Sorry, it ain't that simple.

Martindog

PSI = -17,902 + 1.51 x CUP

However, you have to realize that both measurement systems have considerable random error, and both sources of error are at work in the equation, so individual point conversions can be off a few thousand PSI.

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Originally posted by Zero Drift:
Ditto denton - Somewhere along the way someone stated that the two measurements are not correlated nor convertible. The problem is the random error as denton pointed out.

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I recommend that newcomers to this subject go back and read the earlier thread. In my unhumble but uneducated opinion, the problem is NOT the random error, although that exists. The problem is that they are two somewhat different measurements. Go back and read the thread and form your own opinion, then stick to a trustworthy loading manual.

SAAMI apparently took a position a few years ago that the two measurements (PSI and CUP) are not correlated and cannot be converted from one to the other. This is erroneous.

If CUP and PSI work reasonably well, it is a mathematical requirement that they be highly correlated. Both systems manifestly do work, though not perfectly. They must, therefore, be highly correlated, and there must exist a formula to convert one to the other, plus or minus system errors.

This is confirmed by running either of two statistical processes on the data: Pearson's Correlation Coefficient, or Regression. In the case of Pearson, an outcome near either 1 or -1 confirms high correlation. In this case, we get .963, which is close to 1, and the expected high correlation is confirmed. From Regression, the significant number is R^2, and a result near 1 indicates high correlation. The result is .927. That is near 1. The two measurement systems are highly correlated, as they must be.

If two measurement systems are highly correlated, there must exist a formula to convert from one to the other. That is the formula that I posted.

Now you do have to be careful using this kind of information. First, it only covers the pressure range that I stated. Right now, I can't tell you how the curve behaves outside that range. Second, you have to allow for the fact that measurement errors from both systems are hard at work, and they are significant. They account for about 7.3% of the total observed variation.

The line (formula) itself is derived from 30 data, and that will "average down" the error quite a bit. The line itself is much more accurate than any one pair of points, but don't expect each conversion you do to come out exactly spot on the published values. It is a good approximation, nothing more.

Based on published data, if you could make and fire several identical cartridges, you would find that with the CUP system, you would experience random errors with a standard deviation of about 2,000 PSI. So, if you fire 10 rounds, as the established process requires, the average of those 10 rounds will be within 2,000/(10^.5) (about 650 PSI) of the true value 95% of the time. 5% of the time your answer will be in error by more than that. The piezoelectric PSI system probably has about 1/3 less error than that.

Can you post your chart in here? That usually clears things up.

R-WEST

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"the spotlight of truth will cause the cockroaches of deceit to run for cover every time"
Rush Limbaugh

R-WEST

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"the spotlight of truth will cause the cockroaches of deceit to run for cover every time"
Rush Limbaugh

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Originally posted by denton:
For CUP in the range 28,000 to 65,000:

PSI = -17,902 + 1.51 x CUP

However, you have to realize that both measurement systems have considerable random error, and both sources of error are at work in the equation, so individual point conversions can be off a few thousand PSI.

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Denton,

I've been working with an empirical method to determine maximum permisible breech pressure in the Contender. The method uses the spring-back of the brass case as a way to decide when the redline pressure has been reached. This does not give an actual pressure - rather is an empirical method based on measurements of fired cases. However, I would like to relate the cutoff point to actual pressure, but it seems the calculation methods and pressure limits for some guns are stated in a mix of CUP and PSI units. In trying to sort out the relation between PSI and CUP all I can find is complication and confusion. I have not seen the form you give for the relation before and would like to have a reference for it if you have one.

Thank you,

Don Shearer

The equation is a derived one. SAAMI publishes maximum pressure for most rifles. In some cases, they give the pressure in both PSI and CUP. I know of 30 such cases.

The simple process for finding the equation that links the two is linear regression, and that's what I did.

So what I've posted is a empirical equation, based a fairly simple analysis of SAAMI data. So I guess the reference is "me".

As nearly as I can tell, the firearms industry is relatively unburdened by knowledge of some of the standard statistical methods that should be used in these cases. So, you get confusion and contradiction. It isn't just you.

Just to illustrate: check the Lyman reloading manual (a fine manual!), page 91. They show both CUP and piezoelectric PSI figures for 10 cartridges. The data was actually supplied by Alliant. Neither Lyman nor Alliant picked up on a trend error in the PSI data. If they had caught it, and eliminated the cause, it would improve the precision of their PSI measurement system by about 25%, and that is a worthwhile improvement. Even the manufacturer of the test equipment is probably unaware of the error built into the product.

I don't see an email return address on your post, but if you want to email me, I'll send you a copy of the graph.

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Originally posted by denton:
SAAMI apparently took a position a few years ago that the two measurements (PSI and CUP) are not correlated and cannot be converted from one to the other. This is erroneous.

If CUP and PSI work reasonably well, it is a mathematical requirement that they be highly correlated. Both systems manifestly do work, though not perfectly. They must, therefore, be highly correlated, and there must exist a formula to convert one to the other, plus or minus system errors.

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It is not erroneous. Ask them why, or go back and read the original thread, and you will find out why.

You say that if they both work, it is a mathematical requirement that they be highly correlated. False - my dial caliper and my powder scale both work, and they don't correlate worth a shit. Why? Because they measure different things.

Your argument is very "must"y. No matter how much math you think you know, you should understand the underlying physical facts in order to apply the math correctly.

As far as your "successful" correlation of CUP and piezo PSI, let me assure you that with certain limitations, I can get excellent correlation between a crude tape measure (applied to my waist) and a bathroom scale. That doesn't mean that I can weigh you with a tape measure.

Physical facts? As a matter of fact, I have a degree in Physics, from a well respected university.

The piezoelectric method of measuring chamber pressure produces a voltage, that is proportional to chamber pressure, as a function of time. The electronics typically simply "hold" the peak voltage, and hence the peak pressure. In practice, the piezoelectric method usually measures peak chamber pressure.

The CUP or "crusher" method uses a copper pellet with known properties, and measures how much it is deformed by the chamber pressure. Most of deformation happens at peak chamber pressure. The CUP method gives a decent approximation of peak chamber pressure.

Both the PSI piezoelectric and the CUP copper crusher methods are mainly designed to measure peak chamber pressure, and both are reasonably successful at it. It follows that they must be highly correlated. The correlation betwen CUP and PSI is fundamental, not accidental.

If you use your calipers to measure the the "stretch" of an appropriate spring that is supporting a weight, you will find that the measurement correlates quite well with a good set of scales. In that case, the scales and the calipers are different measurement systems, measuring the same thing, just as PSI and CUP measure the same thing. They must be highly correlated. It cannot be otherwise.

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Originally posted by denton:

The electronics typically simply "hold" the peak voltage, and hence the peak pressure. In practice, the piezoelectric method usually measures peak chamber pressure.

The CUP or "crusher" method uses a copper pellet with known properties, and measures how much it is deformed by the chamber pressure. Most of deformation happens at peak chamber pressure. The CUP method gives a decent approximation of peak chamber pressure.

Both the PSI piezoelectric and the CUP copper crusher methods are mainly designed to measure peak chamber pressure, and both are reasonably successful at it.

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You say that piezo methods measure peak pressure. You say that crusher methods measure deformation which MOSTLY happens at peak pressure. You say that the crusher method gives a decent APPROXIMATION of peak pressure. Probably nobody disagrees with you, except conceivably on the precise definition of "decent."

CUP is a measure of effect of pressure over time. What time? Whatever time during which pressure has effect. Piezo measures peak pressure. Two different things. Do they correlate well? Do my tape measure and bathroom scale correlate well?

I believe I understand your approach, that these are both ways of "measuring" peak pressure, and I don't have a problem with it. What I do have a problem with is your apparent implication that SAAMI and the fellow who posted on the previous thread don't know what thay are talking about. Your concept of measurement is somewhat cruder than what SAAMI apparently endorses, although I suppose it is possible that they are just slobs who stumbled into looking like they know what they are doing. I consider it far more likely that they have physicists and statisticians, too, and know what they are doing.

I have no idea whether random "error*" in the two systems is big enough to mask the differences in what they are really measuring, but I suspect that SAAMI knows, and that's why they write the way they do.

*"Error" in CUP is variation in deformation, or, more properly, measurement thereof. What the peak pressure is is a whole 'nother story, which is one of several reasons people are happy that piezoelectric methods exist.

If you have data and analysis that dispute any of the following, I would be especially interested in seeing them...

1. PSI and CUP must be, and are, highly correlated. An R^2 of .927 effectively ends any discussion on this point. To argue against this is to argue that .927 is much closer to 0 than it is to 1. If you can show that, I think we will all be amazed.

2. If two variables (in this case, PSI and CUP) are highly correlated, and connected by cause and effect, as PSI and CUP are, there must be a formula for converting from one to the other. This comes from the definition of "highly correlated", which, in essence, is that you can predict one variable from the other.

3. Any statement that PSI and CUP are uncorrelated, and cannot be converted from one to the other is false. I have presented the formula, and set the limits on its accuracy and applicable range. The result can be reproduced by any competent person, anywhere in the universe. Once the formula is shown to exist, any claim that it does not exist has been shown to be false.

So if you want to present your conclusions, show us your data, and show us your analysis of the data.

By the way, heavy people do tend to have larger waistlines than light people. For any individual adult, your bathroom scales ARE correlated to a tape measure around your waist, and you CAN estimate one from the other. Both measurements correlate to body mass, and so must correlate to each other. The correlation will not be perfect, and may not be linear, but it's there. You've brought up an example that argues my case, not yours.

I make a good living, in part by teaching companies about measurement systems analysis. I'm not making this stuff up. It's from the most widely accepted texts on the subject, and routinely applied by the most successful companies. So far, I haven't found any evidence that modern measurement systems analysis is practiced in the firearms industry, but I may yet find that it is.

If you think I'm wrong, show us your physics and your statistics. If you have stronger data and analysis than mine, it will prevail, and I will praise your intellect and contribution to the field.

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Originally posted by denton:
By the way, heavy people do tend to have larger waistlines than light people. For any individual adult, your bathroom scales ARE correlated to a tape measure around your waist, and you CAN estimate one from the other. Both measurements correlate to body mass, and so must correlate to each other. The correlation will not be perfect, and may not be linear, but it's there. You've brought up an example that argues my case, not yours.

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No, I haven't. I know that there's a correlation which is why I brought that example up. I also know that they are two different measurements. The correlation is not 1.00. If someone were going to surgically remove the body part of HIS choice equal to your weight over 200#, you would not measure your weight with a tape measure, no matter how good you THINK the correlation is, to use popular language. The correlation can be quite high, but for those unfamiliar with statistics, you might want to make a statement about the probability that the 31st round of ammo with X CUP with a standard deviation of s will exceed a pressure of Y PSI, after you've based your knowledge of CUP and PSI and s on 30 rounds.

I do not need to be more dramatic than necessary - we both know that the world will not likely come to an end because the actual max peak pressure desired is exceeded by a few thousand PSI. But I used to make my living in photographic process control, which uses much cruder processes and much cruder mathematics than I suspect you are used to, and I have had plenty of experience with being preached to about statistics by someone who didn't understand the process. I don't think that is the case here. You probably know more about metallurgy and statistics and internal ballistics, all three, than I do.

However, you're still measuring two different things, and you still haven't shown me that SAAMI doesn't know what they are doing. If you showed it & I missed it, feel free to refer me to it and I will apologise for being sloppy and in a hurry.

A couple of things that I think you are missing (points, not body parts)--

CUP and PSI are, in fact, both attempting to measure exactly the same thing, peak chamber pressure. PSI is a little cleaner from a physics standpoint, but both systems are trying to measure exactly the same thing. The CUP system does respond slightly to the total pressure vs. time curve, but the main thing it indicates, more than any other, is peak chamber pressure.

PSI correlates to peak chamber pressure.

CUP correlates to peak chamber pressure.

Two measurements that correlate with the same thing correlate with each other. That is why PSI and CUP must be highly correlated, why finding them to be highly correlated is not a surprise.

In photo lab terms, think of mercury thermometer vs. thermocouple: Two very different ways to measure liquid temperature. The CUP and PSI situation has a lot in common with that situation.

"Correlated" or even "highly correlated" does not mean "perfectly correlated". One variable does not need to explain ALL the variation in another variable in order to be correlated. It only has to explain PART of the variation. The R^2 value tells you what part of the variation your model describes. An R^2 of .927 means that the CUP vs. PSI model explains 92.7% of the variation, and that the remaining 7.3% is unexplained... often attributable to measurement system errors. In your bathroom scales model, it is possible to introduce more variables, such as gender, height, etc., if you want a more complete picture. But with an R^2 of .927, there is little reason to spend a lot of time on such a thing.

I haven't said that SAAMI doesn't know what they are doing. I have said that they made a major blooper if they said that PSI and CUP are uncorrelated. That statement is manifestly not so. I have also not found any evidence that they have applied modern measurement systems analysis.

30 data is only the point where the Student's T Distribution and the Gaussian Distribution become practically indistinguishable. It is not usually the number of data you need.

Finally, you essentially asked for Confidence Intervals and Prediction Intervals, i.e., an estimate of the random error associated with the measurements. I have already stated those.

From Lyman's reloading manual, speaking of things as they were years ago:

"...the copper crusher method is a valid and useful tool for ammunition evaluation, but it does not actually express pressures in true pounds per square inch....the erroneous term PSI had become so accepted that it was frequently used interchangeably with CUP..."

In short, people were incorrectly calling CUP PSI for many years, before modern equipment capable of giving true PSI became widely available.

Lyman reports that this was causing a lot of confusion and concern. I can easily imagine SAAMI putting out a bulletin that said, in effect, "The crusher method does not produce a measurement in PSI. The proper unit of measure is CUP. PSI measurements are produced by strain gauge and piezoelectric systems, and that is not what the crusher method does. They are different measurement systems. Do not call CUP PSI." That would be a correct statement. Stating that the two are uncorrelated would be an error, and might happen because a writer did not understand the difference between "correlated" and "equal".

Are we possibly dealing with the echos of a long ago SAAMI bulletin that tried to straighten out improper terminology?

[This message has been edited by dsverdrup (edited 04-03-2002).]

I came across this thread last night and thoroughly enjoyed reading your presentation of data and logic. Although I've never met you, I am certain that I would enjoy listening to you present a lecture or seminar (assuming from your posts that your are probably an educator or consultant). Your patience and clarity of logic are outstanding; I will look forward to reading more of your posts.

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Originally posted by denton:

I haven't said that SAAMI doesn't know what they are doing. I have said that they made a major blooper if they said that PSI and CUP are uncorrelated. That statement is manifestly not so. I have also not found any evidence that they have applied modern measurement systems analysis

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OK. If that's the case, I apologise for misunderstanding your criticism of SAAMI, and for beating a dead horse.

A very quick look at an elementary statistics book did not make it immediately apparent to my ignorant self whether it was implicit in an R^2 of .927 what "safety factor" had to be allowed in the conversion to have a ridiculously high confidence that a given peak was not being exceeded. I realize that, in a sense, this is a bass-ackwards question, since SAAMI has for years been using CUP, and now the somewhat more precise (and more direct!) piezo methods are being used, but I'm still curious.

BTW, I briefly reviewed this thread (not the previous one) and see that you relied on SAAMI data apparently equating allowable CUP & allowable PSI. I can certainly understand your considering SAAMI's data, understood as intended, sufficient for refuting SAAMI's claims. I am not so sure I understand your making an independent statement as to conversion from CUP to actual peak PSI, or indicated piezo-measured PSI, for that matter, based on a source that you don't consider particularly competent.

[This message has been edited by Recono (edited 04-05-2002).]

[This message has been edited by MePlat (edited 04-05-2002).]

What??? You don't understand that the fate of civilization itself hangs on this very issue??? :-) Yeah, I think we've given it about enough play.

My interest comes from the fact that I do have pressure instrumentation, and I needed to be able to convert CUP load data to PSI to get an absolute calibration on my setup. It's handy to be able to do that sometimes. I'm in the middle of doing some work to compare the accuracy of CUP vs. piezoelectric vs. strain gauge technology.

Recono, I just got my computer back after having the hard drive scrubbed, to get rid of a virus and install a new operating system. If I can keep my mind on it for long enough, as soon as I get my stats software running again, I'll send you a graph that answers your question.