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Reminders:• Not here last time? Send me an e-mail ([email protected]) so I can put you on
the e-mail list• Assignment 1 due on Monday• Quiz on Monday and Friday – bring calculator; open book• First Lab (Lab 2) is on Sept 18th
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bp
tlmT
Chapter 2Quantities
Ne
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Quantity: five-part definition
1. Name2. Symbol3. Procedural Statement4. Set of numbers collected into a vector5. Units on a defined measurement scale
Lengthl14.5, 10.1, 15.9, 15.4, 14.1, 13.3, 16.1, 12.8, …cm (ratio)
measurement scale
(ratio)
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Types of Measurement Scales
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111100101000100100100010010010010011111001011001001001000101000100101001010010100100101001010101000101000010010100101001000101010010101001011110101101101001010101001010100100111101011001101001010001010100000001011111010010000101101000101001010001000001011011111111001010001001001000100100100100111110010110010010010001010001001010010100101001001010010101010001010000100101001010010001010100101010010111101011011010010101010010101001001111010110011010010100010101000000010111110100100001011010001010010100010000010110111111110010100010010010001001001001001111100101100100100100010100010010100101001010010010100101010100010100001001010010100100010101001010100101111010110110100101010100101010010011110101100110100101000101010000000101111101001000010110100010100101000100000101101111111100101000100100100010010010010011111001011001001001000101000100101001010010100100101001010101000101000010010100101001000101010010101001011110101101101001010101001010100100111101011001101001010001010100000001011111010010000101101000101001010001000001011011111111001010001001001000100100100100111110010110010010010001010001001010010100101001001010010101010001010000100101001010010001010100101010010111101011011010010101010010101001001111010110011010010100010101000000010111110100100001011010001010010100010000010110111111110010100010010010001001001001001111100101100100100100010100010010100101001010010010100101010100010100001001010010100100010101001010100101111010110110100101010100101010010011110101100110100101000101010000000101111101001000010110100010100101000100000101101111000010110100010100101000100000101101111111100101000100100100010010010010011111001011001001001000101000100101001010010100100101001010101000101000010010100101001000101010010101001011110101101101001010101001010100100111101011001101001010001010100000001011111010010000101101000101001010001000
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Ordinal
1st2nd 3rd
No information on the magnitude of the difference
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IntervalDirection
Lat/Long
Units known, but zero does not mean nothing
Temperature (oC)
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Mass
RatioLengthVolume
• Units and zero known• Can take ratios
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Ratio vs. IntervaloC vs. K
• Zero in K means no heat• Zero in oC is freezing
point of water
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Ratio vs. IntervaloC/oF vs. ft/cm
50oF = 10oC, 41oF = 5oC• 50oF / 41oF = 1.2• 10oC / 5oC = 2 • 2 ≠ 1.2
6 ft = 182.88 cm, 2 ft = 60.96 cm • 6 ft / 2 ft = 3• 182.88 cm / 60.96 cm = 3 • 3 = 3
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Methods and scale type
• Measurement procedure scale type– Example
• Record presence/absence of copepods in diet sample Nominal
• Count number of copepods in diet sample Ratio
• Scale type statistics– Example
• Nominal logistic regression• Ratio (count) Poisson regression
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Recording quantities
• Detailing methods is at the heart of the scientific method
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Computer assisted data collection• Becoming more common• Fewer errors, better resolution
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Recording Data
DATE CANADA GOOSE BLACK DUCK
27-May-10 1,35
09-Jun-10 2,52,310
• Save data in common formats (e.g. .txt, .csv)• Avoid matrix-format:
……
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Recording Data• Save data in common formats (e.g. .xls, .txt, .csv)• Use flat-format:
…
date species count27/05/2010 Canada Goose 127/05/2010 Herring Gull 127/05/2010 Great Black-backed Gull 127/05/2010 American Crow 127/05/2010 Savannah Sparrow 127/05/2010 Herring Gull 127/05/2010 Great Black-backed Gull 127/05/2010 Savannah Sparrow 127/05/2010 Herring Gull 127/05/2010 Great Black-backed Gull 127/05/2010 Herring Gull 127/05/2010 Herring Gull 1
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Error checking
• Review your data• Get a second person to review your data• Plot and check
Transcription error: 167 entered for 16.7
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Displaying dataTables• Archival• Provide summaries
Graphs• Display pattern• Discover pattern
Burrow Temperature
bT (oC)
Specific Metabolic Rate E (kcal/kg/day)
0 0.2
1 1.8
2 3.2
3 6.3
4 7.6
5 10.5
6 12.1
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Displaying data
Burrow Temperature
bT (oC)
Specific Metabolic Rate E (kcal/kg/day)
0 0.2
1 1.8
2 3.2
3 6.3
4 7.6
5 10.5
6 12.1
Fully Defined Quantities
Names Units Numbers Symbols (optional)
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Now take a second to read Table 5.2 in the lecture materials
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Critique this graph
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Critique this table
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Ratio continued… Utility
3 years ?30%
• Can do fancy calculations
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Nt = Ni e-Dt
• Extend notation and generate cool plots
% remainingsee Box 2.1
Initial pop. size
Future pop. size
Ni = 1000
Ratio continued… Utility
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Nt = Ni e-Dt
• Predict trend in student registration
% remainingsee Box 2.1
Initial pop. size
Future pop. size
Ratio continued… Utility
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Ratio continued… Utility
• This example highlights four utilities
1. Accurate computation
2. Interpret symbolic notation
3. Combine to make new units
4. Permit analysis across multiple scales
Nt = Ni e-Dt
Ind. = Ind. year/year
…well kinda
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3. Combine to make new units
• 1 hectare = 10,000 m2 = 100 m x 100 m
4. Permit analysis across multiple scales
Ratio continued… Utility
100 m 100 m
50 m 50 m
25 m25 m
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• Ecologists often use transformations…but think about what you’re doing
Note on transformations
100 m
100 m
√10moosehectare
√10 moose
√10,000 m2
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NOTE: Supplementary units defunct since 1995
m s K Amol cdkg
SI Base Units• Seven standard base units
– facilitate comparable results
Unit abbreviations
meter kilogram second Kelvin mole candela ampere
length mass time temp. amt. sub. lum. int. ele. cur.
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Derived units• Combinations of the base units• Which derived units have you used?
Unit (eg.)Acceleration (m·s-2)Area (m2)Energy (N·m; J)Force (kg·m·s-2; N)Frequency (s-1; Hz)Light intensity (W·m-2)Power (J·s-2; W)Pressure (N·m-2; Pa)Velocity (m·s-1)Volume (cm3; cc)Wavelength (m)
Others? (eg.)•••••••••••
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Standard multiples
Is it a coincidence the base is 10 and humans have 10 fingers and toes?
Text Symbol Factor Exampletera T 1000000000000 TWgiga G 1000000000 GWmega M 1000000 MWkilo k 1000 kWhecto h 100 hW
1 Wdeci d 0.1 dWcenti c 0.01 cWmilli m 0.001 mWmicro μ 0.000001 μWnano n 0.000000001 nW
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Unconventional Units
• It is sometimes useful to use non-standard units• Read this excerpt from the notes: If our interest were in the foraging ranges of owls, we might decide to define the range in biological terms, based on the minimum area (in standard units) required to meet daily energy requirements. If we define this area as one unit, we can then examine the problem of foraging area needed by a pair of owls to successfully produce 1 chick, 2 chicks, and so on, relative to the number of minimum foraging units. To phrase this as a question, if 1 owl requires a certain area to meet its own energy needs, then how many of these units will be needed by 2 owls to raise 1 chick? By defining a new unit, we can address this problem with biologically meaningful units, rather than with arbitrary units.
“
”
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Unconventional Units
• Number (#) of entities commonly used in biology
– Base pairs, birds, branches, cells, etc.
• Consequently, unconventional exponents are also common
– Offspring · year-1, fitness, etc.
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Dimensions
• Dimensions are a way of thinking about quantities based on similarity
– Euclidean Dimensions (L, L2, L3)
– Mechanical Dimensions (M, L, T)
– Composite dimensions (M·L2·T-2)
– Additional dimensions ($, A)
– Entities (#)
– Fractal dimensions (LD)
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Euclidean Dimensions (L, L2, L3)
cmm
yards cm2
hectaresacres
ccm3
yards3
Related by an integral change in exponent
(L1, L2, L3)
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Mechanical Dimensions (M, L, T)
MASS LENGTH TIME
kglbstones mfathomftspearlength shyearsyearslight-years millennia
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Composite dimensions (L·T-1)
2 4 6 8 10 12 14 16Distance (cm)
• Ant traveled 16 cm in 2 seconds • 8 cm · sec-1
• L·T-1
• What are the dimensions of the ant’s momentum?• 3 mg · 8 cm · sec-1 = 24 mg · cm · sec-1 • M·L·T-1
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Additional dimensions ($, A)
Gameboard EconomicsMeasuring electric “eel” amps
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Entities (#)
• Biochemical entities: ions, atoms, molecules (including proteins)
• Genetic entities: chromosomes, genes, alleles, mutations
• Cellular entities: nuclei, mitochondria, cells
• Behavioural entities: attempts, successes, modal action patterns (MAPs)
• Population entities: interacting species
• Community level entities: number of taxa (species, order, etc), number of trophic levels.
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Entities (#)
Useful units:
• Pairs
• Kilocounts
• Megacounts
Less useful units:
• Mol
• Dozen
• Murder
• School
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Fractal dimensions (LD)• Fractal dimensions are related to one another by
fractional exponents
• LD where 1 ≤ D ≤ 2• More convoluted than a straight line (D = 1), but not so
convoluted as to fill a plane (D = 2).
• Convolution increases as D 2
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So why are fractals important?Fractals are common in nature
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