Pearson’s R: Pearson Product-Moment Correlation Coefficient Report by Penny Tan

Scatter Plot Used to illustrate the relationship between two (2) variables by displaying data points on a two (2) dimensional graph X-axis – contains the explanatory variable Y-axis – contains the response variable Useful when there are a large number of data points Provides information about the relationship between 2 variables Strength Shape Direction Presence of outliers

Pearson’s R Also called the Pearson Product-Moment Correlation Coefficient (PMCC) Developed by Karl Pearson from a similar but slightly different idea by Francis Galton in the 1880s. The linear dependence between two variables (x and y) It is used to calculate the degree to which variables are related Direct Inverse Null

Direct relationship Returns a Pearson correlation of +1 High values on one variable tend to occur with high levels on the other variable A perfect positive correlation (or perfect direct relationship) happens when data points make a straight line from origin out to high x and y-values

Direct relationship

Inverse relationship Returns a Pearson correlation of -1 High values on one variable tend to occur with low values on the other variable A perfect negative correlation (or perfect inverse relationship) happens when data points make a straight line from a high y-value to a high x-value

Inverse relationship

Null relationship Returns a Pearson correlation of zero (0) When there is no linear (complete absence of a) relationship

Formula for Pearson’s R 𝑟= 𝑥𝑦 − 𝑥 𝑦 𝑛 𝑥 2 − 𝑥 2 𝑛 𝑦 2 − 𝑦 2 𝑛 Or simply: 𝑟= 𝑛 𝑥𝑦 − 𝑥 𝑦 𝑛 𝑥 2 − 𝑥 2 𝑛 𝑦 2 − 𝑦 2

Significance of the Symbols R – correlation ∑ - sum X – value of x Y – value of y N – number of pairs

Sample

Sample: computation of separate parts of the formula 𝑛=6 𝑥=4+3.56+2.4+3.25+3+3.13=19.34 𝑦 =5+6+10+11+11+12=55 𝑥𝑦 = 4 5 + 3.56 6 + 2.4 10 + 3.25 11 + 3 11 + 3.13 12 =171.67 𝑥 2 = 4 2 + 3.56 2 + 2.4 2 + 3.25 2 + 3 2 + 3.13 2 =63.79 𝑦 2 = 5 2 + 6 2 + 10 2 + 11 2 + 11 2 + 12 2 =547 𝑥 2 = 19.34 2 =374.04 𝑦 2 = 55 2 =3,025

Bringing the parts together 𝒓= 𝒏 𝒙𝒚− 𝒙 𝒚 𝒏 𝒙 𝟐 − 𝒙 𝟐 𝒏 𝒚 𝟐 − 𝒚 𝟐 𝑟= 6 171.67 − 19.34 55 6 63.79 −374.04 6 547 −3,025 𝑟= 1,030.02−1,063.70 8.72 257 𝑟= −33.68 47.35 𝑟=−0.71

Steps to Using MS Excel to Compute Pearson’s R Enter your data in a spreadsheet

Steps to Using MS Excel to Compute Pearson’s R In an empty cell, type “=pearson(array1,array2)” and press ENTER on your keyboard Array1 = values of x Array2 = values of y

Steps to Using MS Excel to Compute Pearson’s R The cell will now display the value of r

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