Non-representative sampling in solids processes has costly consequences. A better understanding of the mathematical basis for sampling and of acceptable quality levels can help to alleviate potential sampling issues and more closely approximate reality
Effective sampling in solids-handling processes is critically important to process success and product quality. Conversely, the negative consequences of non-representative sampling are potentially immense. Despite the importance of sampling, the attention placed on ensuring proper sampling and the resources devoted to it are commonly disproportionately small compared to the efforts directed elsewhere. Further, effort aimed at understanding the mathematical basis associated with minimum mass requirements and standard error is generally lacking. For example, the first chapter of Terry Allen’s classic book “Particle Size Measurement”  addresses representative sampling, but is often ignored in pursuit of other more “interesting” chapter topics. Consider that errors caused by non-representative sampling have massive economic impact — one example is quoted at $134 million loss over 10 years in a mining project .
There are several adages, quotations and clichés from the field of statistical sampling that can be aptly applied to solids-handling technology. The concepts held in these adages can help frame the discussion of solids sampling. One particularly insightful quote from solids engineer Mark Murphy states “all samples are wrong — some are more wrong than others.” This quote gets at the error inherent in sampling and the idea that engineers must define their tolerance for error.
Most of us are familiar with the phrase “garbage in equals garbage out,” perhaps more elegantly phrased by Francis Pitard’s quote “Your decisions are only as good as your samples.” This quote expresses the concept that the quality of analytical results depends on the sample size that is analyzed.
The pragmatic engineer needs to get a feel for the minimum sample masses needed to meet a given specification or acceptable quality level (AQL). This practice is part of what this article explores. In addition, the article attempts to help close the gap in appreciation and understanding of representative sampling by investigating the theory and practice of sampling particles for analysis of solids processes.
It was really the end of the 1880s that mining engineers started to take a keen interest in sampling issues. At the time, the realization took hold that separate analyses of gold-bearing ore in particular could have profound effects on the remuneration of the ore seller.
Warwick  quotes an example where Assayer 1 indicates 79 ounces per ton of gold and Assayer 2 indicates 11.5 ounces per ton for the “same (divided) pulp sample.” At that time (1903), this discrepancy amounted to a difference of $1,350 per ton. With hindsight, we can see that both can be right — the gold nugget sitting in one half of the pulp sample obviously affects the difference between the two halves.
The analytical errors are at least two orders of magnitude lower than those potentially introduced by non-representative sampling. This means that we should be spending two orders of magnitude more dollars on the sampling equipment in plants than the analytical tools that deliver high accuracy and high precision on tiny amounts of sample. Transmission electron microscopy springs to mind as a user of tiny sample amounts.
Election poll predictions is another area worth examining as an example that brings a mathematical perspective as a starting point, because it illustrates practical guidelines that help build understanding of other traditionally technical areas.
In the 1948 U.S. presidential election between Thomas Dewey and Harry S. Truman, the polls predicted the results so poorly that a smiling Truman was captured in a now iconic photograph holding up a newspaper trumpeting the incorrect outcome (Figure 1).
What went wrong in this case? The conclusion of many commentators at the time was that “non-representative sampling” caused the incorrect result be predicted. This may or may not be true, and we will revisit this case later in the article. How do voting polls relate to particle size analysis? Noted engineer and statistician W. Edward Deming, a sampling pioneer, and his mentor Walter Shewhart spent a lot of time studying poll analysis in the 1930s (before the Truman election referred to above). These two individuals are well known in the area of quality control circles. So did they get it wrong when it came to the 1948 election?
The answer is “of course not.” In both voter polling examples and in particle size distribution examples in solids handling, the math is relatively simple, and the explanation of what happened requires a look at the concept of standard error (the standard deviation of the sampling distribution).
Standard error in sampling
The standard error is proportional to 1/√n, where n is the number of people interviewed, or the number of experiments, or the number of particles in a particle-size distribution, and so on. Standard error (SE) is the measured standard deviation around a mean value for repeated samplings. The larger the size of the sample that is taken, the nearer the measured result will come to the correct result or “truth.” The rule of thumb for these situations is that there is a roughly two-thirds (68%) probability that the “truth” (actual value) for a given sample will lie within ±1 SE of the measured or sampled mean.
The following example can help illustrate this. Imagine that an election is close — 52% to Candidate A and 48% to Candidate B. It is not possible to determine the actual outcome of the election before all the votes are counted, but by taking a representative sample of the voting results, we can predict the ultimate outcome before all the votes are in. Now imagine a random and representative sample of 1,000 voters in this fictional election. This number sounds a lot — at least it will be a lot of work for the company conducting the voter interviews.
In this case, the SE would be 1/√1,000, or approximately 3.2%. This means that the measured value will have an error associated with it of around 3%. For a normal (Gaussian) distribution, then 68% (roughly two-thirds) will lie within±1 SE of the measured mean. Thus the “truth” will be within this range 68% of the time. For this election example, it means that the margins of error for the 1,000-person representative sample would be 48±3% and 52±3%. In a tight election, this error margin is enough to swing the election the other way. And this margin or error represents only one standard error. To achieve 95% confidence, we would need ±6%. Many elections are decided by smaller margins than this. To reduce the margin of error to ±1%, we would need to have a random and representative sample size of 10,000 people. So whether you are interviewing voters or analyzing a particle size distribution in a solids process, the prerequisite question before sampling begins must be “What precision is required?”
The error margin discussed previously assumes a perfectly representative sample, but what if other factors are at work to increase the error? Continuing with the election example, the inaccuracy of the results can grow substantially if the sample is not truly representative, or if the election polls themselves influence the number of people that vote in the actual election, or if people in the sample have lied to the interviewer about how they placed their vote.
For an example of bias, consider another example. Imagine that pollsters decided to conduct the poll by making phone calls to 1,000 people listed in the telephone directory. Doing so would undoubtedly generate potential problems that would threaten the accuracy of the results. For example, voters in the Millennial generation would be underrepresented because they have lower rates of ownership of “landline” phones, favoring wireless mobile phones. Also, the time of day for the polling call could affect results – daytime calls would miss people who work away from home. Further, the sample would not include voters who could not afford a telephone.
Undecided voters can also affect errors because they may provide one answer in the pre-election interview, only to change their minds by the time they get to the election booth. The pool of undecided voters can be very large in some elections.
And if the sample of interviewed people is not representative in the first place, then the results will be worse. Switching to an analogy in solids handling, the phenomenon of segregation can cause the sample to be non-representative. When this is the case, we enter the “garbage in equals garbage out” scenario, and decisions may be based on faulty data.
Particle size analysis
For solids handling, the question now becomes how to link the sampling information to particle size analysis. With a particle size distribution, it is tempting to think that we can specify it exactly with a sample. The key word is “distribution,” which implies that there is a true value with an associated margin of error. As geochemist F.J. “Father” Flanagan stated in 1979 , most of the methods relating particle size, sampling errors, and the amount of analytical sample to be taken may be used to do the following:
- Calculate the error that might be incurred, given a specified weight of sample
- Specify the sample amount to be taken to keep sampling errors at or below some level
- Specify the grain size to which a sample must be crushed so that significant sampling errors may be avoided
Flanagan further states: “Calculations by most methods should show that errors due to sampling heterogeneous materials may be ignored if the material is powdered to pass a 200# (75 μm) sieve or, for Kleeman (1967), a 230# (63 μm) sieve.” We will examine why this is the case when we address the minimum mass question subsequently.
First, consider a “worst-case” situation. Imagine that we want to specify the x99 point of the size distribution to a precision of 1%. The term x99 refers to the point in the size distribution at which 99% of the particles are smaller than a given size (x) and 1% are greater. Related terms could be x90 (90% less than the size x, and thus 10% greater); x50 (50% above and 50% below size x: the median), and so on.
What implications would this have? As stated previously, the standard error is inversely proportional to the square root of the number of particles (S.E. ∞ 1/√n or n ∞ 1/σ2). For a 1% S.E., n = 1 / (0.01) 2 = 10,000. Therefore, a total of 10,000 particles total would be needed to specify the mean to 1% S.E. To specify the x99 to 1%, 10,000 representative particles would be needed in the x99+ point of the distribution. These 10,000 particles only make up 1/100th of the total mass of the system (because 99% of the volume and mass is below the x99 point). The total minimum mass could therefore be found by calculating the mass of 10,000 particles at the x99+ part of the distribution and multiplying by 100 to arrive at the total minimum mass.
To calculate masses, an assumption must be made about particle shape, so a starting point could be the sphere (the most compact shape). The volume of a sphere is (π/6) × D3 and the mass of a single particle is the volume multiplied by the density. We can calculate the mass of 10,000 particles of the appropriate density at a known or assumed x99 point. It is straightforward to spreadsheet the calculation for silica, for example (~2,500 kg/m3). The cgs (centimeter, gram, second) unit system is slightly easier to use in this regard than the SI mks (meter, kilogram, second) system (Table 1).
This now gives us a convenient framework to estimate the minimum sample masses required for any degree of precision that is desired. Note that we need roughly 1 g of sample at 100 μm (for the x99 point) and that many instrumental techniques take ~1 g or less of sample. Thus, the earlier comment by Flanagan is easy to understand in this context. Also, it is important to note that the largest particles are the most important in sampling issues, and attempts to find a small number of large particles may be akin to the “needle-in-a-haystack” situation.
However, if the x90 (rather than x99) is required to achieve a SE of 5%, then the minimum mass in the last column is reduced by a factor of 250 (the x 90 point, which represents 10% of the total mass of the system). G. Herdan  states: “Sampling by small weights is notoriously conducive to throwing overboard larger pieces” and “… the smaller the weight, the greater, on the whole, the shift in the distribution mode or peak toward the smaller particles.”
For historical reasons, the amount of material needed for statistical validity in sampling has been underestimated. We can set this issue at the foot of Robert Hallowell Richards, an MIT professor. In his classic set of four volumes on ore dressing published in 1908 , there is a very important table (Figure 2) and a comment that have had a great influence throughout the history of sampling. It is worth examining this table and comment in detail.
Richard’s comments underneath the table are the following:
- “The above rule demands finer crushing than practice deems to be necessary and it is therefore more expensive than it is wise”
- (And later on in the chapter) “By adopting the rule that the weight shall be proportional to the square of the largest particles, we obtain a set of figures that will in all probability meet the approval of practicing engineers…”
So Robert H. Richards is essentially stating that weight is proportional to the square of the diameter, and it was this that set sampling theory back for over 50 years, until French statistician and chemist Pierre Gy was able to retrieve the statistical situation in the 1950s and onward.
The powerful nature of Gy’s statistical approach, known as “Theory of Sampling” (TOS), is that it allows us to calculate the best standard error or variability based solely on the heterogeneity of the material. This is simply a reversal of the minimum mass calculation described earlier.
Consider this real-world example. A pharmaceutical manufacturing engineer believed that she had up to 1% of 2,000 μm particle size material in a pharmaceutical slurry containing 20% solids by volume. She wanted to be 99% confident of this assertion. Using a spreadsheet for the calculation as earlier, and assuming a density of 1.5 g/cm 3 for the pharmaceutical solid, the results would be those found in Table 2.
The calculated mass is around 6.3 kg, which represent more than 30 L of 20% slurry. This result can be considered in another way: as the minimum amount of material required to have homogeneity at the 99% confidence level. Each 30 L portion will be statistically equivalent (99% confidence limits) or homogeneous. Even if we are sampling 100 mL at a time (300 samples), we would expect 299 negative tests and only one positive. A single test, or even 10 repeated samplings, could not state with statistical certainty that 1% of the universe of material was at 2,000 μm particle size. The entire 30 L needs evaluating.
This is what statistical quality control is about, hence the use of process analytical technology (PAT) and on-line/continuous verification of production uniformity in the pharmaceutical arena. When informed that 30 L of slurry was required to meet her requirements, the aforementioned pharmaceutical engineer replied, “I can only afford 20 mg for a test.”
A reverse calculation shows the best standard errors based on 20 mg of solid material. The concept is to calculate the mass of a single particle: [π/6] × D3 *Þ, where Þ is the density of the sample — and then to calculate the number of these particles that fall into the appropriate part of the size distribution. Thus for 20 mg of total sample, the x99+ point would represent 1/100 of this total mass (0.002/100 or 0.00002 g). If we calculate the number of appropriately sized particles in this amount of sample, we would have the relationship between the number of particles and the standard error (S.E. ∞ 1/√n or n ∞ 1/ó2) (Table 3).
For a 2,000 μm particle size and 20 mg of sample, the best that can be achieved is 560% standard error because a 20-mg sample of material does not, on average, contain even a single 2,000-μm particle at the x99+ point. This would be considered a specimen, rather than a (representative) sample. If this large particle creates a difficulty for the process or product, it would be far better to try to filter it out before the point of use than to try to “inspect it in.”
Another practical example arose when lunar regolith and simulants were measured for particle size distribution. Because the “moon dust” is a national treasure, researchers are only permitted small quantities to measure. For an x95 point of around 450 μm (a realistic point for the top end of lunar regolith), the best standard error achievable for 20 mg of sample (Þ ~2.73 g/cm 3) is about 36%. It’s no wonder that large variations were seen in replicate particle size determinations.
Sample errors rise markedly as the size of the solid increases – a result of the fact that mass is a cubic function of size. Again, we see the importance that small numbers of large particles may have on the calculated size distribution and standard error.
As South African mine engineering professor RCA Minnitt put it when talking about sampling for mining applications: “… However, that’s not all. At the end of a very careful preparation protocol, the analyst will extract exactly a 30-g aliquot of material from the finely milled powder for fire assay. Twenty-seven such assays would amount to just 810 grams of rock powder. One metric ton (m.t.) is equivalent to 1,000,000 g, so 810 g is 0.00081 m.t. to evaluate 67 m.t. of rock.” Minnitt goes on to say that “Putting that into context, this means that we are trying to represent the 9,000 billion cm 3 in this room using a volume about the size of a pinhead.”
Pierre Gy’s Theory of Sampling (TOS) school has two camps in defining this variability. Some prefer using the term “error,” because it implies that someone is culpable (usually management who has failed to deliver the tools for the job and has over-specified the needs), while others prefer using the term “variability” to express the idea that the standard error numbers reflect the natural variation in the system that cannot be improved upon.
In the author’s work with clients, it is often observed that they want to use only a tiny amount of their “expensive” material as a sample. Using this small amount is akin to sampling the contents of a desk by removing a single pen. If a smaller-than-the-minimum mass is taken, the margin of error increases. The only reasonable and logical action, then, is to widen the specification to accommodate the increased margin of error. However, few clients are receptive to the widening specifications. This situation brings back the “garbage in equals garbage out” rule.
The outlined standard error calculation provides the minimum error based solely on the heterogeneity of the material. All other errors add to this minimum error. Pierre Gy (who sadly died on November 5, 2015) listed six other errors aside from this fundamental sampling error (FSE). These include the “nugget effect” from geo statistics — it is equivalent to the undeterminable x100. It also includes delimitation errors, where certain particles have lowered chances of entering the measurement zone — obviously each particle needs an equal chance of being sampled for statistical validity. There is also the analytical error — normally at least two orders of magnitude lower than the FSE. And there are others that will not be discussed here, except to remind you of segregation, where a representative sample simply isn’t possible unless the whole sample mass is taken. This is the classic “Brazil nut effect,” also known as granular convection. This occurs when granular material of varying sizes shows patterns of movement similar to fluid convection when subject to vibration or shaking, and the largest particles end up on the surface. An example is a container of mixed nuts, where the Brazil nuts (usually the largest sized) end up on top.
The classic question comes down to how to take a sample containing the minimum mass required. Even with careful sample division, the minimum number of particles must be met at the particular point in the distribution. Otherwise, we cannot achieve low sample-to-sample variation. There are two golden rules of sampling expounded in Refs. 1 and 3.
- All samples should be taken when the stream is in motion, rather than stationary
- The objective should be to sample the entire stream for a given period of time, rather than only a portion of the stream for a long time
Allen has provided a classic table in all his editions of his “Particle Size Measurement” books, and it is recreated here in Table 4.
Only with the spinning riffler technique do we get below 1% standard deviation between repeated samples. Indeed, this principle was known in early mining days where wooden buckets and barrels were employed (Figure 3).
The principle of these rotary dividers is also seen in larger-scale variants that retain classic good sampling practice, such as the Vezin sampler and the Burt sampler for slurries and suspensions.
In the slit-type devices favored by the coal industry among others and exemplified in a number of ASTM Committee D-5 standards are the Jones riffler and its variants (Figure 4).
All the above considerations can be obviated if a sample segregates. We may all be familiar with the phrase “The contents may have settled during transit” and the bottom of the potato chip bag containing fragments rather than whole chips. These are simple examples of segregation where one part of the sample is markedly different from another even though each could contain far more than the minimum required sample mass. Early mining illustrations show this segregation and a nice sampler from Jenike and Johanson also illustrates the so-called “Christmas tree effect” (Figures 5 and 6).
In these cases, the only route is to take the entire material, while moving (dropping under gravity from a conveyer belt) and extract a portion of the whole flowing stream for a period of time. This is why samples from mining operations were substantial fractions of railway wagons or large numbers of alternate shovels-full.
To summarize, the analytical laboratory can only be as good as the submitted sample. The origin and transport of the material to the laboratory is crucial in understanding potential sampling errors that may arise. To finish with another quote from the realm of early computing pioneer Charles Babbage:
“On two occasions, I have been asked, ‘Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?’ In one case a member of the Upper, and in the other a member of the Lower House put this question. I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question” .
Edited by Scott Jenkins
1. Allen, T., “Particle Size Measurement” 4th ed., Chapman & Hall, London, U.K., 1990.
2. Minnitt, R.C.A., Sampling: the impact on costs and decision-making, Analytical Challenges in Metallurgy, The Southern African Institute of Mining and Metallurgy, Randburg, South Africa, November 2006.
3. Warwick, A.W., “Notes on Sampling,” Industrial Printing and Publishing Co., Denver, 1903. This book is based on unpublished notes from Henry Vezin after his death.
4. Flanagan, F.J., Reference Samples in Geology and Geochemistry, U.S. Geological Survey Bulletin: 1582, 1986.
5. Herdan, G., “Small Particle Statistics,” 2nd revised ed., Butterworth, Oxford, U.K.,1960.
6. Richards, R., “Ore dressing,” (4 vols. plus index), McGraw-Hill, New York, 1908. See table on p. 850, vol. 2.
7. Brunton, David W., Modern Practice of Ore Sampling, Mining Science, Sept. 1909, pp. 198–200 and 220–224.
8. Babbage, C., “Passages from the Life of a Philosopher,” Pickering and Chatto, London, U.K.,1864, p. 67.
Alan Rawle is the applications manager at Malvern Panalytical Inc. (117 Flanders Road, Westborough, MA 01581; Phone: + 1 508 768 6434; Email: firstname.lastname@example.org; Website: www.malvernpanalytical.com). Rawle has more than 30 years of experience in various aspects of science and technology, and has been with Malvern since 1990. Rawle has also worked extensively on the ISO TC24/SC4 (Particle Characterization) standardization committee, assisting the writing of documentary standards in light scattering, small-angle X-ray scattering, image analysis, zeta potential and dispersion. He has also cultivated an interest in the theory and practice of sampling, and delivers short courses regularly at Pittcon on particle sizing techniques and sampling. Rawle is co-chair of E 56.02, the Characterization Subcommittee of the ASTM E56 Committee on Nanotechnology and is also a member of six other ASTM committees. He holds a bachelor’s degree in industrial chemistry and a Ph.D. in supported alloy catalysts, both from Brunel University (London).
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