Here's the situation:

Bag of 3100 parts. There are 1000 pieces that are bad (32%) .Well mixed by many hands.
I pulled a sample of 150 pieces and measured the problem dimension.
Of the 150, I got 33 bad (22%).

Does this make sense? A 10% difference?

Here's the situation:

Bag of 3100 parts. There are 1000 pieces that are bad (32%) .Well mixed by many hands.
I pulled a sample of 150 pieces and measured the problem dimension.
Of the 150, I got 33 bad (22%).

Does this make sense? A 10% difference?

Yes, it makes sense for a single trial. You shouldn't expect that a single sample should have the same percent defective as the population. In this case, sampling inspection has done its job, and alerted you to the fact that there is substantial number of defectives in the lot.

Here's the situation:

Bag of 3100 parts. There are 1000 pieces that are bad (32%) .Well mixed by many hands.
I pulled a sample of 150 pieces and measured the problem dimension.
Of the 150, I got 33 bad (22%).

Does this make sense? A 10% difference?

Let's see, there are a number of ways to do this. ASSUMING that the likelihood of retrieval of any given bead is the same (and note - that assumption is not always met when it comes to mechanical sampling), then we could go hypergeometric, binomial, or normal approximation to the binomial.

If we go hypergeometric, I need to do 34 calculations in Excel (probability of 0, 1, 2, 3, ... , 33 bad). The answer is 0.0031

Binomial (which assumes an infinite population) is a bit easier.

The Excel formula is =BINOMDIST(33,150,0.32,TRUE) which works out to 0.0045 which seems pretty unlikely.

If we look at the normal approximation, the standard deviation of this binomial is sqrt(.32 * ( 1 - .32) / 150) which is .038. The 22% result is 2.62 standard deviations from the mean.

We don't quite meet the 3 standard deviation criteria, but certainly the significance level is less than 1% and I would be pretty suspicious of this result.

If this were a p-chart, given n=150 and p-bar = .32 the UCL & LCL would be about .44 & .20. So this would fall right on the lower control limit.

A low probability of it happening, but remember low probability events do happen ... just not very often.

Boy, I get here just 30 minutes after the question was posted and there are already 3 good answers! :cool:

I guess I can just go back to work ...


Tim

P.S. I doubt you did this, but could you have mixed 1000 "bad" parts into 3100 "good" parts for a total of 4100 parts? That would be 24% "bad" and much closer to your random sample. I've been know to do such things in the past....

We don't quite meet the 3 standard deviation criteria, but certainly the significance level is less than 1% and I would be pretty suspicious of this result.

Looks like this is closest to the truth.
I just personally did a recount of the parts in the bag and there are significantly less than than 3100 pieces - so it looks like Steve's suspicion is correct.

One of my guys is doing a 100% sort so I'll see actual numbers on Monday. Or as actual as one guy using a digital micrometer to measure that many parts can get.

Here's the situation:

Bag of 3100 parts. There are 1000 pieces that are bad (32%) .Well mixed by many hands.
I pulled a sample of 150 pieces and measured the problem dimension.
Of the 150, I got 33 bad (22%).

Does this make sense? A 10% difference?

Can you let us know how you pulled out that 150 pieces?

When I was doing lab-work in engineering school, we have this thing called 'good sampling practice' and it's called the 'quartering method'. Essentially, you spread the parts (after being well mixed) out and divide them into four quadrants. Repeat the process for one of the quadrants and you get a sample size near 190 pieces for the new quadrant. Then count the total pieces in that quadrant and then the bad pieces. Experience shows that this method gives better results consistently!

If you build a confidence interval (just the upper CI), you would need an approximately 99.5% confidence interal to include the population of 32%. Since we know that the sample came from the population, I can only assume sampling bias or a population rate defective closer to 28% (the 95% CI).

Note: I did the calculations quickly, so someone please check my math.

For the data given, Lot 3,100 pieces, defective 1,000. Sample 150, defective found 33, the probability of exactly 33 is 0.001640.

The probability of exactly 48 defective in the sample is 0.069530

See BINOMIAL_lot_3100.xls attached.

Wes R.

For the data given, Lot 3,100 pieces, defective 1,000. Sample 150, defective found 33, the probability of exactly 33 is 0.001640.

The probability of exactly 48 defective in the sample is 0.069530

See BINOMIAL_lot_3100.xls attached.

Wes R.

Generally, though, we spead of getting 33 or more extreme, or 48 or more extreme, rather than exactly 33 or exactly 48. The chance of getting exactly the average number of defects can be quite low . . .

Can you let us know how you pulled out that 150 pieces?

When I was doing lab-work in engineering school, we have this thing called 'good sampling practice' and it's called the 'quartering method'. Essentially, you spread the parts (after being well mixed) out and divide them into four quadrants. Repeat the process for one of the quadrants and you get a sample size near 190 pieces for the new quadrant. Then count the total pieces in that quadrant and then the bad pieces. Experience shows that this method gives better results consistently!

stuck my hand in the bag, wiggled my fingers around for a second, withdrew some pieces (they're essentially discs about 1/4" in diameter), counted them, set them aside, repeat until I got 150.

stuck my hand in the bag, wiggled my fingers around for a second, withdrew some pieces (they're essentially discs about 1/4" in diameter), counted them, set them aside, repeat until I got 150.

What you do is not 'Random' sampling per se - there's bound to be a certain amount of bias which explains the result you obtained.

What you do is not 'Random' sampling per se - there's bound to be a certain amount of bias which explains the result you obtained.

How is that not random when the parts were already well mixed by several hands?
I should state that you cannot see or feel the defect. To the eye they all look identical and the dimesnion that is off can't be felt with the fingers.

From a theoretical standpoint I can understand what you're saying, but in reality of manufacturing this is about as random as you can get.

How is that not random when the parts were already well mixed by several hands?
I should state that you cannot see or feel the defect. To the eye they all look identical and the dimension that is off can't be felt with the fingers.

From a theoretical standpoint I can understand what you're saying, but in reality of manufacturing this is about as random as you can get.

Yes, that's the short cut that we normally take. What you can do is to take 4 samples (at different localities) instead of one. You will notice that you will get four different answers (and that is proof that however well you think it's mixed, it isn't) and then average it out. This answer will be an improvement over that given by a single sample or rather a better representative of the population as Jim mentioned.

How is that not random when the parts were already well mixed by several hands?
I should state that you cannot see or feel the defect. To the eye they all look identical and the dimesnion that is off can't be felt with the fingers.

From a theoretical standpoint I can understand what you're saying, but in reality of manufacturing this is about as random as you can get.

Dr. Deming always used the example of ore transports to Japan as an example of problems with mechanical sampling. Iron ore was shipped from the Great Lakes area to Japan. Upon arrival, it was sampled and analyzed for iron content. The US suppliers thought the numbers arrived at by the Japanese were always to low.

The sampling method the Japanese used was to take a bucket of ore off the top of the ore bins. Yes, the ore was well mixed in transport, but as you might imagine, all the heavy ore (with lots of iron content) settled to the bottom, and the lighter stuff (less iron) came to the top.

How is that not random when the parts were already well mixed by several hands?
I should state that you cannot see or feel the defect. To the eye they all look identical and the dimesnion that is off can't be felt with the fingers.

From a theoretical standpoint I can understand what you're saying, but in reality of manufacturing this is about as random as you can get.

In this context, "random" means that each member of the population has an equal chance of being selected. If the sampling method doesn't fit that description, then you might get unexpected results.

In this context, "random" means that each member of the population has an equal chance of being selected. If the sampling method doesn't fit that description, then you might get unexpected results.

Understood, but I thought it was pretty clear by my sampling description that the parts were, in fact, well mixed and I was mixing them even more as I drew a sample. I really don't see much chance for bias as the bad and good parts don't separate as heavy/light beads might, I can see this as my guy is measuing them - they are well distributed as he pulls them out of the bag to measure.
I get all the theory, but many theories can't really be applied practially while sitting in a conference room with a bag of parts.
I'd love to have time to experiment though.

Here are the possibilities:

1. This was a random event, a false alarm. Relatively unlikely to occur, and unlikely to recur. Can you rerun the experiment with the same setup and the same person?

2. Something is wrong in the initial data. There aren't that number of items, or that number of defective parts. Could you do a 100% inspection?

3. Something is wrong in how the parts are being categorized - bad parts have been passed as good parts (more often than good parts rejected as bad), decreasing the observed number of bad parts. Could you rerun with a different inspector?

Hokay - it all becomes clearer now. Apparently some information was held back - like some of the bad parts got into the next process (luckily that lot was not sold yet so no recall situation). We don't have count on that yet.

Findings after 100% inpection (rounded for simplicity here): 1400 good, 550 bad = 28% defective.
So both counts were off. Why did I not recognize this gross miscount? It's like guessing jellybeand in a jar. 2000 of these little pieces in a bag just looked like a bag of little pieces. We didn't take them to the counting scale until later in the afternoon on Friday.