Correlation This Chapter is on Correlation We will look at patterns in data on a scatter graph We will be looking at how to calculate the variance and co-variance of variables We will see how to numerically measure the strength of correlation between two variables

Correlation Scatter Graphs Scatter Graphs are a way of representing 2 sets of data. It is then possible to see whether they are related. Positive Correlation  As one variable increases, so does the other Negative Correlation  As one variable increases, the other decreases No Correlation  There seems to be no pattern linking the two variables Positive Negative None 6A

Correlation Scatter Graphs In the study of a city, the population density, in people/hectare, and the distance from the city centre, in km, was investigated by choosing sample areas. The results are as follows: Plot a scatter graph and describe the correlation. Interpret what the correlation means. AreaABCDE Distance Pop. Density AreaFGHIJ Distance Pop. Density Distance from centre (km) Pop. Density (people/hectare) The correlation is negative, which means that as we get further from the city centre, the population density decreases.

Correlation Variability of Bivariate Data We learnt in chapter 3 that: In Correlation: Similarly for y: And you can also calculate the Co-variance of both variables 6B/C (Although remember that this formula changed to make it easier to use) ‘How x varies’ ‘How y varies’ ‘How x and y vary together’

Correlation Variability of Bivariate Data Like in chapter 3, we can use a formula which will make calculations easier BUT: 6B/C

Correlation Variability of Bivariate Data Multiply both sides by ‘n’ The easier formula for variance from chapter 3 For the second fraction, square the top and bottom separately Variability of Bivariate Data Multiplying both fractions by ‘n’ will cancel a ‘divide by n’ from each of them 6B/C

Correlation Variability of Bivariate Data These are the formulae for S xx, S yy and S xy. You are given these in the formula booklet. You do not need to know how to derive them (like we just did!) 6B/C

Correlation Variability of Bivariate Data Calculate Sxx, Syy and Sxy, based on the following information. 6B/C

Correlation Variability of Bivariate Data The following table shows babies heads’ circumferences (cm) and the gestation period (weeks) for 6 new born babies. Calculate S xx, S yy and S xy. We need xy y2y x2x Gestation period (y) Head size (x) FEDCBABaby 6B/C

Correlation Variability of Bivariate Data The following table shows babies heads’ circumferences (cm) and the gestation period (weeks) for 6 new born babies. Calculate S xx, S yy and S xy. We need 6B/C

Correlation Product Moment Correlation Coefficient We can test the correlation of data by calculating the Product Moment Correlation Coefficient. This uses S xx, S yy and S xy. The value of this number tells you what the correlation is and how strong it is. The closer to 1, the stronger the positive correlation. The same applies for -1 and negative correlation. A value close to 0 implies no linear correlation. Positive Correlation Negative Correlation 10 No Linear Correlation 6B/C

Correlation Product Moment Correlation Coefficient Given the following data, calculate the Product Moment Correlation Coefficient. There is positive correlation, as x increases, y does as well. 6B/C

Correlation Limitations of the Product Moment Correlation Coefficient Sometimes it may indicate Correlation between unrelated variables  Cars on a particular street have increased, as have the sales of DVDs in town  The PMCC would indicate positive correlation where the two are most likely not linked  The speed of computers has increased, as has life expectancy amongst people  These are not directly linked, but are both due to scientific developments 6B/C

Correlation Using Coding with the PMCC Calculating the PMCC from this table. 6D xy y2y x2x y x

Correlation Using Coding with the PMCC Calculating the PMCC from this table xy y2y x2x y x 6D

Correlation Using Coding with the PMCC Calculating the PMCC from this table, using coding. 6D pq q2q p2p q p y x

Correlation Using Coding with the PMCC Calculating the PMCC from this table. So coding will not affect the PMCC! pq q2q p2p q p y x 6D

Summary We have looked at plotting scatter graphs We have looked at calculating measures of variance, S xx, S yy and S xy We have also seen types of correlation and how to recognise them on a graph We have calculated the Product Moment Correlation Coefficient, and interpreted it. It is a numerical measure of correlation.