Scatter Graphs Spearman’s Rank correlation coefficient GCSE Statistics Scatter Graphs Spearman’s Rank correlation coefficient

23 May 2019 Learning Objectives Calculate Spearman’s Rank Correlation Coefficient Use Spearman’s rank correlation coefficient as a measure of agreement or for comparisons of the degree of correlation Interpret correlation as a measure of the strength of the association between two variables, including Spearman’s rank correlation coefficient for ranked data

Ranking means giving the largest value the rank one and the next largest rank two and so on, until all observations are ranked. Spearman’s rank correlation coefficient Correlation is a measure of the strength of the linear association between two variables. Spearman’s rank correlation coefficient (rs) is a numerical measure of strength of the association It is given by rs = 1 - 𝟔𝜮𝒅² 𝒏( 𝒏 𝟐 −𝟏) where d is the difference in ranks and n is the number of observations This formula is at the front of every exam paper on the formulae page 

Interpreting Spearman’s rank correlation coefficient rs = -1 is perfect negative correlation - as one variable increases the other decreases rs = 0 is no correlation rs = +1 is perfect positive correlation – as one variable increases the other increases perfect negative strong negative weak negative correlation no positive weak positive strong perfect positive -1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1

As well as interpreting whether it is positive correlation or negative correlation, you also need to interpret in context. A coefficient of 0.7 can be interpreted as fairly strong positive correlation. If you got a coefficient of -0.9 for the relationship between car age and price, the interpretation would be that this is strong negative correlation. As age increases, price decreases. If you got a coefficient of 0.4 for the relationship between car engine size and car running costs, the interpretation would be that this is weak positive correlation. There is some evidence that engine size increases so running costs increase When you comment on correlation remember to put it in the context ot the question.

Example the table shows the marks given by two judges at a competition Judge 1 24 35 18 19 26 Judge 2 36 40 20 16 Rank 1 3 1 5 4 2 Rank 2 d -3 d² 9 Always remember to rank in the same direction. If you start with the highest value being rank one, you will not make a mistake (or so the book tells me!) ∑d² = 14 so rs = 1 - 6 × 14 5( 5 2 −1) = 1 - 84 120 =1 −0.7=0.3 so the Spearman’s rank correlation coefficient for this data is 0.3 Interpret this as fairly weak positive correlation. The judges agree slightly with each other.

Tied Ranks Exam questions will not contain tied ranks but they may occur in the controlled assessment There may be two or more observations that are equal in rank. These are called tied values. When two or more values are tied they are each given the mean of the ranks that they would have had if they were not tied. the notes for this are on page 197

Your turn exercise 5H page 198 exercise 5I page 201 draw tables to show the ranking and subsequent calculations to obtain full marks