Correlation coefficients are used to measure how strong a relationship is between two variables. There are several types of correlation coefficient, but the most popular is . Pearson’s correlation (also called Pearson’s R) is a correlation coefficient commonly used in linear regression. If you’re starting out in statistics, you’ll probably learn about Pearson’s R first. In fact, when anyone refers to the correlation coefficient, they are usually talking about Pearson’s. Show
Watch the video for an overview of the correlation coefficient, or read on below: Intro to the Correlation Coefficient Watch this video on YouTube. Can’t see the video? Click here. Contents:
Correlation Coefficient Formula: DefinitionCorrelation coefficient formulas are used to find how strong a relationship is between data. The formulas return a value between -1 and 1, where:
Meaning
The of the correlation coefficient gives us the relationship strength. The larger the number, the stronger the relationship. For example, |-.75| = .75, which has a stronger relationship than .65.
Types of correlation coefficient formulas.There are several types of correlation coefficient formulas. One of the most commonly used formulas is Pearson’s correlation coefficient formula. If you’re taking a basic stats class, this is the one you’ll probably use: Pearson correlation coefficientTwo other formulas are commonly used: the sample correlation coefficient and the population correlation coefficient. Sample correlation coefficientSx and sy are the sample standard deviations, and sxy is the sample covariance. Population correlation coefficientThe population correlation coefficient uses σx and σy as the population standard deviations, and σxy as the population covariance. Check out my Youtube channel for more tips and help with statistics! What is Pearson Correlation?Correlation between sets of data is a measure of how well they are related. The most common measure of correlation in stats is the Pearson Correlation. The full name is the Pearson Product Moment Correlation (PPMC). It shows the between two sets of data. In simple terms, it answers the question, Can I draw a line graph to represent the data? Two letters are used to represent the Pearson correlation: Greek letter rho (ρ) for a population and the letter “r” for a sample. Potential problems with Pearson correlation.The PPMC is not able to tell the difference between dependent variables and independent variables. For example, if you are trying to find the correlation between a high calorie diet and diabetes, you might find a high correlation of .8. However, you could also get the same result with the variables switched around. In other words, you could say that diabetes causes a high calorie diet. That obviously makes no sense. Therefore, as a researcher you have to be aware of the data you are plugging in. In addition, the PPMC will not give you any information about the slope of the line; it only tells you whether there is a relationship. Real Life Example Pearson correlation is used in thousands of real life situations. For example, scientists in China wanted to know if there was a relationship between how weedy rice populations are different genetically. The goal was to find out the evolutionary potential of the rice. Pearson’s correlation between the two groups was analyzed. It showed a positive Pearson Product Moment correlation of between 0.783 and 0.895 for weedy rice populations. This figure is quite high, which suggested a fairly strong relationship.
How to Find Pearson’s Correlation CoefficientsBy Hand
How to Find Pearson's Correlation Coefficient (by Hand) Watch this video on YouTube. Can’t see the video? Click here. Example question: Find the value of the correlation coefficient from the following table: SubjectAge xGlucose Level y143992216532579442755578765981Step 1: Make a chart. Use the given data, and add three more columns: xy, x2, and y2. SubjectAge xGlucose Level yxyx2y2143992216532579442755578765981Step 2: Multiply x and y together to fill the xy column. For example, row 1 would be 43 × 99 = 4,257. SubjectAge xGlucose Level yxyx2y2143994257221651365325791975442753150557874959659814779Step 3: Take the square of the numbers in the x column, and put the result in the x2 column. SubjectAge xGlucose Level yxyx2y21439942571849221651365441325791975625442753150176455787495932496598147793481Step 4: Take the square of the numbers in the y column, and put the result in the y2 column. SubjectAge xGlucose Level yxyx2y21439942571849980122165136544142253257919756256241442753150176456255578749593249756965981477934816561Step 5: Add up all of the numbers in the columns and put the result at the bottom of the column. The Greek letter sigma (Σ) is a short way of saying “sum of” or summation. SubjectAge xGlucose Level yxyx2y21439942571849980122165136544142253257919756256241442753150176456255578749593249756965981477934816561Σ247486204851140940022 Step 6: Use the following correlation coefficient formula. The answer is: 2868 / 5413.27 = 0.529809 Click here if you want easy, step-by-step instructions for solving this formula. From our table:
The correlation coefficient =
= 0.5298 The of the correlation coefficient is from -1 to 1. Our result is 0.5298 or 52.98%, which means the variables have a moderate positive correlation.
Correlation Formula: TI 83If you’re taking AP Statistics, you won’t actually have to work the correlation formula by hand. You’ll use your graphing calculator. Here’s how to find r on a TI83. Step 1: Type your data into a list and make a scatter plot to ensure your variables are roughly correlated. In other words, look for a straight line. Not sure how to do this? See: Step 2: Press the STAT button. Step 3: Scroll right to the CALC menu. Step 4: Scroll down to 4:LinReg(ax+b), then press ENTER. The output will show “r” at the very bottom of the list. Tip: If you don’t see r, turn Diagnostic ON, then perform the steps again. How to Compute the Pearson Correlation Coefficient in ExcelWatch the video: Find the Correlation Coefficient in Excel Watch this video on YouTube. Can’t see the video? Click here. Step 1: Type your data into two columns in Excel. For example, type your “x” data into column A and your “y” data into column B. Step 2: Select any empty cell. Step 3: Click the function button on the ribbon. Step 4: Type “correlation” into the ‘Search for a function’ box. Step 5: Click “Go.” CORREL will be highlighted. Step 6: Click “OK.” Step 7: Type the location of your data into the “Array 1” and “Array 2” boxes. For this example, type “A2:A10” into the Array 1 box and then type “B2:B10” into the Array 2 box. Step 8: Click “OK.” The result will appear in the cell you selected in Step 2. For this particular data set, the correlation coefficient(r) is -0.1316. Caution: The results for this test can be misleading unless you have made a scatter plot first to ensure your data roughly fits a straight line. The correlation coefficient in Excel 2007 will always return a value, even if your data is something other than linear (i.e. the data fits an exponential model). That’s it! Subscribe to our Youtube Channel for more Excel tips and stats help. Correlation Coefficient SPSS: Overview.Watch the video for the steps: Pearson's Correlation Coefficient in SPSS Watch this video on YouTube. Can’t see the video? Click here. Step 1: Click “Analyze,” then click “Correlate,” then click “Bivariate.” The Bivariate Correlations window will appear. Step 2: Click one of the variables in the left-hand window of the Bivariate Correlations pop-up window. Then click the center arrow to move the variable to the “Variables:” window. Repeat this for a second variable. Step 3: Click the “Pearson” check box if it isn’t already checked. Then click either a “one-tailed” or “two-tailed” test radio button. If you aren’t sure if your test is one-tailed or two-tailed, see: Is it a a one-tailed test or two-tailed test? Step 4: Click “OK” and read the results. Each box in the output gives you a correlation between two variables. For example, the PPMC for Number of older siblings and GPA is -.098, which means practically no correlation. You can find this information in two places in the output. Why? This cross-referencing columns and rows is very useful when you are comparing PPMCs for dozens of variables. Tip #1: It’s always a good idea to make an of your data set before you perform this test. That’s because SPSS will always give you some kind of answer and will assume that the data is . If you have data that might be better suited to another correlation (for example, exponentially related data) then SPSS will still run Pearson’s for you and you might get misleading results. MinitabWatch this video on how to calculate the correlation coefficient in Minitab: How to Find Pearson's Correlation Coefficient in Minitab Watch this video on YouTube. Can’t see the video? Click here. The Minitab correlation coefficient will return a value for r from -1 to 1. Example question: Find the Minitab correlation coefficient based on age vs. glucose level from the following table from a pre-diabetic study of 6 participants: SubjectAge xGlucose Level y143992216532579442755578765981Step 1: Type your data into a Minitab worksheet. I entered this sample data into three columns. Step 2: Click “Stat”, then click “Basic Statistics” and then click “Correlation.” Step 3: Click a variable name in the left window and then click the “Select” button to move the variable name to the Variable box. For this example question, click “Age,” then click “Select,” then click “Glucose Level” then click “Select” to transfer both variables to the Variable window. Step 4: (Optional) Check the “P-Value” box if you want to display a P-Value for r. Step 5: Click “OK”. The Minitab correlation coefficient will be displayed in the Session Window. If you don’t see the results, click “Window” and then click “Tile.” The Session window should appear. For this dataset:
That’s it! Tip: Give your columns meaningful names (in the first row of the column, right under C1, C2 etc.). That way, when it comes to choosing variable names in Step 3, you’ll easily see what it is you are trying to choose. This becomes especially important when you have dozens of columns of variables in a data sheet! Meaning of the Linear Correlation Coefficient.Pearson’s Correlation Coefficient is a linear correlation coefficient that returns a value of between -1 and +1. A -1 means there is a strong negative correlation and +1 means that there is a strong positive correlation. A 0 means that there is no correlation (this is also called zero correlation). This can initially be a little hard to wrap your head around (who likes to deal with negative numbers?). The Political Science Department at Quinnipiac University posted this useful list of the meaning of Pearson’s Correlation coefficients. They note that these are “crude estimates” for interpreting strengths of correlations using Pearson’s Correlation: r value =+.70 or higherVery strong positive relationship+.40 to +.69Strong positive relationship+.30 to +.39Moderate positive relationship+.20 to +.29weak positive relationship+.01 to +.19No or negligible relationship0No relationship [zero correlation]-.01 to -.19No or negligible relationship-.20 to -.29weak negative relationship-.30 to -.39Moderate negative relationship-.40 to -.69Strong negative relationship-.70 or higherVery strong negative relationshipIt may be helpful to see graphically what these correlations look like: The images show that a strong negative correlation means that the graph has a downward slope from left to right: as the x-values increase, the y-values get smaller. A strong positive correlation means that the graph has an upward slope from left to right: as the x-values increase, the y-values get larger. Cramer’s V CorrelationCramer’s V Correlation is similar to the Pearson Correlation coefficient. While the Pearson correlation is used to test the strength of linear relationships, Cramer’s V is used to calculate correlation in tables with more than 2 x 2 columns and rows. Cramer’s V correlation varies between 0 and 1. A value close to 0 means that there is very little association between the variables. A Cramer’s V of close to 1 indicates a very strong association. Cramer’s V.25 or higherVery strong relationship.15 to .25Strong relationship.11 to .15Moderate relationship.06 to .10weak relationship.01 to .05No or negligible relationshipWhere did the Correlation Coefficient Come From?A correlation coefficient gives you an idea of how well data fits a line or curve. Pearson wasn’t the original inventor of the term correlation but his use of it became one of the most popular ways to measure correlation. Francis Galton (who was also involved with the development of the interquartile range) was the first person to measure correlation, originally termed “co-relation,” which actually makes sense considering you’re studying the relationship between a couple of different variables. In Co-Relations and Their Measurement, he said
It’s worth noting though that Galton mentioned in his paper that he had borrowed the term from biology, where “Co-relation and correlation of structure” was being used but until the time of his paper it hadn’t been properly defined. In 1892, British statistician Francis Ysidro Edgeworth published a paper called “Correlated Averages,” Philosophical Magazine, 5th Series, 34, 190-204 where he used the term “Coefficient of Correlation.” It wasn’t until 1896 that British mathematician Karl Pearson used “Coefficient of Correlation” in two papers: Contributions to the Mathematical Theory of Evolution and Mathematical Contributions to the Theory of Evolution. III. Regression, Heredity and Panmixia. It was the second paper that introduced the Pearson product-moment correlation formula for estimating correlation. Correlation Coefficient Hypothesis Test
Sample problem: test the significance of the correlation coefficient r = 0.565 using the critical values for PPMC table. Test at α = 0.01 for a sample size of 9. Step 1: Subtract two from the sample size to get df, degrees of freedom. Step 2: Look the values up in the PPMC Table. With df = 7 and α = 0.01, the table value is = 0.798 Step 3: Draw a graph, so you can more easily see the relationship. r = 0.565 does not fall into the rejection region (above 0.798), so there isn’t enough evidence to state a strong exists in the data. Relationship to cosineIt’s rare to use trigonometry in statistics (you’ll never need to find the derivative of tan(x) for example!), but the relationship between correlation and is an exception. Correlation can be expressed in terms of angles:
More specifically, correlation is the cosine of an angle between two defined as follows (Knill, 2011):
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