Relationship between Linear Algebra and Statistics Linear algebra can be regarded as the arithmetic of linear substitution (Edwards, H. M., 1995). Matrices and linear substitutions are effectively the same. Statistics, on the other hand, in a broad sense is the science of collecting, organizing, analyzing and interpreting data. Statistics find applications in education, research, business, health, engineering, athletics, medicine and a lot more of the fields. Typical examples of statistics are those that deal with average rainfall and temperature, birth and death rates, average snowfall, crime rates, political popularity and much more. Even though statistics is usually studied as a course on its own, understanding basic statistical concepts is requisite for any student pursuing any field of study. This is because the student will be required to conduct research in his own field of study. Hence there will be need to know how to design experiments, gather data, organize, analyze and summarize data to draw conclusions or predictions based on the findings of the research. Statistics are encountered by just about anybody for instance in the magazines, news papers, television and so on. Therefore, basic understanding of statistical vocabulary, procedure and concepts is helpful in avoiding getting mislead by misleading data and information especially when you are a consumer of a product. Statistics as a field has strong relations and dependence on linear algebra. Descriptive statistics, for instance, uses algebraic summation so often (Frank, H., & Althoen, S. C., 1994). The data of various variables are summed up or the probabilities of events are summed. The key areas in statistics that have a stronger bias in linear algebra or applies linear algebra a lot are: problems in multivariate distributions, integrals and distributions, interdependence properties and characterization of distributions, probability inequalities, orderings, and simulations and much more (Johnson, C. R., & American Mathematical Society, 1990). From the look of these statistical topics it is very clear statistics converge with linear algebra in a lot of occasions. In this paper, I am going to study the linear correlation in statistics and show how it uses linear algebra to achieve its statistical objectives. Variance and Covariance of a Statistical Data Variance measures spread or variability in a data set. It is the average of the squared deviations from the mean. The formula is Where Covariance is the measure how corresponding elements from two ordered data sets seem to grow in a common direction. The formula for covariance is Variance-Covariance matrix This is a matrix which presents variances as diagonal elements and co-variances as off-diagonal elements. Variance-Covariance matrix appears as below. To create the variance-covariance matrix; · We transform the row scores from matrix X into deviation score for matrix x as · Computing x’x · Divide each term in th ...

1. Relationship between Linear Algebra and Statistics
Linear algebra can be regarded as the arithmetic of linear
substitution (Edwards, H. M., 1995). Matrices and linear
substitutions are effectively the same. Statistics, on the other
hand, in a broad sense is the science of collecting, organizing,
analyzing and interpreting data. Statistics find applications in
education, research, business, health, engineering, athletics,
medicine and a lot more of the fields. Typical examples of
statistics are those that deal with average rainfall and
temperature, birth and death rates, average snowfall, crime
rates, political popularity and much more.
Even though statistics is usually studied as a course on its own,
understanding basic statistical concepts is requisite for any
student pursuing any field of study. This is because the student
will be required to conduct research in his own field of study.
Hence there will be need to know how to design experiments,
gather data, organize, analyze and summarize data to draw
conclusions or predictions based on the findings of the research.
Statistics are encountered by just about anybody for instance in
the magazines, news papers, television and so on. Therefore,
basic understanding of statistical vocabulary, procedure and
concepts is helpful in avoiding getting mislead by misleading
data and information especially when you are a consumer of a
product.
Statistics as a field has strong relations and dependence on
linear algebra. Descriptive statistics, for instance, uses
algebraic summation so often (Frank, H., & Althoen, S. C.,
1994). The data of various variables are summed up or the
probabilities of events are summed. The key areas in statistics
that have a stronger bias in linear algebra or applies linear
algebra a lot are: problems in multivariate distributions,
integrals and distributions, interdependence properties and
characterization of distributions, probability inequalities,
orderings, and simulations and much more (Johnson, C. R., &

2. American Mathematical Society, 1990). From the look of these
statistical topics it is very clear statistics converge with linear
algebra in a lot of occasions. In this paper, I am going to study
the linear correlation in statistics and show how it uses linear
algebra to achieve its statistical objectives.
Variance and Covariance of a Statistical Data
Variance measures spread or variability in a data set. It is the
average of the squared deviations from the mean. The formula is
Where
Covariance is the measure how corresponding elements from
two ordered data sets seem to grow in a common direction. The
formula for covariance is
Variance-Covariance matrix
This is a matrix which presents variances as diagonal elements
and co-variances as off-diagonal elements. Variance-Covariance
matrix appears as below.
To create the variance-covariance matrix;
· We transform the row scores from matrix X into deviation
score for matrix x as
· Computing x’x
· Divide each term in the deviation sums of squares and cross
product by n
Example
In the table below are the test scores for five students in maths,
French and science. From the table obtain the variance of each

3. test and covariance between the tests.
Student
Maths
French
Science
1
90
60
90
2
90
90
30
3
60
60
60
4
60
60
90
5
30
30
30
Creating a matrix A from the scores,
Solution

4. Step 1: Transforming the raw scores in matrix A to deviation
scores in matrix a using the formula
Step 2: Computing a’a to get deviation score sums of squares
matrix
Step 3: Dividing each element by n to get the variance-
covariance matrix,
From the result, we can conclude that science has the largest
variance of 720 while French has the lowest variance of 360.
Hence Science test scores are more variable than French test
scores. The co-variance between maths and French is positive
(360) and the co-variance between maths and Science is positive
too (180). This implies that these score vary in a positive way
such that as scores in maths rise the scores in French rise too
and the scores in Science rise as well. The covariance between
French and science is zero implying that there is no predictable
relationship between French and Science.
Conclusion
This case study indicates that there is strong relationship
between algebra and Statistics, just as is the case with a lot

5. more disciplines. Statistics is dependent on algebraic concepts
in more than just correlation and regression. A lot more areas
which are mentioned in the theory above but not explored in
details also manifest strong relationship between statistics and
algebra. Further studies can be done to explore other areas in
which algebra is applicable like engineering, geospatial and
geography, computer science and much more.
References
1. Edwards, H. M. (1995). Linear algebra. Boston, Mass:
Birkhäuser.
2. Frank, H., & Althoen, S. C. (1994). Statistics: Concepts and
applications. Cambridge [England: Cambridge University Press.
3. Johnson, C. R., & American Mathematical Society. (1990).
Matrix theory and applications. Providence, R.I: American
Mathematical Society.
Math 208 Application Project Due: 3/20/15
Directions: Using your book, your various professors, the
internet, or other resources, research a

6. particular application of linear algebra to your own field of
study. I want you to find a way that people
use or have used linear algebra to solve problems in their field.
But I want the application you choose to
be one relevant to your interests. So, if you’re an engineer, find
a problem in engineering that can be
solved using linear algebra. If you’re a chemist, find an
application to chemistry. I know what your
majors are so don’t try to fool me. If you have trouble finding a
topic, feel free to come see me. You will
write a small report on your findings. Your report must consist
of the following items.
• Legibility
o Preferably, I’d like you to type your report so I can read it
easily. Typing math is not
always easy, so for your math portion of your report, you are
free to write instead. No
matter what, it needs to be neat and legible.
• 2-3 pages in length.
o Truthfully, the length isn’t that important to me, but I want it
to be something more

7. than a quick paragraph. I want your report to be thoughtful and
hopefully something
you learned from.
• Your paper should be formatted in the following way.
o Section 1: For this part you are to introduce the reader to your
subject. Chances are, I
am not familiar with the particulars of the problem you are
using linear algebra to solve.
So if you’re application is in Physics, spend some time going
over the details of the
problem to the layman. Make sure all ideas are clearly spelled
out. Properly define any
technical jargon and notation. Explain the problem in plain
English. Don’t assume I know
what you’re talking about. Chances are, I don’t. Basically, use
this section to provide the
reader with sufficient background information so the reader can
understand what you
are talking about.
o Section 2: Introduce the linear algebra concepts and
machinery that you will be using to

8. solve your problem. Properly define any relevant material and
begin solving your
problem. The bulk of the work will be done in section 2.
o Section 3: Conclusions. Discuss what findings you have come
to in relation to the
problem you were attempting to solve. Then discuss how using
linear algebra benefited
you in this process. What did you learn? Be thoughtful and
share your insights.
• To receive an A on this paper, your application should be
more than just solving a system. I
would like something more profound than this if possible. There
are many applications of the
determinant, linear transformations, and eigenvectors, for
example. Use the opening flap of
your book to find numerous references to applications
throughout the text. If you are a math
major, come see me for some cool ideas.
Good luck and come see me any time for assistance.