The correlation requires two scores from the same individuals. These scores are normally identified as X and Y. The pairs of scores can be listed in a table or presented in a scatterplot. Example: We might be interested in the correlation between your SAT-M scores and your GPA at UNC. Here are the Math SAT scores and the GPA scores of 13 of the students in this class, and the scatterplot for all 41 students:
The scatterplot has the X values (GPA) on the horizontal (X) axis, and the Y values (MathSAT) on the vertical (Y) axis. Each individual is identified by a single point (dot) on the graph which is located so that the coordinates of the point (the X and Y values) match the individual's X (GPA) and Y (MathSAT) scores. For example, the student named "Obs5" (in the sixth row of the datasheet) has GPA=2.30 and MathSAT=710. This student is represented in the scatterplot by high-lighted and labled ("5") dot in the upper-left part of the scatterplot. Note that is to the right of MathSAT of 710 and above GPA of 2.30. Note that the Pearson correlation (explained below) between these two variables is .32.
In the example above, GPA and MathSAT are positively related. As GPA (or MathSAT) increases, the other variable also tends to increase.
The direction of the relationship between two variables is identified by the sign of the correlation coefficient for the variables. Postive relationships have a "plus" sign, whereas negative relationships have a "minus" sign.
 In this course we only deal with correlation coefficients that measure linear relationship. There are other correlation coefficients that measure curvilinear relationship, but they are beyond the introductory level.
Finally, a correlation coefficient measures the degree (strength) of the relationship between two variables. The mesures we discuss only measure the strength of the linear relationship between two variables. Two specific strengths are:
There are strengths in between -1.00, 0.00 and +1.00. Note, though. that +1.00 is the largest postive correlation and -1.00 is the largest negative correlation that is possible. Here are three examples: Weight and Horsepower
The relationship between Weight and Horsepower is strong, linear, and positive, though not perfect. The Pearson correlation coefficient is +.92. Drive Ratio and Horsepower
The relationship between drive ratio and Horsepower is weekly negative, though not zero. The Pearson correlation coefficient is -.59. Drive Ratio and Miles-Per-Gallon
The relationship between drive ratio and MPG is weekly positive, though not zero. The Pearson correlation coefficient is .42.
For example, we require high school students to take the SAT exam because we know that in the past SAT scores correlated well with the GPA scores that the students get when they are in college. Thus, we predict high SAT scores will lead to high GPA scores, and conversely.
This is a process for validating the new test of intelligence. The process is based on correlation.
A positive correlation is a relationship between two variables that move in tandem—that is, in the same direction. A positive correlation exists when one variable decreases as the other variable decreases, or one variable increases while the other increases. Because these two different variables move in the same direction, they theoretically are influenced by the same external forces.
A perfectly positive correlation means that 100% of the time, the variables in question move together by the exact same percentage and direction. A positive correlation can be seen between the demand for a product and the product's associated price. In situations where the available supply stays the same, the price will rise if demand increases. Additionally, gains or losses in certain markets may lead to similar movements in associated markets. As the price of fuel rises, the prices of airline tickets also rise. Since airplanes require fuel to operate, an increase in this cost is often passed to the consumer, leading to a positive correlation between fuel prices and airline ticket prices. A positive correlation does not guarantee growth or benefit. Instead, it is used to denote any two or more variables that move in the same direction together, so when one increases, so does the other. The existence of a correlation does not necessarily indicate a causal relationship between variables. Correlation is a form of dependency, where a shift in one variable means a change is likely in the other, or that certain known variables produce specific results. A general example can be seen within complementary product demand. If the demand for vehicles rises, so will the demand for vehicular-related products and services, such as tires. An increase in one area has an effect on complementary industries. In some situations, positive psychological responses can cause positive changes within an area. This can be demonstrated within the financial markets, in cases where general positive news about a company leads to a higher stock price.
Correlation among variables does not necessarily imply causation. In statistics, a perfect positive correlation is represented by the correlation coefficient value +1.0, while 0 indicates no correlation, and -1.0 indicates a perfect inverse (negative) correlation. Positive correlation may also be easily identified by graphically depicting a data set using a scatterplot. Each point on a scatterplot represents one sample item at the intersection of the x-axis variable and y-axis variable. A positive correlation on a scatterplot is evidenced by an upward trending series of points that show that as the x-axis variable increases, so does the y-axis variable. When statistically analyzing positive correlation, it is important to understand the dataset's p-value. P-value is the statistical measurement of how statistically significant the findings are. In general, a higher p-value indicates there is greater evidence that two data points are more strongly correlated. A simple example of positive correlation involves the use of an interest-bearing savings account with a set interest rate. The more money that is added to the account, whether through new deposits or earned interest, the more interest that can be accrued. Similarly, a rise in the interest rate will correlate with a rise in interest generated, while a decrease in the interest rate causes a decrease in actual interest accrued. Investors and analysts also look at how stock movements correlate with one another and with the broader market. Most stocks have a correlation between each other's price movements somewhere in the middle of the range, with a coefficient of 0 indicating no relationship whatsoever between the two securities. A stock in the online retail space, for example, likely has little correlation with the stock of a tire and auto body shop, while two similar retail companies will see a higher correlation. This is because businesses that have very different operations will produce different products and services using different inputs. Each of these companies face different risks, opportunities, and operational challenges. Modern portfolio theory is heavily rooted in diversification, the concept that an investor should hold assets that are widely unrelated to reduce portfolio-wide risk. This flies in the face of positive correlation; investing theory usually states that investors should be wary of widespread positive correlation within their portfolio. For most investors, an ideal investing strategy is to avoid positive correlation between assets and asset classes. Though every individual should evaluate their own investing strategy, holding assets with positive correlation tends to increase the risk of loss. Beta is a common measure of how correlated an individual stock's price is with the broader market, often using the S&P 500 index as a benchmark. If a stock has a beta of 1.0, it indicates that its price activity is strongly correlated with the market. A stock with a beta of 1.0 has a systematic risk, but the beta calculation can’t detect any unsystematic risk. Adding a stock to a portfolio with a beta of 1.0 doesn’t add any risk to the portfolio, but it also doesn’t increase the likelihood that the portfolio will provide an excess return. A beta of less than 1.0 means that the security is theoretically less volatile than the market, meaning the portfolio is less risky with the stock included than without it. For example, utility stocks often have low betas because they tend to move more slowly than market averages. A beta that is greater than 1.0 indicates that the security's price is theoretically more volatile than the market. For example, if a stock's beta is 1.2, it is assumed to be 20% more volatile than the market. Technology stocks and small caps tend to have higher betas than the market benchmark. This indicates that adding the stock to a portfolio will increase the portfolio’s risk, but also increase its expected return. Some stocks even have negative betas. A beta of -1.0 means that the stock is inversely correlated to the market benchmark as if it were an opposite, mirror image of the benchmark’s trends. Put options or inverse ETFs are designed to have negative betas, but there are a few industry groups, like gold miners, where a negative beta is also common.
A beta of +1.0 indicates a stock that moves in the same direction as the rest of the market. A beta of -1.0 indicates that a stock moves opposite to the rest of the market. Negative correlation is sometimes described as inverse correlation. In statistics, positive correlation describes the relationship between two variables that change together, while an inverse correlation describes the relationship between two variables which change in opposing directions. Examples of positive correlations occur in most people's daily lives. The more hours an employee works, for instance, the larger that employee's paycheck will be at the end of the week. The more money is spent on advertising, the more customers buy from the company. Inverse correlations describe two factors that seesaw relative to each other. Examples include a declining bank balance relative to increased spending habits and reduced gas mileage relative to increased average driving speed. One example of an inverse correlation in the world of investments is the relationship between stocks and bonds. In theory, as stock prices rise, the bond market tends to decline, just as the bond market does well when stocks are underperforming. It is important to understand that correlation does not necessarily imply causation. Variables A and B might rise and fall together, or A might rise as B falls, but it is not always true that the rise of one factor directly influences the rise or fall of the other. Both may be caused by an underlying third factor, such as commodity prices, or the apparent relationship between the variables might be a coincidence. The number of people connected to the Internet, for example, has been increasing since its inception, and the price of oil has generally trended upward over the same period. This is a positive correlation, but the two factors almost certainly have no meaningful relationship. That both the population of Internet users and the price of oil have increased is explainable by a third factor, namely, general increases due to time passed.
One example of positive correlation is the relationship between employment and inflation. High levels of employment require employers to offer higher salaries in order to attract new workers, and higher prices for their products in order to fund those higher salaries. Conversely, periods of high unemployment experience falling consumer demand, resulting in downward pressure on prices and inflation.
The most common way to determine a positive correlation is to calculate the correlation coefficient. This statistical measurement calculates the strength of the relationship between two variables.
A correlation coefficient of 1.0 means that two variables have perfectly positive correlation. As one variable changes, so does the other. Though this does not mean that one variable directly impacts the outcome or changes to the other, both variables always move in tandem and are most likely highly related.
The correlation between two variables can be evaluated by determining the dataset's correlation coefficient and p-value. Both measurements analyzed together demonstrate the strength of the relationship between the variables and the reliability of the data.
Correlation does not require causation, and it is a common logical fallacy to believe otherwise. When two variables are positively correlated, that does not necessarily mean that one variable causes changes in the other. Both variables may be influenced by an unknown third factor, or the apparent relationship between the variables might be a coincidence. When two variables move in tandem, the two variables are said to have a positive correlation. Though one variable may not directly influence the other, the two variables may at least change in the same direction. Investors trying to minimize portfolio risk often try to shed positive correlation through diversification; this is done by analyzing the correlation coefficient, beta, and other statistical measurements of each of the variables. |
Is a variable that explains how two variables are related in different ways for different people or different situations?
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