Correlation and Regression by K Bhanumoorthy

 

A quote from Dr. Napoleon Hill.

google.com, pub-5194588720185623, DIRECT, f08c47fec0942fa0

“Whatever the mind can conceive and believe, the mind can achieve.”

 Correlation is a measure of relationship or association between two variables say x, y.

There are two types, one is positive and the other is negative. If x increases y also increases, then it is positive or if x decreases, y also decreases this is also positive (+ve).  There may be situation when x increases y decreases or when x decreases y increases, in that case correlation is negative (-ve). There are some instances there are no correlation.

Examples: Positive

 1. One liter petrol costs Rs.102, then the cost of 4 liters.

  2.Cost of bus/ train ticket; If the travel distance is more, cost also goes up.

   3.Consumption of electricity, if one uses longer time, the consumption shall be more, subsequently bill amount will be more.

Eg; Negative

       1.If you increase the speed while driving car (could be even scooter or any vehicle for the matter) the travel time will be less.

      2.In a circuit if the resistance R is more, the flow of current shall be less.  

Francis Galton, during the year 1888, has found out/ introduced this topic in the field of mathematics. This is being used in the field of psychology and education, correlation is used, in business financial analysis and decision making, in Statistics analysis of variates, in research, scientists deal with data analysis, in order to reduce the mistakes.

There 4 types of correlation.

1. Pearson correlation:- It is a correlation coefficient  that measures linear correlation between two sets of data.

2. Kendall rank correlation : It is used to measure the ordinal association between two measured quantities.

3. Spearman Correlation :- It is a nonparametric measure of rank correlation that assesses how well the relationship between two variables can be described using a monotonic function.  It is denoted by the symbol “rho” (ρ) and can take values between -1 to +1. A positive value of rho indicates that there exists a positive relationship between the two variables, while a negative value of rho indicates a negative relationship. A rho value of 0 indicates no association between the two variables.

Spearman’s Correlation formula

4. Point- Biserial Correlation: The point biserial correlation coefficient (r_pb) is a correlation coefficient used when one variable (e.g. Y) is dichotomous. To calculate r_pb, assume that the dichotomous variable Y has the two values 0 and 1. If we divide the data set into two groups, group 1 which received the value "1" on Y and group 2 which received the value "0" on Y, then the point-biserial correlation coefficient is calculated as follows:

${\displaystyle {�}_{��}=\frac{{�}_1-{�}_0}{{�}_{�}}\sqrt{\frac{{�}_1{�}_0}{{�}^2}},}$

where s_n is the standard deviation used when data are available for every member of the population:

${\displaystyle {�}_{�}=\sqrt{\frac{1}{�}\sum _{�=1}^{�}({�}_{�}-\overline{�}{)}^2},}$

 Let me recall what had been learnt in earlier classes. In chapter measures of central tendency, we studied mean, median and mode. These averages give us only a rough idea where the observations centered. We will not get clear idea to what extent observations are scattered or arranged. So, we go further to study about the data, call them measures of dispersion .ie 1. range,2. mean deviation about mean, median,.3, Harmonic Mean, 4. Geometric mean,5. Standard deviation (SD), 6. variance. These measures of dispersion will be able to measure the degree of how the observations are scattered. We use the Standard deviation for our analysis.

Variance: Given a set of numbers. Variance is a measure of, how far each number is from the mean. It is calculated by taking the difference between each number from the mean. The differences are squared.  It is further dividing the sum of squares by the number of values in the given set.

Variance is used in statistical inferences, hypothesis testing. It is also used in investment portfolios, to know, improve investment.

For all types formulae are given, from the given data one has to form the tables, then substitute, in the formulae. It’s a matter of 4 fundamental operations, squares and square roots. One has to remember the formula. let us work out an example.

Before we move to the problems there is another word connected with this topic is Covariance.

Covariance is a measure of directional relationship between the two variables to what extent the variables change together. Correlation coefficient is a mere number lying between -1 and 1 whereas Covariance is measured in units. Variance is a measure of magnitude; it is a number. Variance could be positive (+) or negative (-)

Formula for covariance:     Cov (X, Y) = ∑ { (X_i - `X) ( Y<sub>j -</sub>`Y)}/ n, where X_i denotes the values of variable X, Y_j denotes the values of variable Y, `X the mean of variable X,`Y   mean of variable Y, n, the number of data entries /units.

Variance is square of` Standard Deviation (SD), denoted by s^2 (sigma square). One has to be familiar with the symbols what do we use in

mathematics universally accepted ones.

Correlation coefficient r(X, Y) = Cov (X, Y) / s_x s_y where r(X, Y) correlation between X and Y, Cov (X, Y) covariance between X and Y, s_x Standard deviation of X and  s_y standard deviation of Y.

 

Note: The correlation coefficient always lies  between -1 and 1.  If it is 0 (zero) we can clearly say that there is no correlation between `the two given variables,

Covariance: Examples.

1.Find the Cov (X, Y) between the two variables X and Y:

 Given if   X:  3   4   5   6   7; Y: 8    7   6    5     4.  From the given data ∑XY =140,

 ∑ X =25, ∑Y =30. n=5

(∑XY =24+ 28+ 30+ 30+28). Now the Solution is Cov({X, Y) ={ n∑ XY -  (∑X)( ∑Y) }/ `n^2. Substituting

the values, we get   5×140 - (25×30) / 25 = 700 – 750 / 25 = - 50/ 25 = -2 we can conclude the variables are negatively correlated.

2.Find the correlation coefficient for the data given. Cov (X, Y) = -16.5, Var (X) =2.89, Var(Y) =100,

r(X, Y) = Cov(X, Y) /  Övar(x).Var(Y).=   -16.5 /  Sq root 0f (2.89 ×100)  = -16.5 / (1,7 ×10)  = -16.5/ 17 ,

Correlation coefficient

 is calculated to be   - 0.97, (negative), is the answer.

3.  ∑ X = 15,  .   ∑ Y = 36,.   ∑ XY=110, n =5, find Cov (X, Y)

    Ans: Cov (X, Y) = (1/ n). ∑ X_iY_i -

     (1/ 5) X 110 ---3x 7.2   = 22 ---- 21.6 = 0.4 Answer.

 

4. Cov ( x,y)  = -- 13.5 , Var (X) = 2.25 , Var (Y ) = 100 ,find correlation coefficient . r ( x,y) ?

Ans: r( x,y) = Cov (X, Y )  / Ö Var X . Ö Var Y.

                      =    --- 13.5 / 1.5 x 10   = -- 13,5 / 15 = - 0.9 Answer (negative).     

 

 

                                                                     --------- be continued

 

 

 

 

 

 

GEOMETRY AT A GLANCE -- TRIANGLE THEOREM

 

MORE TO COME 

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JANUARY 11

                                                             JANUARY 11

Let's first begin the article with the historical events occurred on January 11.

1. On Jan 11, 1672 - Issac Newton presented his telescope to the Royal Society of London whereas he was elected the Fellow of Royal Society of London.

2. On Jan 11, 1787, William Herschel discovered the first two moon of Uranus: Titania and Oberon.

3. On Jan 11, 1775, before the Academie des Sciences, Gaspard Monge (1746–1818) presented  a memoir in which he used two planes of projection in his descriptive geometry. 

4. On Jan 11, 1935 at a conference of the Royal Astronomical Society, young astronomer Subrahmanyan Chandrasekhar presented his findings on electron degeneracy pressure and the maximum mass of a stable white dwarf star. 

                                             Mathematicians of Day 

Mathematicians born on 11th January

1545: Guidobaldo del Monte                                                  1707: Vincenzo Riccati 

1734: Achille-Pierre Dionis du Séjour                                    1825: William Spottiswoode 

1826: Giuseppe Battaglini                                                       1845: Victor Bäcklund

1938: Fischer Sheffey Black 

Mathematicians died on 11th January

1757: Louis Castel                                                                    1903: Henry Watson 

1929: Micaiah Hill                                                                    1941: Emanuel Lasker 

1946: J Watt Butters                                                                  1947: Edward Ross

1949: Torsten Carleman                                                            2002: Shanti Swarup Gupta 

2014: Zoltán Dienes                                                                  2019: Michael Atiyah 

 Facts About Number Eleven (11)

1. 11 is the only prime comprising an even number of identical digits.

2.  It is the fourth number that stays the same when written upside down. The first three are 0, 1 and 8. The others such numbers less than 100 are 69, 88, 96

 3. The first four powers of 11 yield palindromic numbers: 

                   11¹ = 11, 11² = 121, 11³ = 1331, and 11⁴ = 14641

4. 11 is the first prime exponent that does not yield a Mersenne Prim. Example , which is a composite number.

5.  A polygon with 11 sides is called - hendecagon. Using a compass and straightedge, you can't make a regular hendecagon. Moreover, it is the first polygon that can't be constructed using angle trisector.

6.  It is the fourth Sophie Germain Prime. A prime number p is a Sophie Germain prime if 2p+ 1 is also a prime. The first ten Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83 and 89.

7.   It is a Stormer number. A Stormer number is a positive integer n for which the greatest prime factor of n² + 1 meets or exceeds 2n. It is named after Carl Stormer. The first eight Stormer numbers are 1, 2, 4, 5, 6, 9, 10, and 11.

Divisibility Test Rule of 11

Here is a link that tells you 3 methods of divisibility by 11

CLICK HERE FOR DIVISIBILITY RULE OF 11

 

 

JANUARY 10

 JANUARY 10, 2024

10th day of Year 2024 ( a leap year). 

1. Hindu Number System contains 10 digits (0 - 9).

2. Pythagoreans loved the number 10 as it is perfectly associated with music. It can be symbolized by the Tetractys. The tetractys is a triangular figure consisting of 10 dots arranged in 4 rows.

The first row represented zero (0) dimensions --- A point

The second row represented one dimension ---- A line is connected by two points.

The third row represented two dimensions. --- A triangle connected by three points.

The fourth row represented three dimensions --- A tetrahedron defined by four points.

The Pythagorean musical system was based on it as the ratios of 4 : 3 , 3 : 2 and 2 : 1 forms the basic intervals of the Pythagorean scale.

3. Here is a mathematical puzzle:- Suppose you have a 10 x 10 square container containing 100 balls arranged as shown below --

Can you insert 5 more balls in the same container, making the count 105?

Here is a solution. Just try it.

The arrangement of balls made the basic difference.

4. This interesting formula came in Srinivasa Ramanujan's dream. 

5. Powers of 10 contain  divisors, where  is the number of digits.

 10 has 2² = 4 divisors ( 1, 2, 5 and 10)

 100 has 3² = 9 divisors ( 1, 2, 4, 5, 10, 20 25, 50 and100)

1000 has 4² = 16 divisors ( 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, and 1000)

5.   It is the fourth triangular number. The first three are 1, 3 and 6.

      6.      It is the sum of square of two consecutive odd numbers.                

                                                  10 = 1² + 3²

     7. Take any triangle and divide each side into three equal parts. Draw lines from these to the opposite corners. A new shape will appear in the middle of the triangle whose area is 1/10^th of the area of whole triangle. This is known as Marion Walter Theorem. This theorem state - the area of the central hexagonal region determined by trisection of each side of a triangle and connecting the corresponding points with the opposite vertex is given by 1/10 the area of the original triangle.

8.  The only known solution to                                                                                                            

             n! = a! b!                                                                 is  10! = 6! X 7!

9. There are only 11 powers of 10 that are products of two integers without any zero. These are--                                                                                                                                           

           10⁰ = 1                         10¹ = 2 x 5                            10² = 4 x 25                             

            10³ = 8 x 125               10⁴ = 16 x 625                         10⁵ = 32 x 3125                                                 10⁶ = 64 x 15625         10⁷ = 128 x 78125                   10⁹ = 512 x 1953125             

10.  It is the first number whose fourth power ( 10 x 10 x 10 x 10 = 10000) can be written as a sum of two squares in two different ways

10000 =   

         =  

11. A math lover K Srinivas Raghava has defined 10 using the Golden number as --- 

                                 Mathematicians born/died on 10th January

Born:

• 1875: Issai Schur 
• 1899: Szolem Mandelbrojt 
• 1905: Ruth Moufang 
• 1906: Rafael Laguardia 
• 1906: Grigore Moisil 
• 1922: Douglas Jones 
• 1938: Donald Knuth

Died:

• 1833: Adrien-Marie Legendre 
• 1843: Louis Puissant 
• 1929: Karl Heun 
• 1941: Issai Schur
• 1944: Thomas Scott Fiske 
• 1976: Alexander Buchan
• 1984: Stephen Bosanquet 
• 1998: Wolfgang Hahn 
• 2014: David Cariolaro 

(Courtesy:- List of Mathematicians have been taken from Mac Tutor Archieve. You can follow the link to access the list --- 

Mathematicians Of The Day Index - MacTutor History of Mathematics (st-andrews.ac.uk)

                                     Interesting Events of Day (10th January)

1. On 10th January 1844, George Boole submitted his first mathematical paper in Royal Society which was rejected as the author of the paper was unknown. Later, it was reviewed by two referees and interestingly, one referee rejected the paper whereas the other recommended it for award. This paper became the first paper to receive the GOLD medal from the Royal Society. 

2. On 10th January 1854, Riemann while presenting a paper at Gottingen challenged the mathematical world to redefine the concept of infinity which should either be endless or unbounded.

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