Correlation/ Relationship/Association between Sales and Advertising expense
Stores Sales [x P1000] Ad Expense [x P1000] XY X2 Y2
(X) (Y)
A 100 25 2500 10000 625
B 150 35 5250 22500 1225
C 200 75 15000 40000 5625
D 158 65 10270 24964 4225
E 50 40 2000 2500 1600
F 75 65 4875 5625 4225
G 120 60 7200 14400 3600
H 130 65 8450 16900 4225
I 110 50 5500 12100 2500
J 500 130 65000 250000 16900
total ΣX =1593 ΣY =610 ΣXY = 126045 ΣX2 =398989 ΣY2 = 44750
∑ ∑ ∑
r= = = = +0.87
√[ ∑ ∑ ][ ∑ ∑ ] √[ ][ ] √
Interpretation: There exists a positive high correlation between sales and advertising expense. High sales are associated with high advertising
expense.
1
Spearman Rho
=correlation between two sets of ranks
Ρ=1- =1- =1 - = - 0.43
Tourist Destinations Rx (Philbert) Ry (Sotomango) D2= (Rx –Ry)2
Siargao 6 3 (6-3)2 = 9
Bohol 2 6 16
Baguio 1 5 16
Palawan 5 4 1
Boracay 4 2 4
Batanes 3 1 4
2
ΣD = 50
There exists a negative moderate correlation between the rankings of Philbert and Karyll of the Tourists destinations. Philbert’s high ranks are
associated with Karyll’s low ranks.
2
Kendall’s Coefficient of Concordance
-to determine the relationship among three or more sets of ranks.
-value ranges from 0 to 1; 0 has no agreement while 1 means a perfect agreement. The nearer the value to 1 the stronger is the agreement.
∑
W= total sum of ranks = average rank = = = 27.5
Sum of
JUDGES D2
Projects Ranks
VIOLA RIZZA MARIAN XYLI ZIAN
A 2 1 2 3 4 12 (12-27.5)2=240.25
B 1 3 1 2 2 9 342.25
C 3 4 4 1 3 15 156.25
D 5 5 5 5 1 21 42.25
E 4 2 6 7 6 25 6.25
F 7 8 3 4 7 29 2.25
G 6 6 8 6 5 31 12.25
H 8 7 7 8 9 39 132.25
I 9 10 10 9 8 46 342.25
j 10 9 9 10 10 48 420.25
2
Σsum of ΣD = 1696.50
ranks= 275
total sum of ranks = = = 275 m=how many sets of ranks are there ; n = number of objects being rated
∑
W= = = 0.82
There exists high correlation among the rankings of the five judges.
3
Point Biserial Coefficient
-determine relationship between a dichotomous (nominal) and continuous (interval or ratio) variable.
∑ ∑ ∑ ∑
rpb =
√∑ ∑ [∑ ∑ ∑ ]
we assign the fp column for the positive response counts in the nominal variable; f w for the negative response counts in the nominal,; Y column
for the responses (scores) in the continuous variable.
Sample. Determine correlation between the results of the interview (pass or fail) and the entrance exams of the applicants.
Entrance Exam fp (# of applicants fw (# of applicants
f (fp +fw) fY fY2 fpY
Scores (Y) who passed) who failed)
10 2 0 2 20 200 20
9 4 0 4 36 324 36
8 6 1 7 56 448 48
7 7 1 8 56 392 49
6 8 2 10 60 360 48
5 6 4 10 50 250 30
4 5 6 11 44 176 20
3 3 8 11 33 99 9
2 2 7 9 18 36 4
1 1 8 9 9 9 1
0 0 9 9 0 0 0
Σfp =44 Σfw =46 Σf =90 ΣfY = 382 ΣfY2 = 2294 Σfp Y=265
∑ ∑ ∑ ∑
rpb = = = = 0.64
√ [ ]
√∑ ∑ [∑ ∑ ∑ ]
4
There exists a positive moderate correlation between entrance test scores and the interview results of the applicants. High scores in
the entrance exam are associated with passing results in the interview.
Partial correlation
-determining the relationship between two variables holding constant (eliminating) the influence of a third variable.
The general formula (correlation between variables 1 and 2 with the effects of variable 3 partialed out).
r12.3 =
√
Correlation between variables 2 and 3 with the effects of variable 1 partialed out
r23.1 =
√
Correlation between variables 1 and 3 with the effects of variable 2 partialed out
r13.2 =
√
Note: r12 = r21 ; r23=r32; r13 = r31
Sample. Suppose
1 = age; 2 = weight 3 =scores in the math test and
r12 = 0.80 r13 = 0.60 r23 =0.50
What is the correlation between weight (2) and scores(3) with the effects of age(1) removed?
5
r23.1 = = r23.1 = = = 0.04
√ √
There exists positive yet negligible correlation between weight and scores with the effects/influence of age removed (partialed
out).
Multiple Correlation
-to get the correlation between one variable and the combined effects of two or more other variables.
-can give only the strength of the association.
Formula for the correlation between variable 1 and the combined effects of variables 2 and 3
R1.23 = √
Correlation between variable 3 and the combined effects of variables 1 and 2
R3.12 == √
Correlation between variable 2 and the combined effects of variables 1 and 3
R2.13 == √
6
Suppose: 1 = age; 2 = weight 3 =scores in the math test and
r12 = 0.80 r13 = 0.60 r23 =0.50
What is the correlation between age(1) and the combined effect of weight(2) and scores(3)?
R1.23 = √ = √ =√ = 0.83
There exists a high correlation between age and the combined effects of weight and scores.
