PEARSON R

Louise Laine P. DayaoJoanne Mae J. de LotaIII-Palladium

Correlation is a statistical measure for

finding

out degree(or strength)of association

between two(or more) variables. If

the

change in one variable affects a

change in

other variable then these variables

are said

to be correlated.

Correlation analysis attempts to measure the strength of such relationships between two variables by means of a single number called a correlation coefficient .

LINEAR CORRELATION

A measure of the linear relationship between the two random variables X and

Y, and denote it by r. That is, r measures the extent to which the points

cluster about a straight line. Therefore, by constructing a scatter diagram

for the n pairs of measurements {( xi , yi ); i= 1, 2, …, n} in

our random sample (see figure1), we are able to draw certain conclusions

concerning r.

Should the points follow closely a straight line of positive slope, we have a

high positive correlation between the two variables. On the

other hand, if the points follow closely a straight line of negative slope, we

have a high negative correlation between the two variables.

The correlation between the two variables decreases numerically as the

scattering of points from a straight line increases. If the points follow a

strictly random pattern, we have zero correlation and conclude

that no linear relationship exists between X and Y.

FIGURE 1

CO

RR

ELA

TIO

N IN

TER

PR

ETA

TIO

N

GU

IDE

CORRELATION INTERPRETATION GUIDE

It is important to remember that the correlation coefficient between two

variables is a measure of their linear relationship, and a value of r=0

implies a lack of linearity and not a lack of association. Hence, if a strong

quadratic relationship exists between X and Y as indicated in Figure 1d, we

shall still obtain a zero correlation even though there is a strong nonlinear

relationship.

The most widely used measure of linear correlation between two variables is

called the Pearson product-moment correlation

coefficient or simply the sample correlation

coefficient.

r

One must be careful in interpreting r beyond what has been stated

above. For example, values of r equal to 0.3 and 0.6 only mean

that we have two positive correlations, one somewhat stronger

than the other. It is wrong to conclude that r = 0.6 indicates a

linear relationship twice as strong as that indicated by the value r

= 0.3.

On the other hand r2, which is usually referred to as the sample

coefficient of determination, we have a number

that expresses the proportion of the total variation in the values of

the variable Y that can be accounted for or explained by the linear

relationship with the values of the variable X.

Thus a correlation of r = 0.6 means that 0.36 or 36% of the total

variation of the values of Y in our sample is accounted by a linear

relationship with the values of X.

EXAMPLE 1. COMPUTE AND INTERPRET THE CORRELATION COEFFICIENT FOR THE FOLLOWING DATA:

X(height)

12 10 14 11 12 9

Y(weight)

18 17 23 19 20 15x  y  x2 y2  xy 

12  18 144 324 216

 10 17 100 289 170

14  23 196 529 322

11 19 121 361 209

12 20 144 400 240

 9 15 81 225 135

∑x = 68 ∑y = 112   ∑x2 =

786 ∑y2 =2128 ∑xy =1292

)112)(68()1292)(6( r

947.0

])112()2128)(6][()68()786)(6[( 22

r

947.0r

A correlation coefficient of 0.947 indicates a very good linear relationship between X and Y. Since r2 = 0.90, we can say that 90% of the variation in the values of Y is accounted for by a linear relationship with X.

TEST STATISTICS

After computing for the Pearson Product Coefficient, we will now determine whether to accept or reject the null hypothesis.

To do this we will compute for the t value.

After computing for the t value, we will compare it to the tabular value obtained by the matrix table at certain level of significance with n-2 degrees of freedom.

Decision Rule: Reject Ho if ltcl ta/2,(n-2) Otherwise, accept.

212 rnrt

29.0126947.0 t

35.4t

SAMPLE NO. 2

 Marks obtained by 5 students in algebra and trigonometry as given below:Algebra 15 16 1

012

8

Geometry 18 11 10 20 17

Calculate the Pearson correlation coefficient.

INTERPRETATION

A correlation coefficient of -0.424 indicates a moderately small negative relationship between X and Y. Since r2 = 0.18, we can say that 18% of the variation in the values of Y is accounted for by a linear relationship with X.

TEST STATISTICS

76.0

18.0125424.0

12

2

2

t

t

rn

rt

Individual

student

Grade in Math (x)

x2Grade in

Science(

y)

y2  xy

1 85 7255 80 6400 6800

2 90 8100 89 2.____. 8010

3 87 7569 84 7056 7308

4 79 6241 86 7396 6794

5 75 1.____. 79 6241 5925

6 80 6400 86 7396 3.____.

7 88 7744 90 8100 4.____.

8 85 7225 90 8100 7650

9 86 7396 87 5.____. 7482

10 80 6400 86 7396 6.____.

n=? 7.___.

∑x =835 ∑x2 =? 8._. ∑y = 857 ∑y2 =?

9.__.∑xy =? 10._.

11-13 Find the correlation coefficient14.Coefficient of Determination15. Interpretation