Hi All,

I need some help with statistical calculations - I can't remember everything from 20 years ago!

Situation:

We have a production process that has to hit a mean size with a tolerance.

Mean = 100mm

Tolerance = +/- 2mm

So therefore, an item is 'good' if it is greater than or equal to 98mm and less than or equal to 102mm, and bad if it is outside that range.

We are willing to accept a maximum 1% chance, that in a run of 10,000 units, there will be at least one item that is bad.

The production process slowly gets out of alignment (due to vibration and other factors), and has to be periodically recalibrated. The misalignment is symmetrical in terms of which way it goes out (it is just as likely to lift the mean upwards as downwards).

In order to maintain the quality, we are proposing to do the following:

Define a range with mean = 100mm, Tolerance = +/- X

where X < 2 mm

We will then sample the production output once every Y items.

If an item sampled falls outside of +/- X then the production process gets stopped, and maintenance is done to recalibrate everything back to the original status, and then restart the production process.

Ultimately, I need to calculate the sampling frequency (Y) we have to use to deliver on the objective.

I would likely be doing the calculations by putting them into an excel spreadsheet, but that seems kind of secondary - that just happens to be the brand of calculator I am using.

I suspect I need to know the Standard Deviation of the actual outputs of the process when it is freshly calibrated. If so, then my first thought is that I would need to get a sufficiently large run (100 items - perhaps more, but it has to be practical, ideally this sample size could be a parameter in the calculation of probability), have them all manually checked, and from that data, calculate an actual mean and Standard Deviation. I have been *told* that the actual output mean is 100mm, but the production people could not give me a standard deviation figure.

What I need from you is:

1) What additional information is required, in order to do the calculation

2) what is the actual calculation that I have to do to determine Y.

Please do post any queries or clarifications required.

Thanks,

Alan.

This hinges on your statement

That is for every 1,000,000 units produced, 990,000 in 99 batches will be within tolerance, and of the remaining units in 1 batch 1 (or more) could be out of tolerance.

For a single out of tolerance item the probability would be 1E-6 (1.0 * 10 to the -6 or 1 in a million).

Assuming that you have independence between units ** then 2 or more defectives in 1,000,000 units is of the order of 1E-12 which is too small to make any change to the solution.

The solution settles down to a question about relative costs. First there is the cost of measurement. Next, when the testing detects that the production process is going out of tolerance there is a cost to manufacturing of stopping production and re-adjusting the machinery.

Those costs need to be balanced against the cost of exceeding your target of more than 1 batch in 100 with 1 (or more) defectives. You might consider this cost is infinite, but if it was truly infinite then the solution would be examine and measure 990,000 items in each 1,000,000 !

Do you have an idea of these (relative) costs?

~~~~

From the manufacturing perspective, is there information about the "speed" of movement of the mean away from 100 mm when the process starts to go wrong. For example your engineers may be able to say that the mean will not move quicker than 0.001 mm each item.

There is a big body of statistical work on process control as well as on sampling of batches for defectives. The process control work develops "charts" where you constantly plot the data as sampled against time. You pick a formula (based on cumulative sums and/or standard deviations) to detect the "out of control" situation.

Ian

** Note on independence: When the process is "in control" it is reasonable to assume that the variation from the mean of 100 mm is independent from unit to unit. However, when the process is straying from the 100 mm value, the likelihood of the next unit having a positive (or negative) error will be related to the size and direction of the error on the current unit. That is, errors will not be independent.