Coefficient de corrélation de Pearson : Calcul + Exemples

If xx is the independent variable and yy the dependent variable, then we can use a regression line to predict yy for a given value of xx. The Pearson Correlation Coefficient (r) is a statistical measure of the strength and direction of a linear relationship between two variables on a scatterplot. It ranges from -1 to 1, with 1 indicating a perfect positive relationship, -1 indicating a perfect negative relationship, and 0 indicating no linear relationship. The formula involves summing products of paired scores and dividing by the square root of the product of the sums of squared scores.

Linear Correlation

Interpretation of correlation coefficients differs significantly among scientific research areas. Therefore, authors should avoid overinterpreting the strength of associations when they are writing their manuscripts. In this context, the utmost importance should be given to avoid misunderstandings when reporting correlation coefficients and naming their strength. In Table 1, we provided a combined chart of the three most commonly used interpretations of the r values. Authors of those definitions are from different research areas and specialties.

Finding the Correlation Coefficient with Pearson Correlation Coefficient Formula

For example, it would be unethical to conduct an experiment on whether smoking causes lung cancer. Correlation does not always prove causation, as a third variable may be involved. For example, being a patient in a hospital is correlated with dying, but this does not mean that one event causes the other, as another third variable might be involved (such as diet and level of exercise). Decide which variable goes on each axis and then simply put a cross at the point where the two values coincide.

While r quantifies the degree of linear association, it doesn’t imply causation. Developed by Francis Galton, Auguste Bravais, and Karl Pearson, it’s foundational in fields like psychology and economics, aiding in the analysis of linear relationships under certain assumptions about the data. To assess linear correlation, examine the graphical trend of the data points on the scatterplot to determine if a straight-line pattern exists (see Figure 4.5). If a linear pattern exists, the correlation may indicate either a positive or a negative correlation. If there is no relationship or association between the two quantities, where one quantity changing does not affect the other quantity, we conclude that there is no correlation between the two variables. The correlation coefficient, rr, tells us about the strength and direction of the linear relationship between xx and yy.

Table 2.

A positive correlation indicates that as one variable increases, the other tends to increase as well. Conversely, a negative correlation suggests that as one variable goes up, the other tends to go down. The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population (Figure 4.8).

User’s guide to correlation coefficients

• An important step in the correlation analysis is to determine if the correlation is significant.
• Correlation allows the researcher to clearly and easily see if there is a relationship between variables.
• Conversely, a negative correlation suggests that as one variable goes up, the other tends to go down.
• If xx is the independent variable and yy the dependent variable, then we can use a regression line to predict yy for a given value of xx.

Often, this phrase is uttered with a tone of interpretation of correlation coefficient disdain, seeming to imply that “correlations” are inferior to “causal” links. This is unfortunate because correlations can be interesting even if they don’t present causal relationships. Correlations can tell us interesting things and can help us understand possible causal links. But we need to be careful and nuanced when understanding and interpreting such correlations. From the scatterplot in the Nike stock versus S&P 500 example, we note that the trend reflects a positive correlation in that as the value of the S&P 500 increases, the price of Nike stock tends to increase as well.

This approach is useful when certain observations carry more significance or have different levels of precision. Each type of Pearson correlation coefficient offers unique insights and analytical tools for various research fields, from statistics and psychology to economics and engineering. Understanding these variations enhances the accuracy and depth of correlation analyses, enabling more informed decision-making and hypothesis testing. Getting creative with our answers to these questions is one way to avoid conflating correlation and causation.

We start to answer this question by gathering data on average daily ice cream sales and the highest daily temperature. Ice Cream Sales and Temperature are therefore the  two variables  which we’ll use to calculate the correlation coefficient. Sometimes data like these are called  bivariate data, because each observation (or point in time at which we’ve measured both sales and temperature) has two pieces of information that we can use to describe it. In other words, we’re asking whether Ice Cream Sales and Temperature seem to move together.

The hypothesis test lets us decide whether the value of the population correlation coefficient ρρ is close to zero or significantly different from zero. We decide this based on the sample correlation coefficient rr and the sample size n. When writing a manuscript, we often use words such as perfect, strong, good or weak to name the strength of the relationship between variables. However, it is unclear where a good relationship turns into a strong one. Therefore, there is an absolute necessity to explicitly report the strength and direction of r while reporting correlation coefficients in manuscripts. Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied.

• Correlation only looks at the two variables at hand and won’t give insight into relationships beyond the bivariate data.
• An automotive engineer is interested in the correlation between outside temperature and battery life for an electric vehicle, for instance.
• A typical threshold for rejection of the null hypothesis is a p-value of 0.05.
• In this case, it’ll look as though people who have higher reported sexual encounters also have higher wages when in fact this relationship is due to them misreporting their sexual activity.
• Similarly, looking at a scatterplot can provide insights on how outliers—unusual observations in our data—can skew the correlation coefficient.

We can also look at these data in a table, which is handy for helping us follow the coefficient calculation for each datapoint. When talking about bivariate data, it’s typical to call one variable  X  and the other  Y  (these also help us orient ourselves on a visual plane, such as the axes of a plot). Causation means that one variable (often called the predictor variable or independent variable) causes the other (often called the outcome variable or dependent variable). Remember, in correlations, we always deal with paired scores, so the values of the two variables taken together will be used to make the diagram. This is done by drawing a scatter plot (also known as a scattergram, scatter graph, scatter chart, or scatter diagram). This article explains the types, methods, and practical examples of correlation analysis, along with its applications and limitations.

Today’s media provide a steady stream of information, including reports on all the latest links that have been found by researchers. Also, add all the values in the columns to get the values used in the formula. In the following example, Python will be used for the following analysis based on a given dataset of (x,y)(x,y) data. These formulas can be quite cumbersome, especially for a significant number of data pairs, and thus software is often used (such as Excel, Python, or R). For example, a doctor observes that people who take vitamin C each day seem to have fewer colds.

The correlation coefficient (r) indicates the extent to which the pairs of numbers for these two variables lie on a straight line. Values over zero indicate a positive correlation, while values under zero indicate a negative correlation. Pearson’s distance measures the dissimilarity or similarity between two data points based on their correlation coefficient. It quantifies the extent of deviation from perfect correlation, providing insights into the relationship between variables. Typically, technology is used to calculate the best-fit linear model as well as calculate correlation coefficients and scatterplot.

Therefore, an endless struggle to link what is already known to what needs to be known goes on. We try to infer the mortality risk of a myocardial infarction patient from the level of troponin or cardiac scores so that we can select the appropriate treatment among options with various risks. We are trying to calculate the risk of mortality from the level of troponin or TIMI score. The most basic form of mathematically connecting the dots between the known and unknown forms the foundations of the correlational analysis.

Correlation analysis is a powerful statistical tool for exploring and quantifying relationships between variables. By understanding its types, methods, and applications, researchers can draw meaningful insights from their data. While correlation analysis is widely applicable, it is essential to use it appropriately, recognize its limitations, and complement it with further analysis for robust conclusions. Partial correlation evaluates the relationship between two variables while controlling for the effects of one or more additional variables. It measures the unique association between variables after accounting for the influence of other factors, allowing researchers to isolate specific statistical relationships.

Note that since rr is calculated using sample data, rr is considered a sample statistic and is used to measure the strength of the correlation for the two population variables. Recall that sample data indicates data based on a subset of the entire population. For example, let’s say that a financial analyst wants to know if the price of Nike stock is correlated with the value of the S&P 500 (Standard & Poor’s 500 stock market index). To investigate this, monthly data can be collected for Nike stock prices and value of the S&P 500 for a period of time, and a scatterplot can be created and examined.

The formula for rr is shown; however, software is typically used to calculate the correlation coefficient. When inspecting a scatterplot, it may be difficult to assess a correlation based on a visual inspection of the graph alone. A more precise assessment of the correlation between the two quantities can be obtained by calculating the numeric correlation coefficient (referred to using the symbol rr). If you are a confused consumer when it comes to links and correlations, take heart; this article can help. You’ll gain the skills to dissect and evaluate research claims and make your own decisions about those headlines and sound bites that you hear each day alerting you to the latest correlation. You’ll discover what it truly means for two variables to be correlated, when a cause-and-effect relationship can be concluded, and when and how to predict one variable based on another.

As mentioned, given the complexity of this calculation, software is typically used to calculate the correlation coefficient. There are several options for calculating the correlation coefficient in Python. The example shown uses the scipy.stats library in Python which includes the function pearsonr() for calculating the Pearson correlation coefficient (rr).