Table Of ContentMany areas of mining engineering gather and use statistical information, provided by observing the
actual operation of equipment, their systems, the development of mining works, surface subsidence that
accompanies underground mining, displacement of rocks surrounding surface pits and underground drives
and longwalls, amongst others. In addition, the actual modern machines used in surface mining are equipped
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with diagnostic systems that automatically trace all important machine parameters and send this information A
to the main producer’s computer. Such data not only provide information on the technical properties of the C
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machine but they also have a statistical character. Furthermore, all information gathered during stand and lab K
investigations where parts, assemblies and whole devices are tested in order to prove their usefulness, have M
a stochastic character. All of these materials need to be developed statistically and, more importantly, based .
on these results mining engineers must make decisions whether to undertake actions, connected with the C
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further operation of the machines, the further development of the works, etc. For these reasons, knowledge A
of modern statistics is necessary for mining engineers; not only as to how statistical analysis of data should be P
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conducted and statistical synthesis should be done, but also as to understanding the results obtained and how
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to use them to make appropriate decisions in relation to the mining operation. K
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This book on statistical analysis and synthesis starts with a short repetition of probability theory and also
includes a special section on statistical prediction. The text is illustrated with many examples taken from mining
practice; moreover the tables required to conduct statistical inference are included.
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STATISTICS FOR MINING ENGINEERING
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JACEK M. CZAPLICKI
an informa business
STATISTICS FOR MINING ENGINEERING
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Statistics for Mining Engineering
Jacek M. Czaplicki
Mining Mechanization Institute, Silesian University of Technology,
Gliwice, Poland
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CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business, London, UK
© 2014 Taylor & Francis Group, London, UK
Typeset by Vikatan Publishing Solutions (P) Ltd., Chennai, India
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All rights reserved. No part of this publication or the information contained herein may be
reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic,
mechanical, by photocopying, recording or otherwise, without written prior permission from the
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Although all care is taken to ensure integrity and the quality of this publication and the informa-
tion herein, no responsibility is assumed by the publishers nor the author for any damage to the
property or persons as a result of operation or use of this publication and/or the information
contained herein.
Published by: CRC Press/Balkema
P.O. Box 11320, 2301 EH Leiden, The Netherlands
e-mail: [email protected]
www.crcpress.com—www.taylorandfrancis.com
Library of Congress Cataloging-in-Publication Data
Czaplicki, Jacek M.
Statistics for mining engineering / Jacek M. Czaplicki, Mining Mechanization Institute, Silesian
University of Technology, Giliwice, Poland.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-138-00113-8 (hardback) -- ISBN 978-1-315-81503-9 (eBook PDF)
1. Mining engineering--Statistical methods. I. Title.
TN272.7.C93 2014
519.5–dc23
2013043750
ISBN: 978-1-138-00113-8 (Hbk)
ISBN: 978-1-315-81503-9 (eBook PDF)
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Contents
Preface and acknowledgements ix
About the author xi
List of major notations xiii
1. Fundamentals 1
1.1. Goal and task of statistics 1
1.2. Basic terms of probability theory 3
1.2.1. Random events and some definitions of probability 3
1.2.2. Random variables, distribution function and probability density function 7
1.2.3. Descriptive parameters of a random variable 8
1.2.4. Relationships between parameters of random variables 13
1.2.5. Probability distributions of discrete random variables 16
1.2.6. Probability distributions of continuous random variables 19
1.2.7. Two-dimensional random variable and its moments 30
1.2.8. Stochastic process 34
1.3. Basic terms of statistical inference 35
1.3.1. Estimator 36
1.3.2. Properties of estimators and methods of their construction 36
1.3.3. Statistical hypotheses and their types 39
1.3.4. Statistical tests and critical region 40
1.3.5. Probability distributions for the verification of statistical tests 43
2. Some areas of the application of mathematical statistics in mining 49
3. Analysis of data 55
3.1. Testing of sample randomness 55
3.1.1. The test procedure 55
3.1.2. Results of a randomness investigation 57
3.2. An outlier in a sample 58
3.2.1. Exponential distribution 60
3.2.2. Erlang distribution 63
3.2.3. Other distributions 65
3.2.4. Result of an outlier analysis 69
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vi Contents
3.3. Stationarity testing of sequences 69
3.3.1. The test procedure 70
3.4. Outcome dispersion testing 78
3.5. Cyclic component tracing 83
3.5.1. The period of cycle is known 84
3.5.2. The period of the cycle is unknown 87
3.6. Autocorrelation analysis 94
3.7. Homogeneity of data 98
3.7.1. The test procedure 102
4. Synthesis of data 111
4.1. Estimation of the parameters of a random variable 112
4.2. Probability distribution description 124
4.3. An example of empirical–theoretical inference about the distribution
of a random variable 135
5. Relationships between random variables 141
5.1. The chi-square test of independence 141
5.2. The Pearson’s linear correlation coefficient 145
5.3. Partial correlation coefficient and multiple correlation coefficient 149
5.4. Non-linear correlation measures 152
6. Synthesis of data—regression analysis 157
6.1. Preliminary remarks 157
6.2. Linear regression 159
6.3. Linear transformations and multidimensional models 167
6.4. Autocorrelation and autoregression models 174
6.5. Classical linear regression for many variables 183
6.6. Regression with errors in values of random variables 187
6.7. Linear regression with additional information 198
7. Special topic: Prediction 207
7.1. Introduction and basic terms 207
7.2. Subject of prediction 210
7.3. Examples 212
8. Explanations of some important terms 231
9. Statistical tables 235
9.1. Distribution function Φ(z) of standardised normal distribution N(0, 1). 235
9.2. Quantiles of the standardised normal distribution N(0, 1) 236
9.3. Critical values of the Student’s t-distribution 238
9.4. Critical values of the Chi-squared distribution 239
9.5. Critical values of the Snedecor’s F distribution for α = 0.10 241
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Contents vii
9.6. Critical values of the Snedecor’s F distribution for α = 0.05 243
9.7. Critical values of the Snedecor’s F distribution for α = 0.025 245
9.8. Critical values of the series distribution 247
9.9. Critical values of the Cochran statistic for α = 0.05 249
9.10. Critical values of the Hartley statistic for α = 0.05 250
9.11. Quantiles of the Poisson distribution 250
9.12. Critical values in Kolmogorov test of goodness-of-fit 252
9.13. Critical values of a linear correlation coefficient and a partial correlation coefficient 254
9.14. Critical values of the Spearman’s rank correlation coefficient 255
9.15. Critical values of a multiple correlation coefficient 255
9.16a. Critical values D (α) in the Smirnov test of goodness-of-fit for two empirical
n,m
distributions 257
9.16b.Distribution of the Smirnov statistic D P{D ≤ k/n} 261
n,m n,m
9.17. Critical values (2n, 2m) of the distribution 264
α
References 265
Subject index 273
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Preface and acknowledgements
The variety of technical devices that is available on the market is quite rich. The items offered
have, in the majority of cases, the broad possibility of realizing their targets. Very often
the devices that are selected are set up together, creating systems to execute predetermined
functions. In mining engineering, mechanized systems are formed in order to excavate rocks,
to haul broken material, to dump waste, to store excavated ore or to carry out ore treatment
by mechanical, chemical or thermal means. The technical objects being used nowadays are
complicated constructions. In order to apply them in an appropriate manner, we should follow
the recommendations formulated by their producers and we should frequently observe the
realization of their operation (exploitation)1 process, which is understood here as the process
of changes in their properties during the use of the object. Running changes are interesting
first of all for the constructor and producer of the object. They are also interesting for the
user of the equipment, who is concerned in its application to attain the best performance of
the item purchased over the long term. Information gathered during the operation of the
object creates statistical data.
Actually, the more complicated machines used in mining (and pieces of equipment from
other engineering areas) are furnished with special systems which diagnose a continuously
running machine, collect information on its exploitation events and send it to the main
producer’s computer. In many cases the information that is transmitted is very rich comprising
a few tens of units of information. Again, this trio: the constructor, producer and user are
interested in the accumulated material. And once again, this information has statistical
character.
The information that is generated through the realization of the operation of a machine
has a twofold meaning. It provides information on the technical parameters that are of
interest at a given moment of time directly, e.g. power consumption, force applied during
retardation, mass accumulated in the shovel etc. But very often there is concern about
the average characteristics of the observed data, i.e. there is concern about the statistical
characteristics of collected material.
It is easy to enumerate at least several further areas of mining engineering interest in which
the important information that is gathered has a statistical character; investigations made
during research on a stand in a lab generating records, collected data from an instrument
with a new device installed and operating in real working conditions, sensors continuously
tracing excessive shocks in a rock mass near mining production areas, information taken
from an ore processing plant and connected with a given stage of ore treatment etc. All of
these examples show areas that generate information of a statistical character. Information
that should be analysed carefully; its synthesis should be conducted and statistical inference
should be carried out.
The main goal of this book is to acquaint readers with a basic knowledge of mathematical
statistics; knowledge that should be known by contemporary mining engineers. For this reason
this book is addressed—first of all—to students of mining faculties and schools. The majority
of the cases considered here are taken from the field of mine mechanization. However, there
1These two terms will be used here interchangeably, i.e. they are synonyms. Grounds for such an approach
can be found in Chapter 1, Czaplicki 2010.
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