do we reconcile that a sample of 36 with no defects is within the common cause variability of a process with 8% defective rate?
I'm not surprised that the LCL is less than zero, since I only calculated to 95%. One could do the same calculation to ensure that the LCL is greater than zero.

But I will defend 36 as a good choice in this example. I can use the 2 out of 3 less than two standard deviations below average (and 4 of 5 one below, and 7 below) to detect decreasing trends. I assume though, that detection of increasing trends is more important. The 36 gives me a more frequent update, but would avoid the issue of just the change of a single defect moving you from 0% to 8% to 16% if I use a 12 point sample size.
 

optomist1

A Sea of Statistics
Trusted Information Resource
#9
Going back to the original question - which appears to be - if the average failure rate is 8%, how many do I need to sample to "ensure" that I detect that there is a failure.

If I replace "ensure" with 95% confidence (a standard threshold out there), then my sample size should be 36

That is - =BINOMDIST(0,36,0.08,FALSE) = 0.049

If the failure rate is 8% and I sample 36 units, there is less than 5% chance I will not find a defect. THIS IS ASSUMING a number of things - such as each unit has an identical probability of failure, and all units are independent from each other.

So, when I do the p-chart, I'd suggest making sure each subset is at least 36 in size.
Thank you both for your thoughtful repsonses, yes ensure is 95%, accepting a 5% producers risk.

The process in question, involves the insertion of one object into another...purely a mechanical process. Yet there are FEW key input variables that can be controlled to any degree, the one key input that does influence the outcome is not covered by a specification, not controlled, nor has it been quantified to a reasonable degree.

Each unit is independent as is the probability of a defect (if I correctly understand your assumption). Of course this "36" merely addresses the most recent observational study or the 8% p-bar. For the other seven studies the p -bar ranged from 19% - 10%, and sample sizes from 38 to 122 units. I do not like attribute or pure binary data, too little insight and too few options for analysis.

Regards,
Marty
 

Bev D

Heretical Statistician
Staff member
Super Moderator
#10
True categorical data has very little informative value...but if it's all you've got at least a p-chart will tell you if and when you get better or worse.

As for sample size for different defect rates: the smaller the defect rate the larger the sample size will/should be. One approach is to simply use the smaller defect rate to set your sample size. This will be the most 'conservative' approach. The other approach is to calculate the grand average defect rate [sum(#defects)/sum(sample sizes)] and use that defect rate to determine the sample size. (the formula Steve used for the Exact Binomial is (without using EXCEL functions): n =LN(1-Confidence level)/LN(1-p_bar). Where LN is the natural log and the confidence interval is 95% or 99% (expressed as a proportion; .95 or .99)

You might also want to simply plot your 'observational studies' in time series on a p chart (using variable control limits based on the varying sample sizes) to assess if the process was stable through your studies or if it was changing.
 
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