What is Pearson’s product-moment correlation coefficient?

• Pearson’s product-moment correlation coefficient (PMCC) is a way of giving a numerical value to a?linear relationship?of bivariate data
  • A?positive value?of?r?describes?positive correlation
  • A?negative value?of?r?describes?negative correlation
  • r?= 0 means there is?no linear correlation
  • r?= 1 means?perfect positive linear?correlation
  • r?= -1 means?perfect negative linear?correlation
  • The closer to 1 or -1 the stronger the correlation

How do I calculate Pearson’s product-moment correlation coefficient (PMCC)?

• You will be expected to use the statistics mode on your GDC to calculate the PMCC
• The formula can be useful to deepen your understanding

• • You?do not need to learn this?as using your GDC will be expected

When does the PMCC suggest there is a linear relationship?

• Critical values?of?r?indicate when the PMCC would suggest there is a linear relationship
  • In your exam you will be given critical values where appropriate
  • Critical values will depend on the size of the sample
• If the?absolute value?of the?PMCC?is?bigger?than the?critical value?then this suggests a linear model is appropriate

What is Spearman’s rank correlation coefficient?

• Each data is ranked from biggest to smallest
  • For?n?data values they are ranked from 1 to?n
  • 1 for the biggest?x-value and 1 for the biggest?y-value
• Spearman’s rank correlation coefficient is a way to measure linear correlation between the rankings of the data
  • A?positive value?of?r_s?describes a?degree of agreement?between the rankings
  • A?negative value?of?r_s?describes a?degree of disagreement?between the rankings
  • r_s?= 0 means there is?no correlation?between the rankings
  • r_s?= 1 means the rankings are in?complete agreement
    • An increase in one variable means an increase in the other
  • r_s?= -1 means the rankings are in?complete disagreement
    • An increase in one variable means a decrease in the other
  • If?r_s?= 1 or?r_s?= -1 then the data is said to have a?monotonic?relationship
    • This means either the points are always increasing or always decreasing
  • The closer to 1 or -1 the stronger the correlation of the rankings

How do I calculate Spearman’s rank correlation coefficient (PMCC)?

• Rank each set of data independently
  • 1 to?n?for the?x-values
  • 1 to?n?for the?y-values
• If some values are equal then give each the average of the ranks they would occupy
  • For example: if the 3^rd, 4^th?and 5^th?highest values are equal then give each the ranking of 4
• Calculate the PMCC of the?rankings?using your GDC
  • This value is?Spearman's rank correlation coefficient

Which correlation coefficient should I use?

• Pearson’s PMCC?tests for a?linear relationship?between two variables
  • It will not tell you if the variables have a non-linear relationship
    • Such as exponential growth
  • Use this if you are interested in a linear relationship
• Spearman’s rank?tests for a?monotonic relationship?(always increasing or always decreasing) between two variables
  • It will not tell you what function can be used to model the relationship
    • Both linear relationships and exponential relationships can be monotonic
  • Use this if you think there is a non-linear monotonic relationship

How are Pearson’s and Spearman’s correlation coefficients connected?

Are Pearson’s and Spearman’s correlation coefficients affected by outliers?

• Pearson’s PMCC?is?affected by outliers
  • as it uses the numerical value of each data point
• Spearman’s rank is?not usually?affected by outliers
  • as it only uses the ranks of each data point