EART20170
Statistics Practical 2
Correlation, Linear Regression and Error Propagation
1. The Hawai'i-Emperor chain of seamounts (volcanoes resting on the ocean floor) comprises about 110
individual volcanoes on the Pacific Ocean floor and is about 6000 km in length. The Hawai'i-Emperor
chain is a classic example of a hot spot track. The hot spot magma source fixed in the deeper mantle,
with a slab of ocean crust and uppermost mantle (called a plate) moving laterally above the hot spot. As
the Pacific Plate moves over the Hawai'ian hot spot, magma erupts through the Pacific Plate, creating an
active volcano. Plate motion carries the active volcano away from the magma source, the volcano
becomes extinct, and a new volcano develops over the hot spot. The Table below shows age and
distance data for the north-trending Emperor chain.
Volcano
Distance
(km)
Age
(million years)
Kilauea
Kohala
East Maui
Kahoolawe
West Maui
West
Molokai
Koolau
Waianae
Kauai
Niihau
Nihoa
unnamed
Necker
La Perouse
Brooks Bank
0
100
182
185
221
280
0.20
0.43
0.75
1.03
1.32
1.90
339
374
519
565
780
913
1058
1209
1256
2.60
3.70
5.10
4.89
7.20
9.60
10.30
12.00
13.00
(a) Calculate the correlation coefficient to assess the relationship between the variables of age and distance
along the volcanic chain.
(b) Plot a graph of age (y-axis) against distance (x-axis).
(c) Perform a classical linear regression of distance on age to find the best-fit line. Plot this line on your
graph.
(d) Calculate the reduced major axis line for distance versus age.
(e) Laysan volcano is situated at a distance 1818 km along the chain. Use your regression equation to
predict the age of this volcano.
2.
A mineralogist is determining the change in volume during heating of a quartz crystal. The change in
volume is given by the equation:
V    V  (T2  T1 )
V = change in volume
V = starting volume of the quartz crystal
 = coefficient of thermal expansion
T2 = final temperature
T1 = starting temperature
For quartz the value of  is 50  10-5 per degree Kelvin (K).
Given that the quartz crystal has a volume of 10  0.1 mm3 at 300  1 K, calculate the new volume when it
is heated to 350  2 K. Propagate the errors given on the starting volume and the temperatures to obtain
an error for the final volume of the quartz crystal after heating.
EART20170
3.
Statistics Practical 2
(a) Find the best combined result and error for the radius of the Earth, R (in metres) from the following
data:
6379020  11294
6377940  15432
6376645  18949
6378004  9900
(b) The density (D) of a spherical body is related to its mass (M) and radius (R) by the relationship:
D
3 M
4    R3
Assuming a spherical shape with  = 3.142, M = 5.977  1024 Kg calculate the average density for
the Earth (in Kg m-3).
(c) Derive an error propagation formula for D assuming that R and M both have errors.
(d) Combine the error in R obtained in part (a), with an error in M = 0.005  1024 Kg, to calculate
a final error in D.
EART20170
Statistics Practical 2
Statistics Practical 2: Answers
1 (a) Calculation of r2
Distance (x) Age (y)
0
0.20
100
0.43
182
0.75
185
1.03
221
1.32
280
1.90
339
2.60
374
3.70
519
5.10
565
4.89
780
7.20
913
9.60
1058
10.30
1209
12.00
1256
13.00
7981
74.02
r
r
r
 xy 
2

  x 2   x 

N

x2
y2
xy
0
0.04
0
10000
0.1849
43
33124
0.5625
136.5
34225
1.0609
190.55
48841
1.7424
291.72
78400
3.61
532
114921
6.76
881.4
139876
13.69
1383.8
269361
26.01
2646.9
319225 23.9121 2762.85
608400
51.84
5616
833569
92.16
8764.8
1119364
106.09 10897.4
1461681
144
14508
1577536
169
16328
6648523 640.6628 64982.92
xy
N
2
 
    y 2   y
 
N
 
64982 .92 




7981  74.02 
15
63696361  
5479 

 6648523 
   640 .7 

15
15 

 
25599 .3
 0.9952
25721 .9
r 2  0.99
99% (100r2) of the variation in age and distance is explained by their linear relationship; the remaining 1%
of the variation is unexplained.
(c) Linear regression of distance on age:
m
15  64982 .92   7981  74.02 
15  6648523   63696361
m
383990 .18
 0.0107
36031484
EART20170
Statistics Practical 2
b
74.02  6648523   64982 .92  7981
15  6648523   63696361
b
 26505012
 0.74
36031484
Equation of the best fit line is: Age  0.0107  Dis tan ce  0.74
Age versus distance for Emperor seamount chain
14
12
Age (Ma)
10
8
6
4
2
0
0
200
400
600
800
1000
-2
Distance (km )
(d) The reduced major axis solution:
m
640 .66  5478 .96 15 
6648523  63696361 15 
m
275 .4
 0.000114  0.0107
2402098 .9
y  4.93; x = 532.1
b = y  mx
b = 4.93-(0.0107  532 .1)=  0.77
(e) The Laysan volcano has a predicted age of 18.7 Ma.
2. The volume of the quartz crystal at 350K is calculated as
1200
1400
EART20170
Statistics Practical 2
V    V  T2  T1 


V  50  10 5  10  (350  300 )  0.25 mm 3
So new volume V2 after heating is
V2  10.25 mm 3
Propagation of errors:
First calculate the error in the temperature
 T2   T21   T22
 T  12  2 2  2.23K
Now calculate the error in the volume
2
2
 T 
  V 
  V 


 
  
 V 
 V 
 T2  T1  
2
2
  V 
 0.1 
 2.24 

 
 

 10 
 50 
 V 
2
2
2
2
 0.1 
 2.24 
3
 V  
 
  0.25  0.011 mm
 10 
 50 
 V  10.25  0.01 mm 3
3. (a) The values of R can be weighted according to their errors to obtain a weighted average:
x
s
 x / s2
 1/ s
2

1
 1/ s2
0.1596
2.50  10 8
 6321 m
R = 6378160  6321 m
(b)
D

3  5.977  10 24

4  3.142  6378160 3
D  5499 Kg m 3
 6378160 m
EART20170
Statistics Practical 2
(c)
2
2
 43R 2  R
 D 
 3 

   M   
3
 D 
 3M 
 4R
2
2
 D 
 
 3 

  M   R 
 D 
 M 
 R 




2
2
(d)
2
 0.005 
 18963 
 D  5499  
 

 5.977 
 6378160 
 D  17 Kg m 3
D  5499  17 Kg m 3
2