1111001010001001001000100100100100
1111100101100100100100010100010010
1001010010100100101001010101000101
0000100101001010010001010100101010
0101111010110110100101010100101010
0100111101011001101001010001010100
0000010111110100100001011010001010
0101000100000101101111111100101000
1001001000100100100100111110010110
0100100100010100010010100101001010
0100101001010101000101000010010100
1010010001010100101010010111101011
0110100101010100101010010011110101
1001101001010001010100000001011111
0100100001011010001010010100010000
0101101111111100101000100100100010
0100100100111110010110010010010001
0100010010100101001010010010100101
0101000101000010010100101001000101
0100101010010111101011011010010101
0100101010010011110101100110100101
0001010100000001011111010010000101
1010001010010100010000010110111111
1100101000100100100010010010010011
Chapter 3
Rescaling
20 %
What is Rescaling?
• Conversion of one measurement scale 
another
• Common technique used in quantitative
biology
111100101000100100100010
010010010011111001011001
001001000101000100101001
010010100100101001010101
000101000010010100101001
000101010010101001011110
101101101001010101001010
100100111101011001101001
20 %
Nominal
Ordinal
Interval
[0,0,1,0,0,0]
[1st, 2nd, 3rd]
[90,180,45]o
Ratio
[0,1.4,3.2]m
More detail
Less detail
12 rescaling options
Why bother?
• Logical rescaling has many applications
– Simplify analyses
– Even out datasets
– Help reveal patterns
– Consolidate explanatory variables
– Run non-parametric statistics
Simplify
Ratio 
Nominal
Other
Gravel
Other
Even out datasets
Net

Harper cuts
Fish census
# by species
10
brown trout
5
smelt
1
Arctic char
15
2005
2006
2007
2008
2009
Salvage
Can
no longer
long
compare
term
analysis:
total
values Nominal
Ratio
6
2010
2011
2012
Help reveal patterns
• Rescaling to a less detailed quantity
sometimes makes it easier to see patterns
• Is the presence/absence of storms associated
with number of vagrant birds observed?
Consolidate explanatory variables
Nominal
forest/barrens
berries present/absent
wet/dry
Ordinal
Habitat rank
…
Run non-parametric statistics
• Interval & Ratio  Rank
[10.5, 15.6, 19.1, 9.8] ml  [3, 2, 1, 4]
• Non-parametrics can be useful when
parametric test assumptions are violated
– Wilcoxon rank-sum test (≈ t-test)
• But…these test are not a staple these days
– Generalized linear models can deal with various
error distributions
Normalization
• Another common technique used in
quantitative biology
• Conversion of quantity to a ratio with no units
𝑄
𝑄𝑟𝑒𝑓
𝛽
Q and Qref have
the same units
• Common Qref values:
Qmax
Qmin
Qsum
Qmean
Qrange
Qsd
Scope
𝑄𝑚𝑎𝑥
Scope(𝑄) =
𝑄𝑚𝑖𝑛
• We can use scope to
– compare the capacity of measurement
instruments,
– compare the information content of graphs,
– compare variability of physical systems, or
biological systems
Physical quantities –
larger scope
Quantity = mass(H2)
mass(Earth)
Scope =
mass(H2)
Biological quantities –
smaller scope
Quantity = body mass
mass(Blue Whale)
Scope =
mass(bacteria)
5.98 ·
g
Scope =
3.3 · 10−24 g
1.8 · 108 g
Scope =
9.5 ∙ 10−13 g
Scope = 1.83 ∙ 1043
Scope = 1.89 ∙ 1020
1019
Scope of measurement instruments
• Defined as the max over min reading
1m
Scope meter stick =
= 100 (𝑖𝑓 𝑚𝑎𝑟𝑘𝑒𝑑 𝑖𝑛 𝑐𝑚)
1 cm
1m
Scope meter stick =
= 1000 (𝑖𝑓 𝑚𝑎𝑟𝑘𝑒𝑑 𝑖𝑛 𝑚𝑚)
1 mm
1 kg
Scope ? =
= 109
1 μg
Survey scope
1. Defining the sample unit
2. Listing all possible units (the frame),
3. Then survey all possible units (complete
census) or sample units at random
Salmon survey
Unit: 100 km transects
1
Scope
2
3
4
Frame: 700 km
5
6
7
Survey scope
Unit: 100 km transects
Frame: sum(rivers)
Scope: # possible transects
Experiment scope
• Unit depends on quantity measured or
sampling interval
– Sampling livers
volume(liver)
– Census
bacteria each day
volume(sample)
10
8
1
2
8
7
= scope
6
4
3
3
4
5
6
7
8
9
Millions of bacteria
recorded each day
10 days
10
Scope =
6
5
= 10
?
1 day
4
Normalization
• Another common technique used in
quantitative biology
• Conversion of quantity to a ratio with no units
𝑄
𝑄𝑟𝑒𝑓
𝛽
Q and Qref have
the same units
• Relative to a statistic:
Qsum
Qmean
Qrange
Qsd
Normalization to a sum
• Taking a percentage
% =
– e.g. Mendel’s experiments
705
224
224
= 0.24
705 + 224
𝑄𝑖
𝑖=1
𝑛 𝑄𝑖
1
Normalization to the mean
• Useful for assessing deviations from the mean
– e.g. Number of plant species on the Canary Islands
Nplant = [ 366 348 763 1079 539 575 391 ] · sp/island
mean(Nplant) = n-1 Nplant
mean(Nplant) = 7-1 · 4061 · species/island = 580
dev(Nplant) = Nplant - mean(Nplant)
dev(Nplant) = [ -214 -232 +182 +498 -41 -5 -189 ] · sp/island
Date
SST anomaly (oC)
SST (oC)
Normalization to the mean
Coefficient of Variation
stdev(𝑄)
CV =
mean(𝑄)
• Unitless ratio that allows comparisons of two
quantities, free of various confounds
– e.g. We can use the CV to compare morphological
variability in mice and elephants
Normalization to a range
• The range is defined as the largest minus
smallest value
• Ranging uses both the minimum and
maximum value to reduce the quantity to the
range 0 to 1
𝑄 − 𝑄𝑚𝑖𝑛
𝑄 ≡
𝑄𝑚𝑎𝑥 − 𝑄𝑚𝑖𝑛
′
Normalization to the stdev
• This is a common form of normalization in
statistical treatments of data
• Returning to example of number of plant
species on 7 Canary Islands:
Rigid Rescaling
• Rigid rescaling replaces one unit with another
0.9114 m
1 km
700 yards ∙
∙
⇒ 0.64 km
yards
1000 m
• Units disappear because any unit scaled to
itself = 1 (no units)
– m/m is notation for metre/metre = 1
– kcal/Joules is a number with no units
– km1.2/m1.2 has no units: it is the number of
crooked m per crooked km
Convert units
• Generic procedure – Three steps
∙
k1 quantity
∙
⇒ Q𝑛𝑒𝑤
1.Q𝑜𝑙𝑑
Write
the
tokbe
rescaled
2
k1 new units
k2 newer units
2.Q𝑜𝑙𝑑
Apply
rigid conversion
factors⇒soQ𝑛𝑒𝑤𝑒𝑟
units
∙
∙
old unit
new units
cancel
0.9114 m
3. Calculate
700 yards
∙
yards
∙
1 km
1000 m
⇒ 0.64 km
• Figure out how much Phelps eats in a day (in lbs)
hamburger
290 g
Kg
2.2 lbs
12,000 Kcal ∙
∙
∙
∙
⇒ 11.43 lbs
670 Kcal hamburger 1000 g
Kg
11.43 lbs
= 5.8 %
195 lbs
How much do you eat in a day, as a % of
body weight?
• 2000 Kcal/day for women not in training
• 2200 kcal/day for men not in training