@whuber So one would think ... but then again, as far as I know (from summaries I have seen, the guidelines themselves are paywalled!) for a standard t-test, APA demands t is quoted to 2 d.p. only, for which it's easy to find examples that wouldn't bound a p-value to 3 d.p. Hence my concern that the "sensible" choice of providing enough precision to reproduce the p-value may not have much correspondence with actual practice... and a bounty for someone who knows what actual practice is!
@Silverfish I agree with you. But are you sure the standard is 2 d.p. or is it 2 significant figures? The distinction is important! I think it would be silly to hold all authors to 2 d.p.; that would eliminate all precision in reporting any p-value less than 0.005!
Anyway, I posted my expanded comment as an answer primarily to document (graphically) the relationship between df and p. It might be useful for authors, in addition to you, who understand and care about the precision of their reporting.
Although I don't have a copy of the style manual, I did find an archetype of a paper on apastyle.org at apastyle.org/manual/related/sample-experiment-paper-1.pdf. The p-value quoted on p. 6, given as ".006", uses 3 d.p. and 1 sig. fig. That would violate a blanket 2 d.p. limitation.
@whuber Many thanks. Sorry if I was unclear: from everything I can glean, t and F statistics in APA are always given to 2 d.p. I think chi-squared is too, from memory. With a huge F or chi-squared the 2nd decimal place is almost always redundant, yet mechanistically it gets quoted! Now, the accuracy p-values get quoted to is a different story: I understand that 2 d.p. and 3 d.p. are both accepted, with "<0.001" where necessary
I think there is also a general APA rule "round to 2 d.p." (not just on test statistics, but on anything - for instance summary stats like means and standard deviations)... very convenient but often a slightly silly level of accuracy. Now if one is applying that general rule, I suppose Welch degrees of freedom should be quoted to 2 d.p. too.
I am trying to find the "most likely" median value for a distribution (or rather a range of distributions) described in a way that is unfamiliar to me. Please look at this PDF: graystarllc.com/file_download/42/…
@whuber Because I am not quite sure what I am asking as you can tell. :^)
That file describes a standard for grading the sizes of (abrasive) particles based on a series of sieves. So you start with the raw granular material and drop it through a stack of screen.... well, OK, then I guess you're surely are the right person to be talking to.
If I were testing a sample of P16 grit, I would first screen it with a 2360 micron screen.
If any amount did not pass through, it would fail: the sample has some unacceptably large pieces.
Then I would take whatever made it through the previous screen and pass it through a 1700 micron screen. Provided less than three percent of the original amount of the sample made it through, it would be ok.
Then I would take whatever remains--which if it passed must constitute between 100-3 = 97% and 100-0 = 100% of the original sample--and put it through a 1400 micron screen.
The sample would be acceptable provided between 100-32 = 68% and 100-20=80% of the original sample were left after these three successive screenings.
At the end, after two more passes at 1180 and 1000 microns, I would want to see no more than 4% of the original sample.
In particular, these percent tolerances do not apply to what has previously passed through each screen, but to the original amount of the sample. (That's a standard way to express grain size distributions.)
We can interpret these particle size tolerances as giving the upper and lower (or, equivalently, left and right) boundaries of a region in which the true distribution's graph must lie.
@Mr.Wizard Yes, the Wikipedia link is exactly right!
The main difference is that these tolerance limits, being discrete, result in somewhat strange looking graphs: they are horizontal everywhere except at a finite number of places, where they jump.
For instance, the jumps in the P16 p-box would occur at sizes of 2360, 1700, 1400, 1180, and 1000 microns.
The first criterion says the true graph lies entirely to the left of 2360 (if we plot screen sizes in the usual ascending direction from left to right).
@Mr.Wizard Yes, except this is a somewhat subtle sense of "fit." It differs a little from the usual sense of, say, "curve fitting."
One thing you can do is consider the collection of all cumulative distributions (CDFs) that could fit within the p-box. Each has a median. The entire collection thereby gives rise to an interval of medians. That interval might the fullest, most honest way of characterizing the median.
It would say "I can guarantee that any sample of P16 that meets all the tolerances has a median within this interval."
To go any further, you have to start making assumptions, such as assumptions about the specific shape of the true CDF.
At this point, it's probably most appropriate for me to ask a question in turn: what do you need these medians for? What information are they intended to convey? What decisions might be made using them?
(Ordinarily when one "fits" a function to data, it is done in some "optimal" way, such as minimizing a sum of squares of residuals. At this juncture we don't have many clues about what to optimize: that's the subtle distinction I'm alluding to.)
We're kind of at the model-building and assumption-making stage still. The more you can say--in at least a semi-quantitative way--about your objectives, the more we can narrow the possible approaches.
The most basic information I wish to extract is the "best" single value that represents the given grit standard, defined by its median particle size by weight. This is with full knowledge that the standard (e.g. P16) can cover a wide range of particle size distributions.
OK, let's see where we can go with that. "Best" suggests you have some kind of objective function to optimize or (using related language) a loss function to minimize.
These convey the costs to you that would be associated with not getting exactly the right value of the median.
The somewhat more complete information is the interval you referred to in: It would say "I can guarantee that any sample of P16 that meets all the tolerances has a median within this interval."
This more complete information we have: conceptually, it's best arrived at graphically, but once you do that it will be obvious how to find it with a (simple) algorithm.
@whuber This is completely over my head, not necessarily on principle but in practical application. All I am attempting it to create the "best" (sorry, can't think of a better way to put that) conversion from this sieve data to a single median micron value, and also generate an interval of possible median values. I do not know what distribution is most likely to apply to abrasive grain or if it varies significantly by the source and type (chemical composition) of the grain.
I realize that this kind of underspecified problem is annoying to those with understanding as I run into that all the time on Mathematica but I hope you can have patience with my limitations.
It could vary. Presumably this is an industrial product and it is created by some process. Understanding of that process could inform your assumptions. Another way to proceed would be by means of more detailed analyses of some samples.
(Underspecified problems are standard in statistics. Accordingly, we have to run our site a little differently than most of SE. Ordinarily the conversation we're having gets carried out in comments and then sent off into chat when it gets long. That's pretty routine.)
@whuber I know that most abrasive is formed in larger pieces and then crushed and sieved to grade it. Other methods are used for finer grains but I already have basic distribution parameters for these (d0, d3, d50, d94).
Right. It comes down to the physical properties of that crushing process. I have no knowledge of such things. Although we could make some reasonable guesses (based on physics and geometry) they would be just that, guesses.
One thing we could do, though, is compare the distributions allowed for the 15 products on this sheet. Maybe they have similar shapes that could be discerned.
@whuber Would your guess be more informed that simply assuming a normal distribution? And since this standard has to apply to all standard ceramic abrasives is there any point in being this specific or attempting that much accuracy?
I would be reluctant to assume a Normal distribution. It's possible, though, that something like the cube or cube root of the particle size could look somewhat Normal.
Yes, I was going to ask whether these tolerances were in terms of number, mass, or volume. I have been assuming all along they were mass (since that is by far the easiest and most reliable to measure).
As illustrated there (which at least I understood beforehand) the shape of the distribution can greatly affect the relative position of mean and median, etc. I don't have a reference for real-world distributions of commercial product, but I can try to look for it.
You are correct: shape will be important, except when the tolerances are narrow.
Anything you can find about actual distributions of product would be great.
Best would be information from the company. Next would be information from other companies using the same equipment and grinding process. Third best would be info from competing companies selling into the same market.
I believe by mass is the most common, but it can also be by volume which gets a bit vague. I believe that the relative height in a sedimentation stack (tube) is used to calculate the volume, but I don't know of the packing density of the particles is then used to correct that to an equivalent mass. All of this is really beyond the scope of what I want to do however.
@whuber I'll bear that in mind. My project however is merely to create an overview of the abrasive standards themselves rather than to compare specific products. The microgrits (above ~220) are specified in terms of distribution limits like this: uama.org/Abrasives101/360.html
What I would like to do is attempt to convert these sieve-graded standards to numbers like that.
Let's consider what is easily accomplished. Look at P16 again. Since at least 20% has to pass the 1400 micron sieve, you know the median grain diameter cannot exceed 1400.
Sorry: since you know that no more than 32% is allowed to be caught in the 1400 micron sieve, that means the median cannot exceed 1400.
OTOH, since at least 66% must be caught in the 1180 micron sieve, the median must exceed 1180. Therefore, with no more ado, we can bound the median by 1180 and 1400 microns.
(Yes: "or above it," too.) Improvement is possible only by making assumptions about the shape of the grain-size distribution. Without such assumptions, the kind of calculation I just did is the best that can be accomplished.
For P20, we know that up to (but not exceeding) 50% of the sample will be caught by the time the 1000 micron sieve is applied. We know that at least 80% will be caught at 850 microns. Therefore [850, 1000] is a valid interval estimate for the P20 median.
How is the microgrit spec related to the macrogrit size tolerances?
@whuber I don't know. I am guessing that the distributions produced by crushing and sieving will have some commonality between low and high grits. I am not sure what else I can go by.
Although that's a reasonable guess, I would be reluctant to rely on it. One can think of all kinds of ways the distribution might change over such a large range of grit sizes. For instance, microgrit particles might tend to be single crystals whereas macrogrit particles might be more heterogeneous assemblages, perhaps causing a systematic shift in distribution shape once a threshold (related to typical crystal sizes) is crossed.
@whuber I can't refute that in any way. I am wondering where to go from here. You wrote:
> One thing you can do is consider the collection of all cumulative distributions (CDFs) that could fit within the p-box. Each has a median. The entire collection thereby gives rise to an interval of medians. That interval might the fullest, most honest way of characterizing the median.
Looking at the microgrit specs, you can see the distribution may be asymmetric: a typical d3 - d50 distance would lie around 42-25=17 or 38-21=17. If the distribution were symmetric, then d50 - d97 would have to be about the same. But 25-17=8 and 21-17=4 seem awfully small.
That rules out using Normal distributions. (But it does not rule out using a power-transformed Normal, which would be a Normal distribution of, say, the square root or cube root of the screen size.)
@Mr.Wizard Yes: that earlier remark about "... fullest, most honest way..." was made in that spirit: it's completely defensible and accurate, requiring no knowledge of the distribution at all.
I doubt you need my assistance: you can emulate the example I did for P16. But I would be happy to review your calculations if you like.
Perhaps another option would be to graph the p-boxes implied by the size tolerance sheet. Although that's merely a way to visualize the numbers there, it could be a very informative visualization.
Well, if it's only medians you're after, you don't need either the d3 or d94 values, because you are given the d50 intervals: that's the median.
Otherwise, for other percentiles you can use the d3 and d94 values in the same way I illustrated with P16--but the intervals you would get would be wide.
There is a difference between what I am personally after, which is a deeper understanding of what I am looking at and which you are helping me with, and what I am pragmatically after which was the "best" single median particle size to represent each grit.
@whuber Suppose I had a distribution illustrated graphically that I wanted to choose to assume widely correct. We have e.g.: azom.com/article.aspx?ArticleID=2794
I haven't even bothered to read what that actually is, it's just an example.
How might I go about applying that to "fit" it to the data?
That's the kind of thing I feared. It has no simple description, but the simplest ones would decompose it as a mixture with perhaps four components centered around 0.5, 4, 35, and 300 microns.
Sort of. It's not that this is so complicated to deal with. The problem is that it's so flexible, you won't get much of an improvement over the conservative intervals we already came up with.
The nice scenario would be if the manufacturer said "oh, we strive to have a homogeneous population of particles, so you wouldn't actually see mixtures like that."
This is actually for cement particles but I think I remember seeing something that looked a bit more like that for actual abrasives.
@whuber Yes, I noted that; I don't think that is graded material. I am going to have to research that more but I also want to know what to do with what I find.
The second figure--as I can just make out--also is on a log scale. Those particles range by over two orders of magnitude! Since the Pxx grits cover much smaller ranges, I surmise they would likely have simpler distributions.
A default assumption in geology, that works fairly well, is to suppose the distribution sort of looks Normal on a log scale. It would be nice to find some evidence to justify such an assumption here.
Exactly how the manufacturer achieves those tolerances may have a strong bearing on the grain size distribution shape.
@Mr.Wizard Yes, this would be interesting. (I'm supposed to be co-authoring a book on interval statistics, although I have been procrastinating all fall, and am looking for good applications. This could very well be one.)
BTW, don't hesitate to post questions on CV. Although the community might put you through a (public) wringer, the diversity of experiences and approaches you get exposed to would likely make the experience worthwhile.
One thing you might enjoy doing--and be quickly successful at--is taking data like those from the spec sheet and having Mathematica graph the p-box. Later on you can superimpose distributional shapes on those p-boxes to see what can be accomplished by specific shape assumptions.
BTW, here's wishing you the best for the holidays. The day is drawing to a close here so I need to get away from the computer and do some real work :-).
