1. Correlation Lesson 2 Aims: To be able to calculate the PMCC, r, given summarised data. To know that the value of the PMCC is unchanged by linear coding To know the limitations of the PMCC

2. Product–moment correlation coefficient where: Usually, the second version of each formula is used. Remember the formula for the PMCC we met last lesson was:

3. Check h/w together!

4. Examination style question: A researcher believes there is a relationship between a country’s annual income per head ( x, in $1000) and the per capita carbon dioxide emissions (c, tonnes). He collects data from a random sample of 10 countries and records the following results: Product-moment correlation coefficient Calculate the value of the product–moment correlation coefficient and comment on the implications of your answer. 5 minutes have a go!

5. So: Therefore, the product-moment correlation coefficient is: Product-moment correlation coefficient Income shows weak positive correlation with CO 2 emissions – emissions are generally higher in wealthier countries. However, as the correlation is low, the result is somewhat inconclusive.

6. PMCC on w/b

8. Effect of coding on the correlation Example: The heights (in cm) of a sample of 11 men and their adult sons were first coded by subtracting 160 cm off each measurement. Then these summarised data values were calculated from the coded data: where u = coded height of father (in cm) and v = coded height of son (in cm). 1.Calculate the value of the product–moment correlation coefficient between the fathers’ and sons’ coded heights. 2.By considering the scatter plots of both coded and original data. State the PMCC of the original data S uv = 1127.55 S uu = 956.727 and S vv = 1492.909 So: 0.943 (to 3 sig. figs.)

9. Effect of coding on the correlation The value of the product–moment correlation coefficient is unaffected by linear transformations of the variables. Do you think the PMCC value would be different if the heights had been measured in meters instead of cm’s? No the relationship between the heights would be the same! We can see the graphs look the same and so the PMCC between the father and sons’ height must also be 0.943

10. The product–moment correlation coefficient (PMCC) measures the strength of a linear relationship. However: Limitations of the PMCC Outliers can greatly distort the PMCC; The PMCC is not a suitable measure of correlation if the relationship is non-linear. Independent study. Do exercise 6B page 155 questions 1 to 5 compulsory others optional. Use your calculator to draw scatter plots and find r.