BSC Mining Engineering
Mining and Metallurgical Engineering Department
Multivariate Statistics
2003
Exercises
1. The data for multivariate analysis should always be examined and tested
for normality, and homogeneity of variances.
a. Why?
b. Explain with sketches two methods to assess the normality of
the variables.
Answer:
(a) The main reason behind these tests are to (1) determine the satiability of the data for
analysis, (2) decide if transformations are necessary, and (3) decide what form of the
data should use.
(b) Testing for normality and homogeneity
Normality of variables is assessed by either statistical or graphical methods. Two
components of normality are skewness and kurtosis. Skewness has to do with the
symmetry of the distribution; a skewed variable is a variable whose mean is not in the
centre of the distribution. Kurtosis has to do with the peakedness of a distribution; a
distribution is either too peaked (with short, thick tails) or too flat (with long, thin tails).
Figure 1 shows a normal distribution, distributions with skewness, and distributions
with non-normal kurtosis. The test of normality may be a normal probability plot on
variables, tests of skewness and kurtosis, chi-square goodness of fit tests, and/or
histograms.
1 coefficient of skewness
The coefficient of skewness is often calculated to determine if the distribution is
symmetrical or whether it tails to the left (negative) or right (positive). Generally, one
can look at departures from symmetry of a distribution using the skewness as a measure
of normality.
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2 coefficient of kurtosis
The coefficient of kurtosis, CK, is a measure of flatness and may be tested. For a normal
distribution, the CK has a value of 0.263.
(a)
Normal
(b)
(c)
Positive skewness
Negative skewness1
(d)
(e)
Positive kurtosis
Negative kurtosis
Figure 1: Normal distribution, distribution with skewness, and distributions with kurtosis
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2
Explain when and why we use the following:
Correlation coefficient – PCA – FA
Answer:
Correlation coefficient ρ
Is the statistic that most commonly used to summarise the relationship between two variables.
It can be calculated from the equation:

1 n
 xi  m x yi  m y 
n i 1
 x y
Where:
n is the number of data values is n
x1, ….xn and y1, …..yn are data values for the first and second variable
mx and my are mean values
σx and σy the standard deviations.
The correlation coefficient measures the strength of the linear relationship between two
variables and takes values from -1.0 (perfect negative or inverse correlation) to +1.0 (perfect
positive or direct correlation). A value of r = 0.0 indicates no linear correlation.
Multivariate statistics Methods
The most known multivariate analysis techniques are; the principal components analysis
(PCA), factor analysis (FA), cluster analysis and the canonical analysis. The first two methods
(PCA) and (FA) are statistical techniques applied to a single set of variables where someone is
interested in discovering which variables in the set form coherent subsets that are relatively
independent of one another. Variables are combined into factors. Factors are thought to reflect
underlying processes that have created the correlation among variables.
Interpreting the results obtained from those methods requires a good understanding of the
physical meaning of the problem.
PCA:
Principal component analysis (PCA) is a multivariate technique for examining relationships
among several quantitative variables by forming new variables, which are linear composites
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of the original variables. The maximum number of new variables that can be formed is equal
to the number of the original variables, and the new variables are uncorrelated themselves. So,
the procedure is used if one interested in summarising data and detecting linear relationships.
In other words, through evaluation of PCA, one seeks to determine the minimum number of
variables that contain the maximum amount of information and determine which variables are
strongly interrelated.
FA
Factor analysis (FA) is a generic name given to a class of multivariate statistical methods
whose primary purpose is data reduction and summarisation. Broadly speaking, it addresses
itself to the problem of analysing the interrelationships among a large number of variables and
then explaining these variables in terms of their common underlying dimensions [factors].
The general purpose of factor analytic techniques is to find a way of condensing
(summarising) the information contained in a number of original variables into a smaller set
of new composite dimensions (factors) with a minimum loss of information.
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3. Summary Statistics Exercise
A frequency table records how often observed values fall within certain interval or class.
Table (1) gives lead grades measured from 1m drill core samples:
Required:
1.
2.
3.
4.
5.
Draw histograms
Calculate the frequency table
Draw the frequency histogram
Draw the inverse cumulative histogram of Pb grades
Determine, minimum, maximum, range, mean, median,
mode, variance, and the standard deviation.
8.72
0.44
2.16
0.64
3.04
6.28
9.76
1.64
1.00
4.12
4.50
1.96
4.84
0.84
0.80
0.90
7.12
7.70
1.20
2.16
0.72
6.30
8.80
1.48
1.10
0.56
0.72
1.40
7.90
0.96
0.44
6.92
1.44
5.96
2.00
1.68
1.30
0.76
8.36
1.32
0.56
7.39
0.56
8.70
7.80
0.60
1.32
1.80
9.80
3.00
2.60
5.20
Answer:
Range
0-0.5
0.5-1.0
1.0-1.5
1.5-2.0
2.0-2.5
2.5-3.0
3.0-3.5
3.5-4.0
4.0-4.5
4.5-5.0
Frequency
2
13
8
5
2
2
1
0
2
1
Cumulative %
3.85
28.85
44.23
53.85
57.69
61.54
63.46
63.46
67.31
69.23
Range
5.0-5.5
5.5-6.0
6.0-6.5
6.5-7.0
7.0-7.5
7.5-8.0
8.0-8.5
8.5-9.0
9.0-9.5
9.5-10.0
Frequency
1
1
2
1
2
3
1
3
0
2
Cumulative %
71.15
73.08
76.92
78.85
82.69
88.46
90.38
96.15
96.15
100.00
Summary Statistics
No of observations
Mean
Median
Mode
Standard deviation
Sample variance
Range
Minimum
Maximum
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3.4475
1.88
0.56
3.07794
9.47372
9.36
0.44
9.8
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