1. Bivariate Data AS 90425 (3 credits) Complete a statistical investigation involving bi-variate data

2. Scatter Plots The scatter plot is the basic tool used to investigate relationships between two quantitative variables. Types of Variables Quantitative (measurements/ counts) Qualitative (groups)

3. What do I see in these scatter plots? There appears to be a linear trend. There appears to be moderate constant scatter about the trend line. Negative Association. No outliers or groupings visible. 4540 35 20 19 18 17 16 15 14 Latitude (°S) Mean January Air Temperatures for 30 New Zealand Locations Temperature (°C)

4. What do I see in these scatter plots? There appears to be a non-linear trend. There appears to be non-constant scatter about the trend line. Positive Association. One possible outlier (Large GDP, low % Internet Users). Internet Users (%) % of population who are Internet Users vs GDP per capita for 202 Countries

5. What do I see in these scatter plots? Two non-linear trends (Male and Female). Very little scatter about the trend lines Negative association until about 1970, then a positive association. Gap in the data collection (Second World War).

6. What do I look for in scatter plots? Trend Do you see a linear trend… straight line OR a non-linear trend?

7. What do I look for in scatter plots? Trend Do you see a linear trend… straight line OR a non-linear trend?

8. What do I look for in scatter plots? Trend Do you see a positive association… as one variable gets bigger, so does the other OR a negative association? as one variable gets bigger, the other gets smaller

9. What do I look for in scatter plots? Trend Do you see a positive association… as one variable gets bigger, so does the other OR a negative association? as one variable gets bigger, the other gets smaller

10. What do I look for in scatter plots? Scatter Do you see a strong relationship… little scatter OR a weak relationship? lots of scatter

11. What do I look for in scatter plots? Scatter Do you see a strong relationship… little scatter OR a weak relationship? lots of scatter

12. What do I look for in scatter plots? Scatter Do you see constant scatter… roughly the same amount of scatter as you look across the plot or non-constant scatter? the scatter looks like a “fan” or “funnel”

13. What do I look for in scatter plots? Scatter Do you see constant scatter… roughly the same amount of scatter as you look across the plot or non-constant scatter? the scatter looks like a “fan” or “funnel”

14. What do I look for in scatter plots? Anything unusual Do you see any outliers? unusually far from the trend any groupings?

15. What do I look for in scatter plots? Anything unusual Do you see any outliers? unusually far from the trend any groupings?

16. Rank these relationships from weakest (1) to strongest (4):

17. 2 1 4 3

18. Rank these relationships from weakest (1) to strongest (4): How did you make your decisions? The less scatter there is about the trend line, the stronger the relationship is.

19. Describing scatterplots Trend – linear or non linear Trend – positive or negative Scatter – strong or weak Scatter – constant or non constant Anything unusual – outliers or groupings

20. Correlation Correlation measures the strength of the linear association between two quantitative variables Get the correlation coefficient (r) from your calculator or computer r has a value between -1 and +1: Correlation has no units

21. Calculating the coefficient of correlation (r) The coefficient of correlation (r) is given by the following formula: Fortunately you will have Excel do all the work for you!

22. What can go wrong? Use correlation only if you have two quantitative variables (variables that can be measured) There is an association between gender and weight but there isn’t a correlation between gender and weight! Gender is not quantitative Use correlation only if the relationship is linear Beware of outliers!

23. Deceptive situations r is a measure of how one variable varies in a linear relation to the other An obvious pattern does not always indicate a high value of the coefficient of correlation (r) A horizontal or vertical trend indicates that there is no relationship, and so r is (close to) zero

24. Always plot the data before looking at the correlation! r = 0 No linear relationship, but there is a relationship! r = 0.9 No linear relationship, but there is a relationship!

27. Tick the plots where it would be OK to use a correlation coefficient to describe the strength of the relationship: 9876543210 400 0 300 0 200 0 100 0 0 Position Number Distance (million miles) Distances of Planets from the Sun Reaction Times (seconds) for 30 Year 10 Students 0 0. 2 0. 4 0. 6 0. 8 00.20.40. 6 0. 8 1 Non-dominant Hand Dominant Hand 4540 35 20 19 18 17 16 15 14 Latitude (°S) Mean January Air Temperatures for 30 New Zealand Locations Temperature (°C) Female ($) Average Weekly Income for Employed New Zealanders in 2001 Male ($) 0 200 400 600 800 1000 1200 0200400 600 80 0

28. Tick the plots where it would be OK to use a correlation coefficient to describe the strength of the relationship: 9876543210 400 0 300 0 200 0 100 0 0 Position Number Distance (million miles) Distances of Planets from the Sun Reaction Times (seconds) for 30 Year 10 Students 0 0. 2 0. 4 0. 6 0. 8 00.20.40. 6 0. 8 1 Non-dominant Hand Dominant Hand 4540 35 20 19 18 17 16 15 14 Latitude (°S) Mean January Air Temperatures for 30 New Zealand Locations Temperature (°C) Female ($) Average Weekly Income for Employed New Zealanders in 2001 Male ($) 0 200 400 600 800 1000 1200 0200400 600 80 0   Not linear Remove two outliers, nothing happening

29. What do I see in this scatter plot? Appears to be a linear trend, with a possible outlier (tall person with a small foot size.) Appears to be constant scatter. Positive association. 2223242526272829 150 160 170 180 190 200 Foot size (cm) Height (cm) Height and Foot Size for 30 Year 10 Students

30. What will happen to the correlation coefficient if the tallest Year 10 student is removed? It will get smaller It won’t change It will get bigger 2223242526272829 150 160 170 180 190 200 Foot size (cm) Height (cm) Height and Foot Size for 30 Year 10 Students

31. What will happen to the correlation coefficient if the tallest Year 10 student is removed? It will get bigger 2223242526272829 150 160 170 180 190 200 Foot size (cm) Height (cm) Height and Foot Size for 30 Year 10 Students

32. What do I see in this scatter plot? Appears to be a strong linear trend. Outlier in X (the elephant). Appears to be constant scatter. Positive association. 6005004003002001000 40 30 20 10 Gestation (Days) Life Expectancy (Years) Life Expectancies and Gestation Period for a sample of non-human Mammals Elephant

33. What will happen to the correlation coefficient if the elephant is removed? It will get smaller It won’t change It will get bigger 6005004003002001000 40 30 20 10 Gestation (Days) Life Expectancy (Years) Life Expectancies and Gestation Period for a sample of non-human Mammals Elephant

34. What will happen to the correlation coefficient if the elephant is removed? It will get smaller 6005004003002001000 40 30 20 10 Gestation (Days) Life Expectancy (Years) Life Expectancies and Gestation Period for a sample of non-human Mammals Elephant