Properties of variances

Most of my life I have worked with William Volk’s Applied Statistics for Engineers. At present I work with his 1980 Reprint Edition. I lost the 1969 Second Edition while I was preaching sound sampling practices and applied statistics around the world. I’m hanging on to my tattered 1958 Original Edition. Volk’s name translates into the Dutch word for “nation” or “people”. That led me to believe William Volk and Jan Visman may share the same roots. Volk holds a 1959 masters degree in mathematical statistics from Rutgers University and an undergraduate degree in chemical engineering from New York University. So he must have written much of his Original Edition before graduating from Rutgers. I took a real liking to Chapter 7 Analysis of Variance. What I like most of all is Section 7.1.4 Variance of a general function. For it was in this section that Volk proved that each function ought to have its own variance.

Volk’s grasp of the properties of variances shows how inspired he was by Fisher’s work. Probability theory had spawned applied statistics by the time Sir R A Fisher was knighted in 1953. And it was the concept of degrees of freedom that empowered applied statistics and set it apart from probability theory. Fisher in 1922 introduced the concept of degrees of freedom to correct Pearson’s χ²-distribution for finite sets of measured values. It did bridge the breach between probability theory and applied statistics. This is why applied statistics deals with finite samples selected from sampling units or sample spaces. What degrees of freedom also did at that time was fuel the legendary feud between those giants of statistics. Fisher was right because the F- and t-distributions both derive from the χ²-distribution once degrees of freedom are taken into account. Volk’s 1958 textbook is of lasting value because it links χ²-, F-, and t-distributions in such a logical manner.

Volk’s symbols and terms are mostly clear and concise. I found Volk’s “central tendency measures” less intuitive than “central values” (of sets of measured values with either constant or variable weights). I avoid terms such as “successive observations” when discussing an ordered set of measured values of a stochastic variable in a sampling unit or a sample space. All it takes in my work is text and context to correctly explain applied statistics and its symbols.

Volk applied Fisher’s F-test to verify whether or not a pair of variances is statistically identical. He applied Bartlett’s χ²-test to verify whether or not a set of variances is homogeneous. He did not show how to apply Fisher’s F-test to verify spatial dependence between measured values in ordered sets. All it would have taken is to apply Fisher’s F-test to var(x), the variance of a set of measured values, and var1(x), the first variance term of the ordered set. Volk, a chemical engineer, may well have worked with some ordered set of measured values in a sampling unit or a sample space of time but he never showed how to derive a sampling variogram.

John von Neumann was a brilliant mathematician at Princeton’s Institute for Advanced Studies when he coauthored Distribution of the Ratio of the Mean Square Successive Difference to the Variance. He seemed unaware in 1941 that a set of n samples gives df=n–1 degrees of freedom, and that an ordered set of n observations gives dfo=2(n–1) degrees of freedom. Had he added all of the terms x1–x2,…,xi–xi+1,…,xn-1–xn, he would have gotten x1–xn, the nth variance term of the ordered set. Had he counted degrees of freedom, he would have gotten the correct number for the ordered set. He may not have noticed that all but x1 and xn are used twice.

Von Neumann deemed working with random numbers a sin of sorts. It explains why he frowned upon heuristic proof. In those days, random numbers were listed in handbooks of statistical tables. That made the mean squared successive difference of a set about as tedious to derive as its variance. He was a pure mathematician, which may well be why his 1941 study did so little to advance mathematical statistics.

Anders Hald, a Professor of Statistics at the University of Copenhagen, pointed out that the correct number of degrees of freedom for the first variance term of an ordered set of n measured values is dfo=2(n–1). He did so in Section 13.5. The Mean Square of Successive Differences of his 1952 textbook on Statistical Theory with Engineering Applications. Hald, too, studied the distribution of r=var1(x)/var(x) rather than Fisher’s F-distribution. Otherwise, he would have noticed that a significant degree of spatial dependence between measured values in some ordered set gives an observed value of F=var(x)/var1(x)>1.

Textbooks on applied statistics such as Volk’s give a table with F-values at 0.05 and 0.01 probability for a matrix of degrees of freedom. Nowadays, Excel’s FINV makes it easy to get the correct F-value at any probability level and with any number of degrees of freedom for either variance. What’s more, Excel’s RAND makes it simple to prove that Standard Uniform Random Numbers (SURNs) and Normally Distributed Random Numbers (NDRNs) do not display a significant degree of spatial dependence. Visit and find out about SURNs and NDRNs under Sampling and statistics explained.

Not all geoscientists know how to test for spatial dependence by applying Fisher’s F-test. In fact, geostatisticians would rather assume than test for spatial dependence. They have also been taught that some functions do not have variances. The problem is that too many geoscientists know too little about sampling and statistics and too much about surreal geostatistics.

Counting degrees of freedom

Geoscientists do not count degrees of freedom quite as well as do statisticians. Way too many have been taught some sort of new of science where spatial dependence need not be verified but may be assumed. Many geoscientists do not grasp why the properties of variances and the concept of degrees of freedom cannot be ignored with impunity. All the same, the world’s mining industry accepted this substitute for applied statistics because it does work miracles with a few boreholes drilled some distance apart.

Professor Dr Georges Matheron (1930-2000) developed his new science at his Centre de Géostatistique (CG) in Fontainebleau, France. The CG has posted much of his seminal work with its On-Line Library. I unscrambled it in Sampling and Statistics Explained, Chapter 2 Sampling Theory. In his 1954 Note Statistique No 3, Matheron derived the length-weighted average grade of a set of core samples with variable lengths but did not derive the variance of this weighted average grade. In his 1960 Note Géostatistique No 28, Matheron derived the length-weighted average grade of a block of in-situ ore but did not derive the variance of this weighted average grade either. Matheron made up the eponym “krigeage” in this 1960 paper to honor D G Krige for his work with weighted average gold grades at the Witwatersrand reef complex in South Africa.

Matheron’s “estimateur” in his 1960 paper turned out to be the length-weighted average grade of a three-dimensional block. Professor Dr Michel David (1945-2000), the author of the first textbook on geostatistics, found infinite sets of kriged estimates. Journel and Huijbregts, the authors of the second textbook, determined that infinite sets of kriged estimates give zero kriging variances. As luck would have it, each infinite set of zero-dimensional points fits in any three-dimensional block. Unluckily, neither the weighted average grade of a zero-dimensional point nor the weighted average grade of a single three-dimensional block does have a variance.

Functionally dependent values such as arithmetic means and weighted averages are not awarded degrees of freedom. By contrast, measured values do give degrees of freedom. It makes no sense whatsoever to test for spatial dependence in a sample space comprised of the first five positive integers of the infinite set. The probability to draw some subset of the infinite set of positive integers is just as immeasurable as the probability to select the least biased subset of an infinite set of kriged estimates. Geostatistians somehow beat such astronomical odds and select so-called BLUEs (Best Linear Unbiased Estimators) of infinite sets of kriged estimates.

Games of chance in the real world are quite different because we need to define some finite sample space of positive integers. For example, the ubiquitous 6/49 lottery is based on selecting a subset of six (6) integers from a sample space that consists of a subset of the positive integers 1, 2,…, 48, 49. The probability to win is P(win)=1/49·1/48·…·1/45·1/44≈ 0.000,000,00001 whereas the probability to lose is P(lose)=1–P(win)≈0.999,999,9999. Such probabilities are cast in stone because games of chance do give discrete outcomes. Unlike sampling practice where measured values are continuous and degrees of freedom are cast in stone. For example, Visman’s sampling experiment gives the arithmetic mean ash contents on dry basis for paired sets of small and large increments and the number of degrees of freedom. In symbols, a set of n measured values gives df=n–1, and the first term for the ordered set gives dfo=2(n–1). The number of degrees of freedom for a set of measured values with equal weights is a positive integer. For a set of measured values with variable weights, the number of degrees of freedom becomes a positive irrational.

In my 1984 textbook on Sampling and Weighing of Bulk Solids I explain how to derive the variance of the mass of metal in a quantity of mineral concentrate from its wet mass, moisture content and metal grade. This methodology requires realistic variance estimates for wet mass, moisture factor and grade factor. The interleaved sampling protocol gives realistic variance estimates at the lowest possible costs. ISO TC183 Copper, lead and zinc concentrates incorporated the methodology in CD13543–Determination of Mass of Contained Metal in the Lot. My son and I did not know the trouble we would see when we derived the variance of the mass of metal in a quantity of in-situ ore from its volume, density, and metal grade.

Coal Feeder for Gasification Plants

Dear Bulkaholics,
I am a new member of your community. My business is Coal to Liquids plants. Is there anyone with experience in feeder technology for coal gasifiers? The problem is to feed dried coal (especially in the case of lignite) into the pressurized (Siemens or Shell) gasifiers. The train: Coal Crushing, Coal Pulverizing, Coal Drying, Coal Bunkering, Coal Feeding has to be optimized and a save handling is required (preventing coal dust explosion). I am interested to hear from someone.     

Leaky Blower Safety Relief Valves in Pneumatic Conveying Systems

Do you have money to burn?? Of course not, yet chances are that you are doing just that within your pneumatic conveying system.? Have you ever noticed the hissing sound of such valves when you tour your plant? If so, you are wasting electric energy and maybe risking future maintenance issues.? I cannot tell you how many cement and food processing facilities I have been to, where not at least some valves leaked severely.? You may think the cause is the valve itself. Think again:? Most likely it is your system that does not need all the air and hence is blowing it off through the valve.? The fix is not necessarily a new valve, but either a new belt drive or motor speed slow-down by means of a variable speed drive.

You do not need to be a blower expert to be able to troubleshoot this issue.? Ask your plant operators and production personnel if any product quality or process problems exist in the area you are reviewing.? Guess what, you can turn down the blower speed, if your valves blow off air from conveying (or aeration) systems that are working fine.?The problem assessment step is the most important one; you can leave it to your system OEM, or the blower manufacturer to help you figure out how much to turn down the speed.

Recently?I was invited to review a 75HP vacuum blower system that we were able to slow down by 20%.? Based on 24/7 operating service and the local electricity rate the customer was able to save over $7,000 per package – they had a total of 8 (!).? This is not only more money in your pocket, but it is also easier on the blowers and drives.  Besides of increasing the total service life, this measure also helps reduce the overall CO2 envelope of your plant – you can help the environment by using less energy.

Especially positive pressure relief valve create another issue in plant areas where blowers are installed in building or sound enclosures:? The hot air escaping the valves literally super heats the area around the blowers.? This tends to cause long term reliability issues as the hotter than normal blower intake air lets the machine run even hotter.? Lube oil viscosity decreases, which can cause premature bearing failures.? And it usually reduces the oil service life too (mineral oil in particular).? Elastomeric (rubber) blower shaft seals tend to harden, which can cause oil leaks:? Catastrophic failures follow quickly.? I am sure you get the idea, it is even more money in your company’s budget.

Ralf Weiser

Testing for spatial dependence

When I went from sampling shipments of coal, concentrate, potash and sulphur in the Port of Vancouver to sampling coal, concentrate and ore at Cominco’s operations in Canada and abroad, the concept of spatial dependence in sampling units and sample spaces started to grow on me. In March 1978, SGS had send me a draft of Gy’s Unbiased Sampling from a Falling Stream of Particulate Matter, and asked my opinion on its content and language. I took the task seriously not only because SGS wanted to distribute Gy’s paper among selected clients but even more so because I was a member of the Canadian Advisory Committee to ISO Technical Committee 102 on iron ore. The objective of Gy’s experiment was to derive the optimum width and speed of the primary sampler as a function of the top size of the material in bulk. His experiment was technically brilliant but its symbols and terms were characteristically his own. I defined accuracy and precision my way, and Pierre sent me an autographed copy of his 1979 Sampling of Particulate Materials: Theory and Practice.

At Cominco I met many a geologist and metallurgist who struggled with spatial dependence between ordered sets of measured values, and who bought all kind of textbooks for guidance. Autocorrelation is a somewhat dated term that implies a significant degree of associative dependence between measured values in ordered sets. In the Index of his 1979 textbook Gy does not refer to autocorrelation, associative dependence, or to degrees of freedom for that matter. In the Index of their 1976 Time Series Analysis, Box and Jenkins refer to autocorrelation function but not to associative dependence or degrees of freedom. They worked mostly with sets of measured values ordered in the sample space of time, and with covariances between measured values in ordered sets. Degrees of freedom were mentioned only when they discussed F-, t- and χ²-distributions in the text. What these authors did not do was apply Fisher’s F-test to two variances to verify whether they are statistically identical or differ significantly.

Any set of ordered data may be used to show how to apply Fisher’s F-test. For example, the variance of a data set that consists of the numbers 1, 2, 3, 4 and 5 equals var(x)=2.50. The first variance term of the ordered set equals var1(x) =∑(ni-ni+1)2/2(n-1)=4/8=0.50. The observed F-value of F=2.50/0.50=5.00 exceeds the tabulated F-value of F0.05;df;df(o)=3.84 at 5% probability. Hence, the ordered set displays a significant degree of spatial dependence in its sample space. Set up a simple Excel spreadsheet template and derive this F-statistic. Use Excel’s FINV-function with p=0.01, df=4 and df(o)=8 to find out if the observed F-value is significant at 1% probability. This example shows why degrees of freedom play a key role in Fisher’s F-test.

Some scientists are taught to assume spatial dependence rather than verify it by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set. It sounds convenient but makes bad science. Others may not know how to derive a sampling variogram, a simple graph that shows where spatial dependence in a sample space or sampling unit dissipates into randomness. This is why I want to show how to derive and interpret sampling variograms. It may not make our world a cooler place but we can measure how hot is too hot in a scientifically sound manner.

Expanding Conveyor Technology

Through study analysis and experience the writer will attempt to continuously rationalize and expand the conventional conveyor technology revealing the link between theory and practical issues and in the process, through a deeper understanding, take the technology beyond its presently perceived limits. This approach has, in the past 20 years, yielded a greater understanding, a broader application of the principles, and even new technologies to the market place.

This Post:
Experiences/Frustrations with Conveyor Belt Specs:

Why don’t Belt Manufacturers understand what they publish?


At Dos Santos International we are experts in the Sandwich Belt High-Angle Conveyor Technology. The Sandwich belt system works on the principle of hugging bulk material continuously in a sandwich between two smooth surfaced rubber belts. Hugging pressure on the bulk material develops its internal friction, which resists any back sliding tendencies, allowing the material to convey at any high angle up to vertical. The writer rationalized this technology as an expansion of the conventional conveyor technology during the period 1979 thru 1982. This work culminated in the landmark article “The Evolution of Sandwich Belt High-Angle Conveyors” by Dos Santos and Frizzell.

All Dos Santos Sandwich Belt Conveyors start with a troughed bottom belt that receives the bulk material load in a conventional manner. The bottom belt is joined by a top belt which sandwiches the bulk material between. The belt sandwich is then supported along a convex curve of inverted troughing idlers along which the conveying angle is increased up to the ultimate high angle. In the case of the DSI Snake Sandwich System, shown on Figure 1, the profile is made up of alternating convex curves where the inner belt is supported on the convex curve of troughing idlers and the outer belt hugs itself and the conveyed material up against the inner belt according to the relation:

Pr = T/R,

where Pr is a radial load, T is the belt tension and R is the radius of curvature.

A moment is induced at the troughed belt section according to the equation:

M = EI/R,

where M is the moment on the troughed belt section, E is the elastic modulus of the belt in the warp direction, I is the belt section moment of inertia.

Belt stress due to the induced moment is:

Fb = My/I = Ey/R,

where y is the distance from the troughed belt section’s neutral axis.

Since a conveyor belt cannot be subjected to compression, as it will buckle, at a minimum the belt tension must counter the compressive bending stresses, at the inside of the curve. Furthermore, when the belt tension is added to the tensile bending stresses, at the outside of the curve, the combined stress must not exceed the belt’s tension rating.

So, the induced bending stresses are directly related to the belt’s elastic modulus. The lower the elastic modulus the lower the induced bending stresses, permitting tighter convex curves and a more compact transition from the low (conventional) loading angle to the ultimate high angle. Nylon warp fabric belting offers the best solution for tight convex curves.

Indeed, these curvature constraints apply to all convex curves along the DSI Snake Sandwich conveyors. These curvature constraints also apply to convex curves along conventional conveyors though in this case there is typically no great incentive to make such curves tight.

The curvature constraint equations for troughed belt conveyors, based on the basic equations above, are published in the engineering manuals of all major belt manufacturers. The all important Belt (elastic) Modulus is determined for the belt’s long term behavior according to the ISO 9856 Belt Modulus test.

Because of its importance, Dos Santos International always strictly specifies the belt modulus (not to exceed) value. Such specified values are typically comfortably above those already published by the belt manufacturers.

In this light it is frustrating to find, after its manufacture, that the belting we ordered exceeds the specified belt modulus and even more frustrating when the manufacturer claims that they guarantee its performance. Indeed performance is guaranteed to fail in such a case unless measures are taken to compensate for the higher modulus, such as increasing tension to offset the higher induced compressive bending stresses. Higher tension however, may not be possible if such, when combined with the already higher tensile bending stresses, exceed the belt’s tension rating. Indeed DSI design criteria attempts to allow ample margin in case of such mishaps which occur all too often.

Such a mishap, recently, is the source of my frustration and prompted this writing

Visman’s sampling experiment

Visman’s work is based on the additive property of variances. His sampling experiment showed that the variance of the primary sample selection stage (the sampling variance) is the sum of the composition variance and the segregation variance. When I met Visman for the first time in Canada in the 1970s, we talked about his sampling theory. He agreed that the adjective segregation seems to suggest that some quantity of coal may have been more homogeneous at an earlier stage, and that term distribution variance more succinctly describes this component of the sampling variance. Thus, the composition variance is a measure for variability between particles in primary increments, and the distribution variance is a measure for variability between primary increments in the complete set that constitutes the sampling unit. I work with the distribution variance because it is an intuitive measure for intrinsic variability in sample spaces such as in-situ coal seams and ore blocks.

I was aware that ASTM Committee D-5 on Coal and Coke wanted to include Visman’s sampling experiment in ASTM D2234 Standard Practice for Collection of a Gross Sample of Coal. In fact, this ASTM Standard was the first internationally recognized document that specified the precision for ash content. Visman’s sampling experiment with small and large increments may still be found in Annex A1. Test Method for Determining the Variance Components of a Coal. It is based on taking pairs of small and large increments side-by-side from a stopped conveyor belt such that pairs are evenly spaced in the sampling unit. Each large increment was selected with a sampling frame, and its paired small increment was taken next to that frame. Each increment was weighed, air-dried, prepared and tested for ash on dry basis. In those early days, the variance of the set of small increments was called random variance, and the variance of the set of large increments was called segregation variance.

What ASTM D2234 did not determine but Visman defined in his 1947 PhD thesis is the variance of sample preparation and analysis. This variance is small when compared with the variance of the primary sample selection stage. In fact, Visman’s C turned out to be about 5%. It is possible to optimize sampling protocols by applying analysis of variance to the variances of the primary sample selection stage, the sample preparation stage, and the analytical stage.

But there’s so much more to Visman’s seminal sampling experiment than meets the eye. For example, Annex A1 in ASTM D2234 reports highly variable weights of both small and large increments. So much so that the mass-weighted average dry ash contents and its variance make more sense than the arithmetic means and its variance. Visman himself knew how to derive the weighted average and its variance but ASTM D-5 kept it simple. What’s more, degrees of freedom for sets of measured values with variable weights are positive irrationals rather than positive integers. This fundamental concept in applied statistics and sampling practice is most annoying to those who want to do more with less and think degrees of freedom are for the birds. But I digress!

Mechanical Innovations

Dear Sir,

I read much of the Powder/Bulk Portal and the Bulk On-line is on my computer desktops. I commend your site as one of the best sites to gain contacts and information.

I do not have an engineering degree but was for many years the senior conveyor technologist supervisor with Ace Conveyor Services (Australian Conveyor Engineering), later (Continental Ace Services) (Continental Conveyor & Equipment) [Australia] and have worked in the mining industry since 1950s.

I have quite a number of conveyor and other mechanical innovations in my repertoire with the latest “patent application” throughout the PCT and remaining major countries of the world. The innovation is viewable on the company’s website, which is the company in which I hold a 25% shareholding.

The innovation has now been sold into major coal mines with other major Coal Mine operators arranging site measuring to retrofit into their existing systems. One Port Authority Services in Australia have purchased Stainless Steel units for their surge bins and further orders to be negotiated in the next financial year’s AFCs. Other Bulkcoal Port Authorities have been measured to retrofit into their Yard, Stacker/Reclaimers, Ship Loader and all Feeder conveyors. The Bulk handling Port Authorities of the Queensland Government and other private owners have expressed interest in using the products. Expressions of interest from major materials handling conveyors miners throughout the world and local in-country conveyor manufacturers attest to the quality of the innovation. The initial models have been in service since March 2006 flawlessly and many more are being installed to complete the conveyor system.

About the innovation. The design is the culmination of my desire to address the OH&S issues that plague the present conveyor systems with hazards involved in conveyor idler roll maintenance and also to ease and reduce the plant, equipment but mainly manpower required to maintain conveyor idler roll changeout/s. The ‘NO Bullshit’ spiel is available to any prospective clients but the technology is under Standard Patent Applications and as such is of a confidential nature.

I am also enquiring as to how one might sell the IP as this is the first that I have started commercially exploiting. Any comments or information would also be greatly appreciated.

I look forward to hearing replies from Bloggers.


Leslie (Les) D. Dunn
TECMATE Mine Services Pty Ltd
Ph/Fax: 07 4984 9122 Mobile: 0417 619 362


Serendipity played a key role in building a bridge between sampling theory with its homogeneous populations and sampling practice with its heterogeneous sampling units and sample spaces. Jan Visman, a mining engineer at the Dutch State Mines, surfaced after the Second Word War with a massive amount of test results determined in samples selected from heterogeneous sampling units of coal. Visman had gathered so much valuable information that he was encouraged to write a PhD thesis on the sampling of coal. Dr Jan Visman defended his PhD thesis titled “De Monsterneming van Heterogene Binomiale Korrelmengsels, in het Bijzonder Steenkool” at the Technical University of Delft on December 17, 1947.

Visman proved that the variance of the primary sample selection stage (the sampling variance) is the sum of the composition variance and the distribution variance. The composition variance is a measure for variability between particles within primary increments. In contrast, the distribution variance is a measure for variability between primary increments in a sampling unit, and, thus, for its degree of heterogeneity. He described a simple experiment to estimate the composition and distribution components of the sampling variance.

Visman worked briefly in Ottawa after immigrating to Canada in 1951. Until his retirement in 1976, he headed the Western Regional Laboratories of the Department of Mines and Technical Surveys, which is nowadays known as Energy, Mines & Resources. These laboratories were located in Calgary until 1955 when they moved to Edmonton and are still operating alongside the Alberta Research Council at the Coal Research Centre in Devon, Alberta.

Visman wrote Towards a Common Basis for the Sampling of Materials, Mines Branch Research Report R 93, which was published in July 1962. He participated in ASTM Committees D-5 on Coal and Coke and E-11 on Statistics. ASTM D 2234 Standard Practice for Collection of a Gross Sample of Coal was the first internationally recognized standard to specify a precision estimate for a measured variable. Visman’s sampling experiment with small and large increments is described in ASTM D 2234, Annex A1. Test Method for Determining the Variance Components of a Coal. Visman’s paper titled A General Sampling Theory was published in the November 1969 issue of Materials Research & Standards.

After my transfer to Vancouver, Canada, in October 1969, I met Dr Jan Visman on a number of occasions. I shall remember him as the accidental sampling expert because his true calling was coal processing but he became interested in sampling because of his need to understand this process. His innovative work inspired me a great deal when I wrote Sampling and Weighing of Bulk Solids. His brilliant mind succumbed long before he passed away on February 19, 2006. Dr Jan Visman’s contribution to the bridging of the breach between sampling theory and sampling practice for material in bulk should be remembered.

Sound science in sampling and statistics

When I worked in the Port of Rotterdam many years ago, I wondered about the risk international commodity traders encounter because of small samples in sealed bottles. How could the quality of a cargo aboard a bulk carrier possibly be confined in a few little bottles? This question has preoccupied me since I found out that sampling and statistics are inseparable subjects in science and engineering. So much so that most disciplines teach the basics behind sampling and statistics. But not all! In fact, geostatistics is one discipline where inviolable requirements of classical statistics are routinely ignored. I shall explain why!

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