1 Further Maths Chapter 4 Displaying and describing relationships between two variables

2 Bivariate Data Often we are interested to see if a relationship exists between two variables, i.e. – Hours studied and study score, – Gender and resting pulse rate – Hair colour and Favourite Sport

3 Dependent and Independent Variable The first thing that needs to be considered is if there is some sort of dependency relationship between the two i.e. which of the two variables is likely to depend on the other? Does a person’s study score depend on number of hours studied or does the number of hours a person studies depend on their study score? Complete the other examples

4 Dependent and Independent Variable In the first case, the study score is the dependent variable, hours studied is the independent variable. In the second case, the resting pulse rate is the dependent variable, gender is the independent variable In the third case you would not expect there to be any relationship

5 Dependent and Independent Variable The variable that can be controlled is referred to as the independent variable. The variable that is measured in response is referred to as the dependent variable.

6 3 possible cases There are three possible situations that may occur when considering two variables at the one time Both variables are categorical One variable is categorical and the other is numerical Both variables are numerical

7 Categorical and Categorical One hundred people were randomly selected and surveyed as to whether they were in favour of lowering the speed limit in suburban streets to 40 km/h.

8 Categorical and Categorical When the data was tabulated it was found that from the males interviewed, 25 were in favour and the rest against. Of the females, 20 were in favour and the rest against.

9 Categorical and Categorical The group consisted of 65 males and 35 females. Each person voted for or against the proposal. The independent variable is The dependent variable is

10 Categorical and Categorical A two way frequency table is an appropriate way to display this data The independent variable should be put in the columns. The dependent variable should be put in the rows.

11 Categorical and Categorical Gender MaleFemale OpiOpi In favour niOnniOn Not in favour Total

12 Categorical and Categorical Gender MaleFemale OpiOpi In favour 2520 nionnion Not in favour 4015 Total6535

13 Categorical and Categorical Does this table appear to indicate that men are more in favour of lowering the speed limit than women? Discuss

14 Categorical and Categorical Two way frequency table (appropriately percentaged) Since the independent variable is in the columns, we need to calculate column percentages.

15 Categorical and Categorical Gender MaleFemale OpiOpi In favour 25 / 65 * 100 = 38.5% 20 / 35 * 100 = 57.1% niOnniOn Not in favour 40 / 65 * 100 = 61.5% 15 / 35 * 100 = 42.9% Total100%

16 Categorical and Categorical This can then be displayed using a percentaged segmented barchart.

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18 Categorical and Categorical Report 57.1% of females are in favour of lowering the speed limit compared to 38.5 % of men. Women are clearly more in favour of lowering the speed limit.

19 Exercises Exercise 4A Pages 88 – 89 all Exercise 4B Page 91 all

20 Numerical and Categorical An investigation was carried out to see if there was a relationship between gender and resting pulse rate. Data was collected of the resting pulse rates of 23 boys and 23 girls. Dependent variable is Independent variable is Numerical variable is Categorical variable is Pulse rate Gender Pulse rate Gender

21 Numerical and Categorical Males Females

22 Back to Back Stem and Leaf Plot A back to back stem and leaf plot can be used to display the relationship between a numerical variable and a two valued categorical variable. malesfemales

23 Parallel Boxplots Parallel box plots can be used to display the relationship between a numerical variable and a two or more level categorical variable. Calculator Display Exercise 4C Page 93 Questions 1-3

24 Numerical and Numerical The following data was collected from 10 students Average hours of study for Further Mathematics per week Study Score

25 Numerical and Numerical The independent variable is The dependent variable is The appropriate way to display this data is by use of a scatterplot. The independent variable should always be found on the horizontal axis The dependent variable should always be found on the vertical axis.

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27 Calculator Calculator Display Exercise 4D Page 96 all

28 Interpreting a Scatterplot When describing a scatterplot the following four features should be discussed Direction Form Strength Outliers

29 Direction The direction of a scatterplot can either be positive or negative

30 Form The form of a scatterplot can either be linear or non-linear

31 Strength The strength of a scatterplot can be either perfect, strong, moderate or weak.

32 Strength

33 Outliers Outliers are any points that are separated from the general body of points

34 Discuss

35 Discuss

36 Discuss

37 Correlation coefficient (r) r is a measure of the strength of the linear relationship between two numerical variables. The value of r is between –1 (perfect negative linear relationship) and 1 (perfect positive linear relationship). It is not appropriate to calculate a correlation coefficient if there are outliers in the data.

38 Estimating r Board Work Exercise 4E Page all

39 Calculating Pearson’s r r =  ( x – x ) ( y – y ) ( n –1 ) s x s y The key assumptions when using Pearson’s r is that the data is linear and that there are no outliers. Calculator demonstration Exercise 4F Pages Question 1,3

40 Coefficient of Determination (r 2 ) The coefficient of determination is the square of Pearson’s correlation coefficient. It is used to explain the degree to which one variable can be predicted from another variable

41 Coefficient of determination The coefficient of determination gives the percentage variation (r 2 * 100) in the dependent variable that is explained by the variation in the independent variable.

42 Example Average hours of study for Further Mathematics per week Study Score

43 Example Calculate r and interpret Calculate r 2 Interpret r 2

44 Example of the variation in the study score can be explained by the variation in the number of hours studied per week. The other is due to other factors.

45 Warning Pearsons r can be positive or negative depending on the direction of the scatterplot. The coefficient of determination will always be positive and is normally expressed as a percentage.

46 Warning If r 2 is equal to 0.36, then what will r be?

47 Exercise Complete Exercise 4G Pages all

48 Correlation and Causation For a number of rural towns the number of nightclubs was recorded, as were the number of churches. The following data resulted Graph this relationship as a scatterplot Number of churches Number of nightclubs

49 Correlation and Causation

50 Correlation and Causation Interpret this graph. As the number of churches increases, so does the number of nightclubs. Therefore an increase in the number of churches will lead to or cause an increase in the number of nightclubs. As people become more religious they will tend to visit nightclubs more. Nightclubs will be full of super religious people.

51 Correlation and Causation An increase in one variable will not always cause an increase in the other. In this situation there is a third variable that is hidden I.e. Population Therefore we never use the word cause when describing a relationship. We say As the number of churches increases, the number of nightclubs tends to increase. Exercise 4H Page 108 – 109 all

52 Chapter 4 Exercise 4I Page 109