linear algebra

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 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 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

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 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.