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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

bp

tlmT

Chapter 2Quantities

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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)

Types of Measurement Scales

111100101000100100100010010010010011111001011001001001000101000100101001010010100100101001010101000101000010010100101001000101010010101001011110101101101001010101001010100100111101011001101001010001010100000001011111010010000101101000101001010001000001011011111111001010001001001000100100100100111110010110010010010001010001001010010100101001001010010101010001010000100101001010010001010100101010010111101011011010010101010010101001001111010110011010010100010101000000010111110100100001011010001010010100010000010110111111110010100010010010001001001001001111100101100100100100010100010010100101001010010010100101010100010100001001010010100100010101001010100101111010110110100101010100101010010011110101100110100101000101010000000101111101001000010110100010100101000100000101101111111100101000100100100010010010010011111001011001001001000101000100101001010010100100101001010101000101000010010100101001000101010010101001011110101101101001010101001010100100111101011001101001010001010100000001011111010010000101101000101001010001000001011011111111001010001001001000100100100100111110010110010010010001010001001010010100101001001010010101010001010000100101001010010001010100101010010111101011011010010101010010101001001111010110011010010100010101000000010111110100100001011010001010010100010000010110111111110010100010010010001001001001001111100101100100100100010100010010100101001010010010100101010100010100001001010010100100010101001010100101111010110110100101010100101010010011110101100110100101000101010000000101111101001000010110100010100101000100000101101111000010110100010100101000100000101101111111100101000100100100010010010010011111001011001001001000101000100101001010010100100101001010101000101000010010100101001000101010010101001011110101101101001010101001010100100111101011001101001010001010100000001011111010010000101101000101001010001000

Ordinal

1st2nd 3rd

No information on the magnitude of the difference

IntervalDirection

Lat/Long

Units known, but zero does not mean nothing

Temperature (oC)

Mass

RatioLengthVolume

• Units and zero known• Can take ratios

Ratio vs. IntervaloC vs. K

• Zero in K means no heat• Zero in oC is freezing

point of water

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

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

Recording quantities

• Detailing methods is at the heart of the scientific method

Computer assisted data collection• Becoming more common• Fewer errors, better resolution

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:

……

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

Error checking

• Review your data• Get a second person to review your data• Plot and check

Transcription error: 167 entered for 16.7

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

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)

Now take a second to read Table 5.2 in the lecture materials

Critique this graph

Critique this table

Ratio continued… Utility

3 years ?30%

• Can do fancy calculations

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

Nt = Ni e-Dt

• Predict trend in student registration

% remainingsee Box 2.1

Initial pop. size

Future pop. size

Ratio continued… Utility

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

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

• Ecologists often use transformations…but think about what you’re doing

Note on transformations

100 m

100 m

√10moosehectare

√10 moose

√10,000 m2

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.

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.)•••••••••••

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

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.

“

”

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.

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)

Euclidean Dimensions (L, L2, L3)

cmm

yards cm2

hectaresacres

ccm3

yards3

Related by an integral change in exponent

(L1, L2, L3)

Mechanical Dimensions (M, L, T)

MASS LENGTH TIME

kglbstones mfathomftspearlength shyearsyearslight-years millennia

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

Additional dimensions ($, A)

Gameboard EconomicsMeasuring electric “eel” amps

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.

Entities (#)

Useful units:

• Pairs

• Kilocounts

• Megacounts

Less useful units:

• Mol

• Dozen

• Murder

• School

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

So why are fractals important?Fractals are common in nature