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Chapter 10 Relationships between variables

Chapter 10 Relationships between variables. Definition

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Chapter 10 Relationships between variables

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  1. Chapter 10Relationships between variables • Definition • A Scatter Plot is a picture of bivariate numerical data in which each observation ( ie each pair of values (x,y)) is represented by a point located on a rectangular co-ordinate system. The Horizontal Axis is identified with values of x and the vertical axis with values of y.

  2. Example:Draw a Scatter Plot to represent the following dataset:x: 1, 3, 2, 4, 7, 6, 5 y: 4, 2, 5, 6, 9, 8, 7

  3. Another Example:Draw a Scatter Plot to represent the following dataset:x: 1, 3, 2, 4, 7, 6, 5 y: 4, 6, 1, 3, 2, 4, 1

  4. QuestionAny comments on these two datasets? Is there anything special about them? • Looking at a scatter plot can sometimes allow us to determine if a relationship exists between two variables. • But in general we need to go beyond pictures and develop a numerical measure of how strongly the two variables x and y are related.

  5. Definition • Pearson’s Sample Correlation Coefficient, r, is a measure of the strength of the linear relationship between two variables x and y.

  6. Properties of r • The correct interpretation of r requires an appreciation of some general properties: • The value of r does not depend on the unit of measurement for either variable, nor does it depend on which variable is labelled x or y. • The value of r is between -1 and 1. • A positive value of r indicates a positive linear relationship between the variables. So as x increases so does y. • A negative value of r corresponds to a negative relationship. As x increases y decreases.

  7. The value r = 1, which indicates the strongest possible positive relationship between x and y results only when all points in the scatter plot lie exactly on a straight line that slopes upward. • The value r = -1, which indicates the strongest possible negative relationship between x and y results only when all points in the scatter plot lie exactly on a straight line that slopes downward.

  8. The value of r is a measure of the extent to which x and y are linearly related i.e. the extent to which the points in the scatter plot lie close to a straight line. • A value close to zero does not rule out any strong relationship between x and y; there could still be a strong relationship but one that is not linear.

  9. Examples For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation or no correlation. • Minimum daily temperature and heating costs • Interest rate and number of loan applications • Incomes of husbands and wives when both have full-time jobs • Ages of boyfriends and girlfriends • Height and IQ • Height and shoe size • Your Maths score in the Leaving Cert and your Irish score in the Leaving Cert

  10. Correlation and causation • Years of research have established several facts: • There is a strong correlation between the numbers of storks in a country and the number of births in that country. Countries with many storks have a high number of births and countries with low stork counts have low numbers of births. • There is a high correlation among primary school children between vocabulary and numbers of tooth fillings. Children with many fillings have a larger vocabulary than children with only a small number or with no fillings.

  11. Correlation and causation • What should we conclude from these facts? • That storks really are responsible for bringing babies. • That eating Mars bars will increase your vocabulary. • No, these examples illustrate a very important point. • Correlation is not the same as causation.

  12. Correlation and causation • Larger countries have larger stork populations and usually have higher human populations as well and so there will be higher numbers of babies born than in smaller countries. • Young children have very few fillings because they have only been around for a few years whereas older children have had time to eat lots of sweets, get a lot of bad teeth and learn a lot of new words. • So be careful before you interpret a correlation as causation. It may be that a third confounding variable is causing the correlation: Size of country, Age of child.

  13. Least Squares Introduction • We have just mentioned that one should not always conclude that because two variables are correlated that one variable is causing the other to behave a certain way. However, sometimes this is the case, eg: interest rate and number of loan applications. • In this section we will deal with datasets which are correlated and in which one variable, x, is classed as an independent variable and the other variable, y, is called a dependent variable as the value of y depends on x.

  14. Least Squares • We saw that correlation implies a linear relationship. Well a line is described by the equation • y = a +bx • where b is the slope of the line and a is the intercept i.e. where the line cuts the y axis. • The intercept a is just the value that y takes when x is zero. • The slope b is how much y increases by when x increases by one unit.

  15. Suppose we have a dataset which is strongly correlated and so exhibits a linear relationship, how would we draw a line through this data so that it fits all points best? • We use the principle of least squares, we draw a line through the dataset so that the sum of the squares of the deviations of all the points from the line is minimised.

  16. Regression • Suppose we have a dataset and we have calculated the equation of the Least Squares Line • y = a +bx • Then we can use this line to predict a value for Y if we know a value for X. • Note we should only predict for values of X which are bigger than the smallest X value in the dataset and smaller than the largest value in the dataset.

  17. Example of Regression: • A study performed in the UK examined the relationship between husband’s and wives’ ages. • The data were analysed and a Least Squares Line computed: Y = 3.6 + (0.97) X • Where • Y is Husband’s age • X is Wife’s age • Predict the age of the husband of a 20 year old woman. • Predict the age of the husband of a 25 year old woman.

  18. Regression Answers: • 20Yr old Woman Y = 3.6 + (0.97) 20 Y = 23.0 So Husband is probably 23 years old • 25Yr old Woman Y = 3.6 + (0.97) 25 Y = 27.9 So Husband is probably 27.9 years old

  19. Congratulations! • It’s over! • You have survived the dreaded course on STATISTICS. • Hopefully none of you have died of Boredom.

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