5. correlation between interval and any nominal variables (1) (1)

SPEARMAN RANK CORRELATION COEFFICIENT

(rs)

SPEARMAN RANK CORRELATION COEFFICIENT (rs)

When the variables of interest are measured in an ordinal scale, the

spearman rank correlation coefficient (rs) maybe used instead of the Pearson r. To obtain the Spearman rs , apply the following steps summarized as follows:


STEP 1. Rank the scores in distribution X giving the lowest a rank of 1 and the highest a rank of n. Repeat the process for the scores in distribution Y.


STEP 2. Obtain the difference (di) between the two sets of ranks.STEP 3. Square each difference and then take the sum of the squared di


STEP 4. Complete the formula


STEP 5. If the proportion of tries in either X or the Y observations is large, use the formula.

*optional


where:

tx = number of observations in X tied at a given rankty = number of observations in Y tied at a given rank


STEP 6. To test whether the observed rs value indicates an association between variables , the following maybe applied:For n from 4 to 30, critical values of rs for .05 and .01 level of significance are shown in the table.For n>30, significance of the observed rs under the null hypothesis can be determined using the t-test using the formula:


The sampling distribution of the test is the student t distribution with n-2 degrees of freedom.


Math Rank X Stat Rank

Y di di2

1 18 7 24 4 3 92 17 6 28 6 0 03 14 5 30 7 -2 44 13 4 26 5 -1 15 12 3 22 3 0 06 10 2 18 2 0 07 8 1 15 1 0 0

COMPUTATION FORMAT:


This test of significance of the null hypothesis using the computed rs in the example above is:H0: r=0(There is no relationship between math scores and statistics scores of students)Ha: r(There is significant relationship between math scores and statistics scores of students)


At α = .05 the tabular t is tα/2.(n-2)=t.025.5=2.571

Decision: The null hypothesis is accepted because the computed t-value is less than the tabular value.Conclusion: The math scores is not significantly correlated to the statistics scores obtained by students.

Correlation Between Interval and Any Nominal Variables

The use of Correlation Rho Formula

E=∑Niyi

2– Ny2

∑y2 – Ny2 Where: Ni = number of sample

per categoryyi = average obtained

per categoryN = total no. of samplesy = over-all averagey = individual item

y

2

Let us measure the degree of relationship between the civil status and the anual salary of the given samples.

Single 65 83 81 69 73 89 76 60

Married 70 67 90 84 78

Widowed 89 64 78

Solution:N1 = 8 y1 = 596/ 8 = 74.5

N2 = 5 y2 = 389/5= 77.8

N3 = 3 y3 = 231/3= 77

N = 16 y = 1216/16 =76.0

y2 = (65)2 + (83)2 + (81)2 + ... + (89)2 + (64)2 + (78)2 = 93,792

[8(74.5) 2 + 5(77.8) 2 + 3(77)2] – 16(76) 2E2

= 93,792 – 16(76) 2 E2 =

0.03

INTERPRETATION: There is a very small positive relationship between the civil status and the annual salary of the given samples

Let us measure the degree of relationship between the subjects and the scores of the given samples.

Science 15 20 9 3 12 16

Math 5 5 14 6

English 23 13 12

Solution:N1 = 6 y1 = 75/ 6 = 12.5

N2 = 4 y2 = 30/4= 7.5

N3 = 3 y3 = 48/3= 16

N = 13 y = 153/13 =11.77

y2 = (15)2 + (20)2 + (9)2 + ... + (23)2

+ (13)2 + (12)2 = 2239 [6(12.5) 2 + 4(7.5) 2 + 3(16) 2] – 13(11.77) 2E2

= 2239 – 13(11.77) 2 E2 =0.3

INTERPRETATION: There is a very small positive relationship between the subjects and the scores

Let us measure the degree of relationship between the performance rank obtained by the trainees during the first and second evaluation period.

Student Trainee Rank During 1st Evaluation

Rank During 2nd Evaluation

A 8 7B 2 5C 7 10D 1 4E 4 2F 9 6G 3 1H 6 9I 10 8J 5 3

SOLUTION:Student Trainee

Rank During 1st Evaluation

Rank During 2nd

Evaluation D D2

A 8 7 1 1B 2 5 -3 9C 7 10 -3 9D 1 4 -3 9E 4 2 2 4F 9 6 3 9G 3 1 2 4H 6 9 -3 9I 10 8 2 4J 5 3 2 4

∑D2= 62

ρ = 1- 6(62)10(102

-ρ = 1- 37210(102-1)

ρ = 0.62

1)

Interpretation:

There is a high positive correlation between the

student trainees’ performance rank during the first and

second evaluation period.

THANK YOU

CREDITS TO: Ian EstradaReported by: Christen Diana Gallinera

Jocelle Mella

5. correlation between interval and any nominal variables (1) (1)

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