Physics for Geologists, Second edition

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13 Some dangers of mathematical statistics Before we look at these dangers we should look at one of the hidden dangers of computers and electronic calculators in general. How many significant figures? The number of significant figures in a number is largely a matter of common sense. If you read that Brazil's coastline is 7491 km long, common sense tells you that 7500 is a reasonable figure. If the measurement was made some years ago, the natural changes have certainly altered the figure (even if it could be measured to the nearest kilometre). In Chapter 3 in the context of sea-level we asked what we mean when we say that Mount Everest is 8 840 or 8 850 or 8 835m above sea-level. If you do not know the level of the datum with precision, clearly there is no meaning in quibbling about metres, but if all the elevations of the Himalayas are referred to some datum, then they have relative values. Likewise if you measure elevation with an aneroid barometer, it makes no practical sense to give elevations to the metre even if you can read the instrument to the nearest metre. In Chapter 8 we looked at the positions of the magnetic poles in decimal degrees, 0.01 of a degree being 0.6 nautical miles or a little more than a kilometre. With the advent of calculators and computers it has become possible to calculate with great apparent precision - almost to as many decimal places as you wish. But to do so can be a danger to understanding. For example, you want to know the monthly interest payments on a loan you made of $1000 for a year at 4.45% per annum, so you write in your spreadsheet $1000 x 0.0445112 = $3.71. The number your computer or calculator actually uses is 3.7083 (the 3 recurring). You would gain less than a cent each month and the error is insignificant. Lend a million dollars on the same terms and the monthly interest would be $3 708.33. The point here is that you may be using precision without being aware of it. If you measure a rectangular field and find the sides are 246.52 x 197.32 m - that is, to the nearest centimetre - it may seem reasonable to believe the computer that gives its area as 48 643.326 4m2 rounded to 48 643.33. But the long side might have been 246.516 and the short side 197.316, giving Copyright 2002 by Richard E. Chapman
Some dangers of mathematical statistics 13 1 a figure for the area of 48 641.55 m2. Or the measurements might have been 246.524 and 197.324, and so an area of 48 645.10m2. You may format the cell to have no decimal figures, and so believe the figure you see in the cell, but the computer uses the full quantity unless you tell it not to. At best you could only say that the area is 48 643 f 2 m2. Not even the last whole number 3 can be relied upon. You may, however, convert this to 4.864 ha as the probable area. The use of prefixes to SI units requires attention in this context. For example, 145.53 kN may seem excessive, but if you can measure the pressure to &4 N it may be acceptable. In the parameters of mathematical statistics (such as average, standard deviation, etc.) which by their nature seek to create some order out of dis- order, the rule of thumb is to use no more than two unless there are good reasons to use more. But again, your computer will use many more but it will rarely be worth the bother of doing more than formatting for two decimal places. Linear regression Mathematical statistics and statistical methods pervade modern science, but there are great dangers in using techniques that you do not thoroughly understand. Statistics is a background consideration of the most common geological operations, such as sample taking, fossil collecting and estimat- ing the proportions of different constituents in thin-sections of rocks. Is the sample you have just taken representative? You went for the freshest part of the outcrop - the freshest part that you could reach - so really it is most unlikely to be representative. But if you don't have a fresh sample, you may not know sufficiently well what the rock really is or was! The fundamental problem with statistics in the physical and natural sci- ences lies in the concept of randomness. When you toss a coin, it is reasonable to assume that it is a matter of pure chance whether it comes to rest heads or tails upwards - each is equally likely if the spin of the coin is fast. If large meteorites hit the Earth from time to time, it would be a reasonable assump- tion that the rotation of the Earth on its axis and around the Sun, coupled with the indeterminacy of the positions of the larger meteorites, means that each square kilometre of the Earth is as likely to receive an impact as any other. If that is found not to be the case, then one conclusion (there may be others) will be that meteorites do not approach us from all directions. If they lie in the plane of the solar system or ecliptic (as they do, f 30°), impacts will be more likely on a square kilometre in the tropics than at the poles.* Already the idea of meteorites randomly distributed in space would have to be qualified. 1 Meteorites have been found in polar snow and ice, so the poles are not immune. Perhaps some of these would have missed but for the force of gravity. Copyright 2002 by Richard E. Chapman
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Physics for Geologists Second editi
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Copyright 2002 by Richard E. Chapma
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... vlii Contents Polarization 47 L
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Figures Copyright 2002 by Richard E
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Tables Copyright 2002 by Richard E.
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xiv Preface waves diminishes with d
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xvi Preface provides. This book is
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xviii Acknowledgements to K. Whaler
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xx Symbols mass refractive index bu
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2 Basic concepts Table 1.1 Dimensio
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4 Basic concepts Table 1.2 Basic SI
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6 Basic concepts pascal (Pa) and si
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8 Basic concepts all the relevant d
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10 Basic concepts relating them by
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2 Force The dimensions of force are
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14 Force The ratio of the Moon's ac
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16 Force Figure 2.1 The sum of the
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18 Force When a motorcycle takes a
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20 Force Figure 2.3 Torque is the p
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22 Force How does momentum differ f
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24 Force the weights - that is, the
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26 Force Figure 2.6 Wall-sticking i
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3 Gravity Universal gravity When di
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30 Gravity are significant to geolo
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32 Gravity Figure 3.2 The tractive,
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34 Gravity Mean sea level It might
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36 Gravity The benefits of space fl
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38 Gravity Figure 3.6 Profile of th
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40 Gravity Figure 3.8 The siphon at
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4 Optics Light is an electromagneti
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44 Optics Figure 4.2 Refraction. (a
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46 Optics The coeficient of reflect
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48 Optics only the light oscillatin
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50 Optics called Iceland Spar. Two
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52 Optics Diffraction If you shine
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54 Optics Figure 4.7 Geometrical op
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56 Optics Figure 4.9 The mirrors in
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5 Atomic structure and age-dating P
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60 Atomic structure and age-dating
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66 Electromagnetic radiation radiat
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68 Electromagnetic radiation Source
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Heat and heat flow The noun heat ha
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72 Heat and heat flow Second Law: A
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8 Electricity and magnetism Electri
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Electricity and magnetism 77 Figure
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Page 97 and 98: Electricity and magnetism 8 1 defin
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Page 101 and 102: 9 Stress and strain Pressure, p, is
Page 103 and 104: Stress and strain 87 than some natu
Page 105 and 106: These three elastic constants are r
Page 107 and 108: Stress and strain 91 Figure 9.1 New
Page 109 and 110: Stress and strain 93 Figure 9.2 Com
Page 111 and 112: Stress and strain 95 Figure 9.3 Ide
Page 113 and 114: Fracture Stress and strain 97 We fo
Page 115 and 116: Stress and strain 99 from which we
Page 117 and 118: 10 Sea waves The nature of water wa
Page 119 and 120: Sea waves 103 Figure 10.1 Wave moti
Page 121 and 122: Sea waves 105 refraction by islands
Page 123 and 124: Acoustics: sound and other waves 10
Page 125 and 126: Acoustics: sound and other waves 10
Page 127 and 128: 12 Fluids and fluid flow Introducti
Page 129 and 130: Fluids and fluid flow 1 13 Figure 1
Page 131 and 132: Fluids and fluid flow 115 Figure 22
Page 133 and 134: Fluids and fluid flow 117 and the f
Page 135 and 136: water above a vertical thickness h
Page 137 and 138: Fluids and fluid flow 121 where V i
Page 139 and 140: Fluids and fluid flow 123 ardentes,
Page 141 and 142: Fluids and fluid flow 125 Figure 12
Page 143 and 144: - 14 $ 12 E 10 K .- 3 8 r cll + 6 0
Page 145: Fluids and fluid flow 129 the liqui
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Page 155 and 156: Appendix 139 What would happen if y
Page 157 and 158: Notes 141 and since x is very large
Page 159 and 160: References Allum, J. A. E. (1966) P
Page 161 and 162: References 145 -(1950) 'Mechanism o
Page 163: Further reading 147 Trigg, G. L. an
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