Statistics question

Factual Questions
1

Sorta long, sorry.

My work is in the area of very very high-reliability electronics. So we have a strict rule: exposed pure tin (Sn) is not allowed, and pure Sn finishes/platings are not allowed. This is because “tin whiskers” can grow from pure Sn, causing short circuits. To prevent this from happening, we require solder to contain at least 3% lead (Pb) by mass. (Traditional solder is 63% Sn and 37% Pb.) Also, the plating/finish on component terminals/leads cannot be pure Sn; the finish must contain at least 3% Pb. (A traditional solder finish is best, as it contains 37% Pb. But we allow all the way down to 3% Pb. But not lower.)

A contractor assembled a bunch of printed circuit boards (PCBs) using traditional solder (63% Sn / 37% Pb), which is good. But… some of the capacitors that were soldered to the PCBs had a pure Sn finish on the leads/terminals. :roll_eyes:

Now, in the areas where the solder covered the terminals during soldering, it’s fine. But the solder did not completely cover the top of the terminals. This explains it better:

The contractor said, “Nothing to worry about. Even though it doesn’t look like it, some of the solder did make its way to the top of the terminal, which means some Pb made its way to the top of the terminal.” To prove this, they measured the percentage of Pb at the top of the terminals for each capacitor using XRF. They took five measurements on each terminal. Here is data from one terminal for one of the capacitors:

(That’s not the actual data or actual photo, but close to it.)

Contractor is saying, “See, look! The lowest value on that terminal is 3.6% Pb, which is higher than the 3% minimum. All is good!”

So I thought, hmm, let’s do some statistics on this. So I estimated the population mean by doing this:

N = 5
Degrees of freedom (DF) = 4
Sample average = 6.26
sample standard deviation (sn-1) = 3.494
Confidence level (CL): 95%
t_value = 2.776 (based on DF and CL)

Therefore, and with 95% confidence, the population mean is estimated to be:

6.26 ± (t_value * sn-1)/sqrt(N)
= 6.26 ± (2.776 * 3.494)/sqrt(5)
= 6.26 ± 4.34, or between 1.92 and 10.60

So with 95% confidence, the population mean is estimated to be between 1.92% Pb and 10.60% Pb.

Is the above correct and valid? And is this the best way for me to do this? I’m thinking it would be better to compute the probability the population mean is less than 3%. Or perhaps even better, the percentage of the population that is less than 3%. But I don’t know much about statistics, and don’t know how to do that.

2

I would suggest testing more than a single capacitor. For example, you could pass/fail each one, and if you test, let’s say, 60 of them and they all pass then you could be 95% confident that at least 95% are good.

You seem to be suggesting that the data off each one is suspect, however, as in you took five measurements of a single terminal and got wildly differing results each time. Therefore it sounds like you really need to know the accuracy and precision of the XRF instrumentation in use. ±20%? 30%? Is there a way to prepare your own (known) samples and get an accurate number?

3

Looking at this from a different point of view, why are you allowing the contractor to try to talk his way out of using pure Sn tips when you specified a minimum 3% Pb?

4

I think I’d have to have more industry knowledge to speak about how acceptable that is, however I just want to note that a 3% mixture is not necessarily the same as uniform 3%. Like if you samples an area that used your acceptable 3% lead, you’d get some readings above and some below, right?

5

Echoing the others sorta.

So they took 5 samples from a single exemplar installed capacitor and are claiming somehow that’s adequate proof of a production run measured in dozens to thousands? And you don’t / can’t know if they cherry-picked that one?

I am most definitely not a statistician. But that doesn’t even pass the laugh test.

Statistically speaking what they demonstrated is that they are utterly clueless as to everything about statistical process control. Or they’re bad at lying. Or they think you & your organization are fools. Probably a mix of all three, but none are a good look.

Assuming I accurately understand the situation. ETA: Which based on the response below I evidently do not.

6

Each PCB contains five of these “risky” capacitors, and there are 1,500 PCBs. They took XRF measurements on a few of them. The data I presented is just one of the data sets they provided.

That’s a good point; I don’t know how accurate their XRF tool is. I guess I was just trying to show that simply looking at the minimum value of five measurements is not valid from a statistical POV, hence the reason for estimating the population mean. But as mentioned in my OP, I’m thinking it would be better to compute the probability the population mean is less than 3%. Or perhaps even better, the percentage of the population that is less than 3%. But I don’t know much about statistics, and don’t know how to do that. Regardless, I am going to recommend the PCBs be reworked to fix the problem; was just looking for better statistical treatment.

I am not sure what was originally specified – I haven’t seen the contract. I am working on the assumption that the contract didn’t specify a minimum Pb content for the finishes on component terminals. (If it did, you’re correct, and we simply tell them to fix the problem.)

7

Strictly speaking, if you have a population, and some variable has a mean, then there is no “probability the population mean is less than 3%” — it has some definite value, either less than 3% or not less than 3%! What you can do is hypothesis testing with a p-value.

I am just not positive that I grok what the XRF measurements are telling us. It seems that, looking at a single capacitor, both the amount of lead varies from point to point [in some way, but how exactly?] , and that the measurements taken at any given fixed point are also noisy? So you will have to take both of those things into account.