Upon completion of this chapter, you should be able to:

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1 1 Chaptter 7:: CORRELATIION Upon completion of this chapter, you should be able to: Explain the concept of relationship between variables Discuss the use of the statistical tests to determine correlation Interpret SPSS outputs on correlation tests CHAPTER OVERVIEW Introduction What is correlation coefficient? o Pearson product moment Range of values o Positive correlation o Negative correlation o Zero correlation Calculation of the correlation coefficient SPSS correlation coefficient Correlation and causation Summary Key Terms Chapter 1: Introduction Chapter 2: Descriptive Statistics Chapter 3: The Normal Distribution Chapter 4: Hypothesis Testing Chapter 5: T-test Chapter 6: Oneway Analysis of Variance Chapter 7: Correlation Chapter 8: Chi-Square This chapter introduces the concept of correlation and how it is used in analysing educational data. The correlation coefficient is a useful statistical tool in showing the relationship between two variables. The relationship can range from 1.00 to ; though in the behavioural sciences seldom is there a perfect positive or negative correlation between two variables. However, it should be emphasised that correlation is not causation. In other words, even though there is a high correlation between A and B; it does not mean that A caused B. Introduction

2 2 Researchers are often concerned with the way two variables relate to each other for a given groups of persons such as students in schools, workers in a factory or office. For example, do students who have higher scores in mathematics also have higher scores in mathematics? Is there a relationship between a person's self-esteem and his or her personality? Is there a relationship between attitudes towards reading and the number of books read? Is there a relationship between years of experience as a teacher and attitudes towards teaching? These are some of the questions asked by educational researchers. To answer these questions, you must make observations or collect data for each variable for a group of persons. What is Correlation Coefficient? The correlation coefficient a concept from statistics is a measure of how well trends in the predicted values follow trends in past actual values. It is a measure of how well the predicted values from a forecast model "fit" with the real-life data. The correlation coefficient is a number between 0 and 1. If there is no relationship between the predicted values and the actual values the correlation coefficient is 0 or very low (the predicted values are no better than random numbers). As the strength of the relationship between the predicted values and actual values increases so does the correlation coefficient. A perfect fit gives a coefficient of 1.0. Thus the higher the correlation coefficient the better. a) Pearson Product Moment Correlation Coefficient Pearson's product moment correlation coefficient, usually denoted by r, is one example of a correlation coefficient. It is a measure of the linear association between two variables that have been measured on interval or ratio scales, such as the relationship between amount of education and income levels. If there is a relationship between amount of education and income levels, the two variables co-vary. b) Assumptions Testing Correlational analysis has the following underlying assumptions: (S. Coakes and L. Steed, 2002, SPSS Analysis Without Anguish. Brisbane: John Wiley & Sons) Related Pairs the data to be collected from related pairs: i.e. if you obtain a score on an X variable, there must ne a score on the Y variable from the same subject. Scale of Measurement data should be interval or ration in nature Normality the scores for each variable should be normally distributed Linearity the relationship between the two variables must be linear

3 3 Homogeneity of Variance the variability in scores for one variable is roughly the same at all values of the other variable; i.e. it is concerned with how the scores cluster uniformly about the regression line. c) Strength of the Correlation The strength of a relationship is indicated by the size of the correlation coefficient: the larger the correlation, the stronger the relationship. A strong relationship exists where cases in one category of the X variable usually have a particular value on the Y variable while those in a different value of X have a different value on Y. For example, if people who exercise regularly nearly always have better health than those who do not exercise, then exercise and health are more strongly correlated. If those who exercise regularly are just a little more likely to be healthy than the nonexercisers then the two variables are only weakly related. How high does a correlation coefficient have to be to be called strong? How small is weak correlation? The answer to these questions varies with the variables being studied. For example, if the literature shows that in previous research, a correlation of 0.51 was found between variable X and variable Y, but in your study you obtained a correlation of 0.60; then you might conclude that the correlation between variable X and Y is strong. However, Cohen (1988) has provided some guidelines to determine the strength of the relationship between two variables by providing descriptors for the coefficients (see Table 7.1). Keep in mind that in education and psychology it is rarely that the coefficients will be very strong or near perfect since the variables measured are constructs involving human characteristics which are subject to wide variation. Trivial Low to Moderate to Substantial to Very Strong Near Moderate Substantial Very Strong Perfect > 0.90 Table 7.1 General guidelines on the strength of the relationship between variables

4 4 EXAMPLE: Data was gathered for the following two variables from a sample of 12 students. Student No. IQ Test Scores (Science Test) (X) (Y) Each unit or student is represented by a point on the scatter diagram. A dot is placed for each student at the point of intersection of a straight line drawn through his IQ score perpendicular to the X axis and through his Science score perpendicular to the Y axis. For example, a student who obtained an IQ score of 120 also obtained a Science score of 24. The intersection between these lines is represented by the dot 'A'. The scatter diagram (see Figure 7.1) which shows a moderate positive relationship between IQ Scores and Science Scores. However, we do not have a summarised measure of this relationship. There is need for a more precise measure to describe the relationship between the two variables. You need a numerical descriptive measure of the correlation between IQ scores and Science scores which will be discussed later.

5 5 A Figure 7.1 Scatter Diagram Showing the Relationship between IQ Scores (X axis) and Science Score (Y axis) for 12 Students Range of Values (rxy) Note that rxy can never take on a value less than - 1 nor a value greater than + 1. The following are three graphs showing various values of rxy and the type of linear relationship that exists between X and Y for the given values of rxy.

6 6 a) POSITIVE CORRELATION Value of rxy = = Perfect & Direct Relationship 5 English Score [y axis] Attitude Towards English [x axis] Figure 7.2 Perfect Correlation See Figure 7.2. If Attitudes (x) and English Achievement (y) had a positive relationship than the Slope (β1) will be a positive number. Lines with positive slopes go from the bottom left toward the upper right. i.e. and increase from 1 to 2 on the x axis is followed by an increase from 3 to 3.5 on the y axis.

7 7 b) NEGATIVE CORRELATION Value of rxy = 1.00 = Perfect Inverse Relationship 5 Score [y axis] 4 English Attitude Towards English [x axis] Figure 7.3 Negative Correlation If Attitudes (x) and English Achievement (y) have a negative relationship than the Slope (β1) will be a negative number. Lines with negative slopes go from the upper right to the lower left. The above graph has a slope of -1. An increase of 1 on the X axis is associated with a decrease of 0.5 on the Y Axis; i.e. an increase from 1 to 2 on the x axis is followed by a decrease from 5 to 4.5 on the y axis.

8 8 c) ZERO CORRELATION Value of rxy =.00 = No Relationship 5 English Score [y axis] Attitude Towards English [x axis] Figure 7.4 No Correlation If Attitudes (x) and English Achievement (y) have NO relationship than the Slope (β1) will be ZERO (see Figure 7.4). In other words, there is NO SYSTEMATIC RELATIONSHIP between X and Y. Some students with high Attitude scores have positive low English scores while some students have low Attitude score have high positive English scores..

9 9 Correlation of the Correlation Coefficient (r or rxy) A researcher conducted a study to determine the relationship between verbal and spatial ability. She was interested in finding out whether students who scored high on verbal ability also scored high on spatial ability. She administered two 15 item tests measuring verbal and spatial ability to a sample 12 primary school students. The results of the study are shown in the table below: Student Verbal Test Spatial Test x y x² y² xy Σx = 135 Σy = 114 Σx² = 1566 Σy² =1139 Σxy =1305 a) Illustration Of The Calculation Of The Correlation Coefficient (R or Rxy) for the Data in the Table Above. The Pearson Correlation Coefficient (called the Pearson r) is the commonly used formula in computing the correlation between two variables. The formula measures the strength and direction of a linear relationship between variable X and variable Y. The sample correlation coefficient is denoted by r. The formula for the sample correlation coefficient is:

10 10 (Σ x) (Σ y) SSxy = Σ xy = n (Σ x)² SSxx = Σ x² = n (Σy)² SSyy = Σ y² = n Using the formula to obtain the correlation coefficient : = (47.50)(56.00) = To Obtain A Bivariate Pearson Product-Moment Correlation Using SPSS A study was conducted to determine the relationship between reading ability and performance in science. A reading ability and science test was administered to 200 lower secondary students. The Pearson product-moment correlation was used to determine the significance of the relationship. The steps for using SPSS is shown below:

11 11 SPSS Procedures: 1. Select the Analyze menu. 2. Click on Correlate and then Bivariate... to open the Bivariate Correlations dialogue box. 3. Select the variables you require (i.e. reading and science) and click on the button to move the variables into the Variables: box. 4. Ensure that the Pearson correlation option has been selected. 5. In the Test of Significance box, select the One-tailed radio button. 6. Click on OK. SPSS Output: Reading Science Reading Pearson Correlation ** Sig. (1-tailed) N Science Pearson Correlation 0.630** Sig. (1-tailed) N To interpret the correlation coefficient, you examine the coefficient and its associated significance value (p). The output show that the relationship between Reading and Science scores is significant with a correlation coefficient of r = 0.63 which is p <.05. Thus higher reading scores are associated with higher scores in science. NULL HYPOTHESIS The null hypothesis (Ho:) states that the correlation between X and Y is ρ = 0.0. What is the probability that the correlation obtained in the sample came from a population where the parameter ρ = 0.0? The t-test for the significance of a correlation coefficient is used. Note that the correlation between Reading and Science (r = 0.630) is significant at p < Hence, the null hypothesis is REJECTED which affirms that the two variables are positively related in the population.

12 12 Coefficient of Determination r = the correlation between X and Y = and r² = the coefficient of determination = (0.630)² = Hence 39.6% of the variance in Y that can be explained by X. TO OBTAIN A SCATTERPLOT USING SPSS SPSS Procedures: 1. Select the Graph menu. 2. Click on Scatter... to open the Scatterplot dialogue box 3. Ensure Simple Scatterplot option is selected. 4. Click on the Define command pushbutton to open the Simple Scatterplot subdialogue box. 5. Select the first variable (i.e. science) and click on the button to move the variable into the Y Axis: box.. 6. Select the second variable (i.e. reading) and click on the button to move the variable into the X Axis: box. 6. Click on OK.

13 13 SPSS Output SCIENCE READING Figure 7.4 Scatterplot As you can see from the scatterplot (Figure 7.4) there is a linear relationship between reading and Science scores. Given that the scores cluster uniformly around the regression line, the assumption of homogeneity of variance has not been violated. Causation And Correlation Causation and correlation are two concepts that has been wrongly interpreted by some researchers. The presence of a correlation between two variables does not necessarily mean there exists a causal link between them. Say for instance that the there is a correlation (0.60) between "teachers salary" and "academic performance of students". Does this imply that a well-paid teaching staff "cause" better academic performance of students? Would the percent of academic performance increase if we increased the pay of teachers? It is dangerous to conclude causation just because there is a correlation or relationship between two variables. It tells nothing by itself about whether "teachers salary" causes "achievement".

14 14 Significance Of The Correlation Coefficient We introduced Pearson correlation as a measure of the strength of a relationship between two variables. But any relationship should be assessed for its significance as well as its strength. The significance of the relationship is expressed in probability levels: p (e.g., significant at p =.05). This tells how unlikely a given correlation coefficient, r, will occur given no relationship in the population. It assumes that you have a sample of cases from a population. The question is whether your observed statistic for the sample is likely to be observed given some assumption of the corresponding population parameter. If your observed statistic does not exactly match the population parameter, perhaps the difference is due to sampling error. To be useful, a correlation coefficient needs to be accompanied by a test of statistical significance. It is also important for you to know about the sample size. Generally, a strong correlation in a small population may be statistically nonsignificant, while a much weaker correlation in a large sample may be statistically significant. For example, in a large sample, even low correlations (as low as 0.06) can be statistically significant. Similar sized correlations that are statistically significant with large samples are not significant for the smaller samples, This is because with smaller samples the likelihood of sampling error is higher.

15 15 LEARNING ACTIVITY A researcher conducted a study which aimed to determine the relationship between self-efficacy and academic performance in geography. A 20 item selfefficacy scale and a 25 item geography test was administered to a group of 12 students. The following are the results of the study: Self-Efficacy Scale Geography Test a) What is the correlation coefficient? b) What is the mean for the self-efficacy scale and the mean or the geography test? c) Comment on the scatter plot. SUMMARY

16 16 The correlation coefficient a concept from statistics is a measure of how well trends in the predicted values follow trends in past actual values. Pearson's product moment correlation coefficient, usually denoted by r, is a measure of the linear association between two variables The null hypothesis (Ho:) states that the correlation between X and Y is ρ = 0.0. The presence of a correlation between two variables does not necessarily mean there exists a causal link between them. The strength of a relationship is indicated by the size of the correlation coefficient: the larger the correlation, the stronger the relationship. The scatterplot is a graphical representation of the intersection of a point on the x-axis with the point on the y-axis. The presence of a correlation between two variables does not necessarily mean there exists a causal link between them. The coefficient of determination is the proportion of variance in Y that can be explained by X. KEY WORDS: Correlation Correlation coefficient Pearson product moment Range of values Positive correlation Negative correlation Zero correlation Scatterplot Causation Coefficient of determination