CHAPTER NINE
MEASURES OF
ASSOCIATION
9.1Introduct
i
on
In the p
r
evious chapt
ers discussio
n were advanc
ed that permits the Researcher to describe data, compute measurements of relative locations, measures of variability, and measures of central tendency.
You will discover the measurements of link (association) between variables in this chapter. Correlation and regression are connection measures that involve comparing two or more sets of data to determine whether or not they have any common traits. A perfect positive relationship between two or more variables is shown by a value of 1, whereas a perfect negative relationship is shown by a value of –1, and there is no relationship at all by a value of 0.
For example, a Researcher may be interested to determine relationship between Students’ Anxiety and Academic Performance in a course, the relationship between Parents’ Socio-economic status and Students’ Performance, peer-Group influence and Students’ performance, among others.
9.2Objectives
At the end of this chapter, you should be able to:
demonstrate all types of relationships
compute the Pearson ‘r’ and
compute the spearman rho.
explain the meaning of correlation
list the importance of correlation
9.3Importance of Correlation
The followings are some of reasons why we measure the correlation between different variables.
These are.
To determine the causes and effect relationship
To find out whether a relationship exist or not
To examine whether it is significant
To determine the direction of relationship; and
To conclude or analyse usefulness of data.
9.4Types of Correlation
Correlation is of different types; linear and non-linear(curvilinear)correlation. The difference between linear and non-linear correlation is rated upon ratio of change between the variables. Linear correlation is type in which the change in one variable leads to change in another variable in the proportion. Consider this example.
A | 5 | 7 | 9 | 11 | 13 | 15 | |
B | 50 | 70 | 90 | 110 | 130 | 150 |
To plot this point on a graph, you will have straight line. Non-linear correlation is a type of relationship in which the change in the value of one variable does not cause a proportional change in the value of the other variables. The change for the two variables is not proportional (i.e. are different).
Positive correlation and negative correlation.
Examples to show relationship between positive and negative correlation.
Positive X | Correlation Y | |
10 | 60 | |
9 | 59 | |
8 | 58 | |
7 | 57 | |
6 | 56 | |
5 | 55 | |
Negative X | Correlation Y | |
10 | 55 | |
9 | 54 | |
8 | 53 | |
7 | 52 | |
6 | 51 | |
5 | 50 | |
Zero Correlation
When there is no relationship between two variables then it is said to be zero correlation.
Student | X | Y |
1 | 10 | 60 |
2 | 9 | 59 |
3 | 8 | 58 |
4 | 7 | 57 |
5 | 6 | 56 |
From example, it shows that, there is no relationship exists X and Y. The student that obtained highest value in X, has lowest in Y.
All these relationships can be illustrated in the 9.1a, 9.2b and 9.3c.
'‘Thus, the relationship or correlation with a numerical value that is coefficient of correlation’‘. This indicates the extent or degree of relationship between the variables. Positive one (+1) means perfect positive correlation, negative one (–1) means perfect negative correlation and zero means there is no correlation. Between the variables. These are various method, in calculating coefficient of coefficient of correlation. But only two methods will discuss.
These methods are Pearson’s product moment correlation (R) and Spearman (rho).
Product moment method was developed by Karl Pearson. It is denoted by symbol (r). The value of '‘r’' may be regarded as an arithmetic mean of standard scores. While the other method was developed by the British psychologist, Charles Edward Spearman. It is known as Spearman’s coefficient of correlation. It is also denoted by P (rho). Rank difference method is used when ordinal scale data I the form of ranks are given. This method is only possible when the number of observations is small. Also, if only individual scores are given and frequently distribution.
9.5Computation of Pearson Product-Moment Correlation Coefficient
The most widely used measure of association is correlation known as Pearson ‘r’, which was named in Honor of the man Karl Pearson that developed it. There are two methods of computing the Pearson (r); mean deviation and raw score. The result obtained is the index of relationship between two variables, which refers to correlation coefficient.
Let us use mean deviation formula to compute the correlation between two variables (X and Y).
The formula is stated as: \\frac{\\mathbf{\\Sigma }\\left(\\mathbf{X}-\\overline{\\mathbf{X}}\\right)\\left(\\mathbf{Y}-\\overline{\\mathbf{Y}}\\right)}{\\sqrt[]{\\mathbf{\\Sigma }\\left(\\mathbf{X}-\\overline{\\mathbf{X}})^{2}\\mathbf{\\Sigma }\\left(\\mathbf{Y}-\\overline{\\mathbf{Y}})^{2}}}or \\frac{\\mathbf{\\Sigma }\\mathbf{X}\\mathbf{Y}}{\\sqrt[]{(\\mathbf{\\Sigma }\\boldsymbol{X}^{2})(\\mathbf{\\Sigma }\\boldsymbol{Y}^{2})}}
Example 9.1: The Researcher that conducted a study on the influence of problem-solving attitude on academic performance of pre-service teachers in department of Mathematics obtained the following data.
Table 9.1: Problem-solving Attitude and Academic Performance Scores of Pre-service Teachers.
Pre-service Teachers | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
Problem-solving Attitude Score (X) | 11 | 12 | 13 | 13 | 14 | 15 | 16 | 16 | 17 | 18 | 18 | 19 | 19 |
Academic Performance Score (Y) | 25 | 40 | 45 | 20 | 35 | 30 | 40 | 45 | 50 | 50 | 60 | 70 | 65 |
The formula for computation of the correlation coefficient between two variables X and Y scores using mean deviation is stated as:
r = \\frac{\\mathbf{\\Sigma }\\left(\\mathbf{X}-\\overline{\\mathbf{X}}\\right)\\left(\\mathbf{Y}-\\overline{\\mathbf{Y}}\\right)}{\\sqrt[]{\\mathbf{\\Sigma }\\left(\\mathbf{X}-\\overline{\\mathbf{X}})^{2}\\mathbf{\\Sigma }\\left(\\mathbf{Y}-\\overline{\\mathbf{Y}})^{2}}} =\\frac{\\mathbf{\\Sigma }\\mathbf{X}\\mathbf{Y}}{\\sqrt[]{(\\mathbf{\\Sigma }\\boldsymbol{X}^{2})(\\mathbf{\\Sigma }\\boldsymbol{Y}^{2})}}
Where r = index of correlation coefficient
x = Deviation of each (x) score from their mean
y = Deviation of each (y) score for the mean
∑ = sum of.
Let us use the data in Table 9.1 to obtain the values following the step by step using mean deviation methods.
Step 1: Write down the scores in column A and B
Sep 2: Sum up the two scores X and Y and compute the mean values for the scores.
Step 3: Record the mean deviation for X scores in column C and that of Y scores in column D
Step 4: Square each deviation scores (X) and (Y) and record them in E and F column respectively.
Step 5: Find the product of x by its corresponding y to obtain xy scores. Write down the results in G.
Step 6: Substitute In the formula to obtain ‘r’
S/N | A (X) | B (Y) | C \\left(\\mathbf{X}-\\overline{\\mathbf{X}}) X | D (\\mathbf{Y}-\\overline{\\mathbf{Y}}) Y | E (x2) | F (y2) | G (xy) |
1 | 11 | 25 | –4.5 | –19.2 | 20.25 | 368.64 | 86.4 |
2 | 12 | 40 | –3.5 | –4.2 | 12.25 | 17.64 | 14.7 |
3 | 13 | 45 | –2.5 | 0.8 | 6.25 | 0.64 | –2.0 |
4 | 13 | 20 | –2.5 | –24.2 | 6.25 | 585.64 | 60.5 |
5 | 14 | 35 | –1.5 | –9.2 | 2.25 | 84.64 | 13.8 |
6 | 15 | 30 | –0.5 | –14.2 | 0.25 | 201.64 | 7.1 |
7 | 16 | 40 | 0.5 | –4.2 | 0.25 | 17.64 | –2.1 |
8 | 16 | 45 | 0.5 | 0.8 | 0.25 | 0.64 | 0.4 |
9 | 17 | 50 | 1.5 | 5.8 | 2.25 | 33.64 | 8.7 |
10 | 18 | 50 | 2.5 | 5.8 | 6.25 | 33.64 | 14.5 |
11 | 18 | 60 | 2.5 | 15.8 | 6.25 | 249.64 | 39.5 |
12 | 19 | 70 | 3.5 | 25.8 | 12.25 | 665.64 | 90.3 |
13 | 19 | 65 | 3.5 | 20.8 | 12.25 | 432.64 | 72.8 |
201 575 \\overline{\\mathbf{X}} = 15.5 \\overline{\\mathbf{Y}} = 44.2 | 87.25 2692.32 404.6 | ||||||
Substituting the formula
\\mathrm{r}=\\frac{404.6}{\\sqrt[]{87.25\\times 2692.32}}=\\frac{404.6}{\\sqrt[]{234904.92}}=\\frac{404.6}{484.67}=0.84
Let us use the same data In Table 9.1 to calculate the Pearson (r) using the raw score method.
r =\\frac{\\mathbf{N}\\mathbf{\\Sigma }\\mathbf{X}\\mathbf{Y}-\\left(\\mathbf{\\Sigma }\\mathbf{X}\\right)\\left(\\mathbf{\\Sigma }\\mathbf{Y}\\right)}{\\sqrt[]{\\left(\\mathbf{N}\\mathbf{\\Sigma }\\boldsymbol{X}^{2}-(\\mathbf{\\Sigma }\\boldsymbol{X})^{2}\\right)\\left(\\mathbf{N}\\mathbf{\\Sigma }\\boldsymbol{Y}^{2}-(\\mathbf{\\Sigma }\\boldsymbol{Y})^{2}\\right)}}
Where
X and Y = Scores of variables
N = The number of scores.
The following steps can be adopted for calculating Pearson using the raw score method.
Step 1: Enumerate X and Y scores in column A and B
Step 2: Find the sum of X and Y scores to obtain ∑X and ∑Y
Step 3: Square X and Y scores to have X2 and Y2 and write in the C and D column
Step 4: Sum the square X2 and Y2 to have ∑X2 and ∑Y2
Step 5: Multiply the X and Y scores to obtain XY and write the result in column E
Step 6: Obtain ∑ XY in the formula
Step 7: Substitute the values in the stated formula to obtain ‘r’.
Table 9.2: Required Data to compute Pearson ‘r” using Raw scores method
S/N | A (X) | B (Y) | C
\\mathbf{X}^{2} | D
\\mathbf{Y}^{2} | E XY |
1 | 11 | 25 | 121 | 625 | 275 |
2 | 12 | 40 | 144 | 1600 | 480 |
3 | 13 | 45 | 169 | 2025 | 585 |
4 | 13 | 20 | 169 | 400 | 260 |
5 | 14 | 35 | 196 | 1225 | 490 |
6 | 15 | 30 | 225 | 900 | 450 |
7 | 16 | 40 | 256 | 1600 | 640 |
8 | 16 | 45 | 256 | 2025 | 720 |
9 | 17 | 50 | 289 | 2500 | 850 |
10 | 18 | 50 | 324 | 2500 | 900 |
11 | 18 | 60 | 324 | 3600 | 1080 |
12 | 19 | 70 | 361 | 4900 | 1330 |
13 | 19 | 65 | 361 | 4225 | 1235 |
TOTAL 201 575 9295 3125 28125 | |||||
2012 = 40401 5752 = 330625 | |||||
\\mathbf{r}=\\frac{\\mathbf{N}\\mathbf{\\Sigma }\\mathbf{X}\\mathbf{Y}-\\left(\\mathbf{\\Sigma }\\mathbf{X}\\right)\\left(\\mathbf{\\Sigma }\\mathbf{Y}\\right)}{\\sqrt[]{\\left(\\mathbf{N}\\mathbf{\\Sigma }\\mathbf{X}^{2}-(\\mathbf{\\Sigma }\\mathbf{X})^{2}\\right)\\left(\\mathbf{N}\\mathbf{\\Sigma }\\mathbf{Y}^{2}-(\\mathbf{\\Sigma }\\mathbf{Y})^{2}\\right)}}
=\\frac{13\\times 9295-\\left(201\\right)\\left(575\\right)}{\\sqrt[]{(\\left(13\\times 31925)-(201)^{2}\\right)(\\left(13\\times 28125)-(575)^{2}\\right)}}=\\frac{120835\\hbox{--}11572}{\\sqrt[]{(41535-40401)(365625\\hbox{--}330625}\\right)}=\\frac{5263}{\\sqrt[]{(1134)(35000)}}=\\frac{5263}{\\sqrt[]{(39690000)}}=\\frac{5263}{6300}=0.84
Since the r value is the same, the researcher has the liberty to use any of formula to calculate Pearson ‘r for interval or ratio data obtained.
9.5 Computation of Spearman Rank Order Method
Another method of calculating correlation coefficient is Spearman rank order correlation developed by Spearman and Brown. It is known as Spearman rho, it is used by ranking each score in variable through the same direction with respect to magnitude, getting the compare the rankings, then square the difference. Apply the formula below:
Rho = 1- \\frac{6\\sum \\mathbf{D}^{2}}{\\mathbf{N}(\\mathbf{N}\\hbox{--}1)}
Where rho = Spearman rank-order correlation coefficient
d = Difference between ranks
N = Number of ranks
Let us illustrate the calculation of Spearman rho (rs) by using the data obtained by a Researcher who desires to investigate the relationship between student’s test-anxiety and academic achievement in Geometry.
Table 9.3: Test-anxiety and Academic Achievement Score of Five Students
Test-Anxiety (X) | 50 | 40 | 60 | 30 | 20 |
Academic Achievement (Y) | 80 | 70 | 50 | 60 | 40 |
In solving the above example using Table 9.3, the following procedures should be adopted:
Step 1: List out the scores X and Y
Step 2. Rank the scores X and Y from the highest to the lowest. That is to have Rx and Ry
Step 3: Determine the square of each difference (d) Add the d2 to obtain ∑d2
Step 5: Substitute in the formula and calculate the rho value.
(X) | (Y) | Rx | Ry | D | D2 |
50 | 80 | 2 | 1 | 1 | 1 |
40 | 70 | 3 | 2 | 1 | 1 |
60 | 50 | 1 | 4 | –3 | 9 |
30 | 60 | 4 | 3 | 1 | 1 |
20 | 40 | 5 | 5 | 0 | 0 |
Rho = 1- \\frac{6\\sum \\boldsymbol{D}^{2}}{\\mathbf{N}(\\mathbf{N}\\hbox{--}1)} = 1 - \\frac{5\\mathrm{x}12.00}{5(5-1)}
= 1\\hbox{--}\\frac{60}{5\\left(25\\hbox{--}1\\right)}
= 1 - \\frac{60}{5(24)}
= 1 - \\frac{60}{120}
= 1—0.5
= 0.5
By the result obtained Rho = 0.5, there is positive correlation between test-anxiety and academic achievement.
The type and strength of correlations between the two or more variables under study are always determined by a very excellent research study. Correlation refers to the strength of the link that exists when comparing variables, and the correlation coefficient is the statistical index used to measure the relationship.
Student Activity
Using mean deviation and raw score method to compute the correlation coefficient of the data obtained by researcher and comment on the result.
S/N | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
X | 12 | 17 | 62 | 27 | 53 | 33 | 26 | 52 | 14 | 47 | 49 | 11 |
Y | 05 | 07 | 57 | 13 | 46 | 31 | 23 | 27 | 13 | 32 | 43 | 03 |
Motivational Level and Academic Performance Scores of Secondary Students
Students | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Motivational Level Score (X) | 5 | 8 | 10 | 12 | 13 | 8 | 20 | 15 | 17 | 18 |
Academic Performance (Y) | 40 | 45 | 50 | 55 | 60 | 50 | 65 | 55 | 62 | 60 |
Calculate the correlation coefficient using any practical correlation approach.
Using the data below, to calculate the Spearman Rank order correlation(rho).
Students | A | B | C | D | E | F | G | H | I | J |
X Score | 51 | 44 | 80 | 45 | 71 | 70 | 32 | 65 | 19 | 67 |
Y Score | 49 | 41 | 75 | 21 | 64 | 45 | 31 | 50 | 11 | 61 |
Below are scores in multiple choice test in each of a unit in Mathematics and Physics. By 10 students: scores are X and Y respectively
S/N | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Mathematics (X) | 5 | 8 | 10 | 6 | 17 | 19 | 12 | 9 | 12 | 8 |
Physics (Y) | 10 | 8 | 7 | 9 | 9 | 6 | 6 | 8 | 6 | 9 |
Compute the Pearson r between X and Y using mean deviation method.
Calculate the Pearson ‘r’ between X and Y using raw scores approach.
Compute the Spearman rho based on the data provided above.
References
Maruf, O.I & Aliyu, Z. (2013). Measurement and Evaluation in Education printed by: Stevano Printing press, General Printers and publishers.
National Teachers Institute, Kaduna & National Open University of Nigeria (2016). Basic Research Methods in Education.