Math Review #2
“I just got lost in thought. It was
unfamiliar territory”
“What happens if you
get scared half to
death twice?”
Math Review
Friday June 4 2003
A) Introduction
a. Symbols
b. Operations
c. Central Tendencies
B) Linear Algebra
C) Correlation/Regression Analysis
D) System of Equations: Linear/Quadratic
E) Applied Calculus
b. Operations
Basic Math Review
Why logarithms?
Power and product rules:
logb(xy) = logb(x) + logb(y)
logb(xn) = nlogb(x)
These rules motivated the introduction of
logarithms (by Napier, in early 17th Century)
and motivated their use in scientific
computation until… computers!
b. Operations
Basic Math Review
Why logarithms?
12
2
Example: Calculate 5
7
First use logs, then use log tables:
y = 75 / 212
 Log y = Log (218 / 75)
http://www.sosmath.com/tables/logtable/logtable.html
b. Operations
Basic Math Review
a) Solve for x: ln(ea) = bx
b) Solve for y using common
logarithms (base 10): y = 175
c) Find the exponent of 10 that solves
for x: x2 = 5.5.10-12
Basic Math Review
c. Central Tendencies
The most commonly used descriptive statistics
are measures of central tendency
The sample mean (: pronounced “x bar”) is:
x

x
n
i
1
  xi
n
Where Sxi represents the sum of all values in
the sample and n represents the sample size

Basic Math Review
c. Central Tendencies
 Mean: arithmetic average
 Median: middle value of a set of values
 Mode: the data value that occurs most often
Basic Math Review
c. Central Tendencies
Let’s assume we have a student population (n = 47)
Frequency Distribution
14
12
Frequency
10
8
6
4
2
0
22 23 24 25 26 27 28 29 30 31 32 33 34
Age
But what happens if we have an outlier
(skewed distribution )?
Basic Math Review
c. Central Tendencies
Let’s assume we have a real student population
MPA ('04) - Age Freq. Distribution
10
Frequency
9
8
Frequency
7
6
5
4
3
2
1
0
20
22
24
26
28
30
32
Age
34
36
38
40
42
Basic Math Review
B) Linear relationships
Let’s play…
Address#
1372
1585
1656
2252
2379
2681
1460
1841
2045
2600
2851
3410
2521
2929
Street #
37
48
51
81
87
103
42
61
71
99
111
139
95
115
Starbucks anyone?
Basic Math Review
B) Linear Relationships
160
140
Street Number
120
100
80
60
40
20
0
0
1000
2000
Broadway Address
3000
4000
Basic Math Review
B) Linear Algebra
The “slope” (m) of a line is its rate of change:
160
140
Slope:
or
(y2-y1)/(x2-x1)
Street Number
Dy/Dx
120
100
80
60
40
20
0
0
1000
2000
Broadway Address
3000
4000
Basic Math Review
B) Linear Realtionships
The “intercept” (b) of a line is the point where x = 0
160
140
Street Number
120
100
80
60
40
20
0
0
1000
2000
Broadway Address
3000
4000
Basic Math Review
B) Linear Relationships
The function f(x) = y = mx + b
You can use it to make predictions
160
140
y = 0.05x - 31.317
Street Number
120
2
R = 0.9999
100
80
60
40
20
0
0
1000
2000
Broadway Address
3000
4000
Basic Math Review
B) Graphing Linear Relationships
Let’s assume we have a real fish population
Weight (lb)
1.18
1.35
1.71
1.72
1.99
2.02
2.58
4.26
4.5
7.31
7.99
8.1
Length (in)
13.53
14.5
13.5
16.03
16.42
15.83
15.72
21.1
21.47
22.96
24.39
23.17
Any question regarding this data set?
Basic Math Review
B) Correlation
The sample mean is:
x

x
i
n
1
  xi
n
Sum of squares for variable x. This statistics
quantifies the spread of variable x:

n
SSXX   (x i  x)
i1
2
Basic Math Review
B) Correlation
Sum of squares for variable y. This statistics
quantifies the spread of variable y:
n
SSYY   (y i  y)
i1

2
Basic Math Review
B) Correlation
Sum of the cross-products. This statistics is
analogous to the other sums of squares
except that it quantifies the extent to which
the two variables go together or apart:
n
SSXY   (x i  x)(x i  x)
i1

Basic Math Review
B) Correlation
Fish Data:
Weight (lb)
1.18
1.35
1.71
1.72
1.99
2.02
2.58
4.26
4.5
7.31
7.99
8.1
Length (in)
13.53
14.5
13.5
16.03
16.42
15.83
15.72
21.1
21.47
22.96
24.39
23.17
SSxx: 78.5
SSyy: 182.0
SSxy: 113.8
The correlation coefficient is:
SS XY
r
( SS XX )( SSYY )
Here r = 0.95
Basic Math Review
B) Correlation: Fish Data
25
Length (in)
13.53
14.5
13.5
16.03
16.42
15.83
15.72
21.1
21.47
22.96
24.39
23.17
20
Length (in)
Weight (lb)
1.18
1.35
1.71
1.72
1.99
2.02
2.58
4.26
4.5
7.31
7.99
8.1
15
10
5
0
0
2
4
6
8
Weight (lb)
The correlation coefficient is positive
10
Basic Math Review
B) Correlation:
the correlation coefficient has no inherent value,
and in the exception of strong relationships as in
the case presented, r is hard to use to determine
correlational strength. Another statistics is much
more useful: the coefficient of determination (r2)
Weight (lb)
1.18
1.35
1.71
1.72
1.99
2.02
2.58
4.26
4.5
7.31
7.99
8.1
Length (in)
13.53
14.5
13.5
16.03
16.42
15.83
15.72
21.1
21.47
22.96
24.39
23.17
Basic Math Review
B) Correlation:
25
Here
r2
= 0.91
Length (in)
20
y = 1.4492x + 12.819
2
R = 0.9058
15
10
5
0
0
2
4
6
8
10
Weight (lb)
This statistic quantifies the proportion of the
variance of one variable that is explained by the
other – Functional?
Basic Math Review
B) Linear Algebra
Forgot a section of the fish data set
Weight (lb)
0.015
0.05
0.06
0.07
0.08
0.09
0.12
0.15
0.16
0.25
0.27
0.33
0.42
0.44
0.5
0.53
0.6
0.83
Length (in)
3.16
6.07
5.72
6.57
4.32
5.52
8.39
8.32
7.79
6.05
8.11
8
10.13
10.97
9.72
11.02
11.33
13
Basic Math Review
B) Correlation:
2
Here r = 0.82
15
Length (in)
12
9
y = 10.339x + 5.1588
2
R = 0.815
6
3
0
0
0.2
0.4
0.6
Weight (lb)
0.8
1
Basic Math Review
B) Whole data set
25
20
y = 1.4492x + 12.819
Length (in)
R2 = 0.9058
15
10
y = 10.339x + 5.1588
5
2
R = 0.815
0
0
2
4
6
Weight (lb)
8
10
Basic Math Review
B) Correlation: Linear?
25
Length (in)
20
15
y = 2.2819x + 8.3152
10
R2 = 0.8172
5
0
0
2
4
6
Weight (lb)
8
10
Basic Math Review
B) Non-linear relationship
25
Length (in)
20
15
10
5
0
0
2
4
6
Weight (lb)
8
10
Basic Math Review
B) Non-linear relationship
Let’s make a statements about the relationship:
-) The weight is  to the volume
WV
Where:
V=AxL
A = a x L2
V = a x L3
W=rxV
Therefore
W = a x r x L3
25
Length (in)
20
15
10
5
0
0
2
4
6
Weight (lb)
8
10
Basic Math Review
B) Non-linear relationship
W = a x r x L3
L W 
3
L W
1
ar
L  k 3 W  kW
ar
25
20
Length (in)
3
1
1
3
15
10
5
0
0
2
4
6
Weight (lb)
8
10
Basic Math Review
B) Non-linear relationship
1
L3 W
ar
1
L  k 3 W  kW
ar
25
20
Length (in)
L W 
3
15
10
0.307
y = 12.797x
2
R = 0.9462
5
0
0
2
4
6
Weight (lb)
8
10
1
3
L  k 3 W  kW
Log L = Log (k x W1/3)
Log L = Log k + 1/3 Log W
y = b + mx
1
3
1.6
Length (in)
1.2
0.8
y = 0.307x + 1.1071
2
R = 0.9462
0.4
0.0
-2.0
-1.6
-1.2
-0.8
-0.4
Weight (lb)
0.0
0.4
0.8
Monday
 Lamont orientation (LDEO Exec. Director and DEES
Chair)
 Math Review #3: System of Equations: Linear/Quadratic
- Applied Calculus