Lecture 14 Show
Association is one of the fundamental tools of scientists. Francis Bacon, for instance, discovered that heat is a form of motion by compiling lists of items that were hot and cold. Ivan Pavlov, who was originally studying the digestive system, discovered an important rule of learning, classical conditioning, by observing that dogs salivated when he rang their dinner bell. In both instances, an association was noted between two variables. As one variable increases, so does the other. The statistical index of the degree to which two variables are associated is the correlation coefficient. Developed by Karl Pearson, it is sometimes called the "Pearson correlation coefficient". The correlation coefficient summarizes the relationship between two variables. Let's take an example. Did you ever wonder whether the person that took the longest on the test did very well or very poorly? It might be that the students who take the longest on the exam are the most careful, and they score the highest. This would be an example of a positive correlation, because high values of one variable (e.g., time spent on the test) are associated with high values on the other variable (e.g., better performance on the test). Or it might be the other way around: longer time on the test is associated with poorer scores. The latter is an example of a negative correlation, because high values on one variable are associated with low values on another variable. A person who scores highly usually finishes quickly. To examine whether there is a positive or negative association between grades on an exam and time spent on an exam, one has to look to see if individuals who did well on the exam also spent longer on it. Here are some hypothetical grades on an exam and the amount of time each student spent on the exam. One way to find out if there is a positive or negative relationship is to examine the list and see if the highest grades are associated with the shortest or longest time spent on the exam. But it is difficult to easily see if there is a relationship between the variables this way. A better method is to create a bivariate scatter plot (bivariate meaning two variables). If we plot the scores from the table above with the time on the x-axis and the grades on the y-axis, we would get something that looks like this: Each point represents one student with a certain score for time on the exam, x, and grade, y. The scatter plot reveals that, in general, longer times on the exam tend to be associated with higher grades. Notice that there is a kind of stream of points moving from the bottom left hand corner of the graph to the upper right hand corner. That indicates a positive association or correlation between the two variables. About r Take a minute to look at some examples of scatter plots with different correlations, by clicking here. In these graphs, the r values are in parentheses. Notice that for the perfect correlation, there is a perfect line of points. They do not deviate from that line. For moderate values of r, the points have some scatter, but there still tends to be an association between the x and the y variables. When there is no association between the variables, the scattering is so great that there is no discernable pattern. Correlations can be said to vary in magnitude and direction. Magnitude refers to the strength of association--higher r values represent stronger relationship between the two variables. Direction refers to whether the relationship is positive or negative, and hence the value of r is positive or negative. About r2 The percentage of shared variance is represented by the square of the correlation coefficient, r2. Another way to visualize this is with a Venn diagram that represents the amount of shared variance, or overlap of variation, of two variables. Click here to see some examples. Because r-square is interpreted as the percentage of shared variance, it is best to compare two r2s rather than two rs. For instance, a correlation of .8 seems to be twice as large as a correlation of .4. But the larger coefficient actually indicates there is 4 times as much shared variance. .64 vs. .16. Occasionally, shared variance is called the variance accounted for in one variable by another variable. An r-square of .64 suggests that x accounts for 64% of the variance in y. Example
The formula for correlation is really just a computational one. It does not make much sense as is, but will give us a correlation coefficient more quickly. To test r for significance, we test the null hypothesis that, in the population, the correlation is zero. To do that we compute a t statistic. The d.f. for the test is n - 2 =18, and we use the usual t table. The critical value is 2.101 at alpha=.05, so the correlation is significantly greater than zero. In other words, there is a statistically significant linear relationship between the grades and time spent on the exam. If we were to measure exams grades and time spent on test in the population, we expect that the correlation between the two would be greater than 0. Standardized Relationship Also, it turns out that correlation can be thought of as a relationship between two variables that have first been standardized or converted to z scores. Correlation Represents a
Linear Relationship Correlation tells you how much two variables are linearly related, not necessarily how much they are related in general. It is true that the most common measure of association is correlation, and, hence, whether or not there is a relationship is usually determined by whether or not there is a correlation. However, there are exceptions. A curvilinear relationship is one example. In some cases, two variables may have a strong, or even perfect, relationship, yet the relationship is not at all linear. In these cases, the correlation coefficient might be zero. Take for example, a well know psychological relationship between arousal and performance. This is referred to as the Yerkes-Dobson law. If someone has very low arousal (e.g. half-asleep), performance on a test will be very poor. If one is moderately aroused, the performance on the test will be high because of stronger motivation. If that arousal becomes too high, as it is with extreme test anxiety, performance on an exam will be very poor. So, overall, there is not a linear relationship between arousal and performance, because there is no general tendency to do better as arousal increases. Here is a graph of the Yerkes-Dobson curve, in which the correlation between arousal and performance is zero, but there is a strong curvilinear relationship. The best way to make sure that your correlation coefficient is not misleading about the relationship between the two variables is to look at a bivariate plot. Restricted range These graphs illustrate restricted range. So, sometimes a small or zero correlation may obtained, because of restricted range rather than because there is not really a true relationship between the two variables. What kind of correlation indicates that two sets of scores tend to rise and fall together?A positive correlation is a relationship between two variables that tend to move in the same direction. A positive correlation exists when one variable tends to decrease as the other variable decreases, or one variable tends to increase when the other increases.
Which correlation indicates the strongest relationship between two variables quizlet?The correlation coefficient, often expressed as r, indicates a measure of the direction and strength of a relationship between two variables. When the r value is closer to +1 or -1, it indicates that there is a stronger linear relationship between the two variables.
What type of correlation occurs when both variables increase in the same direction quizlet?A Positive Correlation is a steady relationship between two variables in the same direction, meaning that as the value of one variable increases, the value of the other increases as well. (And as the value of one variable decreases, so does the other).
What type of measure is positive if two sets of scores such as height and weight tend to rise or fall together?A correlation is positive if two sets of scores, such as for height and weight, tend to rise or fall together. A correlation is negative if two sets of scores relate inversely, one set going up as the other goes down.
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