UNIT 2 BIVARIATE DATA

BIVARIATE DATA – THIS TOPIC INVOLVES….

Comparing two sets of data -o One Variable is the INDEPENDENT variableo The other is the DEPENDENT variable

Analysis of the data on a SCATTERPLOT

The strength of the relationship between the 2 variables are compared using both

o Pearson’s Product Correlation coefficient, o Coefficient of Determination,

Finding a line of best fit called the LEAST SQUARES REGRESSION and interpreting the result based on the equation of this line.

y-ax

isDE

PEND

ENT

VARI

ABLE

x-axisINDEPENDENT VARIABLE

INDEPENDENT VS DEPENDENT VARIABLES

The value of the DEPENDENT variable depends on the INDEPENDENT variable.

eg. Identify the INDEPENDENT and DEPENDENT variable in each case:

a. The time spent filling up a swimming pool with water compared to the size of the pool.

b. The hours per week spent doing laundry compared with the number of children living in the house.

c. The last time you went to the hairdresser and the length of your hair. d. A child’s height compared to their age.

INDEPENDENT VS DEPENDENT VARIABLESIf you need more help with this, view the following video tutorial

http://www.vcefurthermaths.com/2011/01/tutorial-12-independent-and-dependent-variables/

SCATTERPLOTS

https://www.youtube.com/watch?v=VlJ85vE97lA

If you need more help with this, view the following video tutorial

SCATTERPLOTSIf you need more help with this, view the following video tutorial

http://www.vcefurthermaths.com/2011/02/interpreting-scatterplots/

SCATTERPLOTS We always put the INDEPENDENT variable on the x-axis and the DEPENDENT variable on the y-axis.

We can plot these by hand but we can also use our calculator to create a scatterplot for us.

eg. Results on a Maths test were compared to the time spent studying the day before the test.

The following data was obtained from a sample of 10 students.

CALCULATOR: Statistics Enter Data ‘Study’ into List 1, Enter Data ‘Results’ into List 2.

Study (hrs)

1 4 4 1 0 3 3.5 4.5 5 2

Result (%)

53 82 100 44 40 72 81 90 94 63

.CALCULATOR: Statistics Enter Data ‘Study’ into List 1, Enter Data ‘Result’ into List 2. To plot the SCATTERPLOT, now press

Choose - Type: Scatter - Xlist: Independent Variable List (Time) - Ylist: Dependent Variable List (Result) Click SET Now click graph

Study Time(hrs)

1 4 4 1 0 3 3.5 4.5 5 2

Result (%) 53 82 100 44 40 72 81 90 94 63

The following scatterplot is produced.

Comparing this scatterplot to the graphs on our worksheet given in last lesson,

we can say that there is a Strong Positive relationship between the number of hours

studying and the score achieved on the test.

Study Time(hrs)

1 4 4 1 0 3 3.5 4.5 5 2

Result (%) 53 82 100 44 40 72 81 90 94 63

Study Time

Resu

lt

NOW TRY WORKBOOK SECTION ONE

PEARSON’S PRODUCT CORRELATION COEFFICIENT

What is it?o Correlation between sets of data is a measure of how well they are related. The most common measure of correlation in stats is the Pearson Correlation. o It shows the linear relationship between two sets of data.o The correlation coefficient can be represented as ro Calculated using the equation:

o Luckily, our calculator can solve this for us!

PEARSON’S PRODUCT CORRELATION COEFFICIENT What does it look like? What values can we expect?

o The value will be between -1 and 1. o 1 tells us there is a perfect positive relationshipo -1 tells us there is a perfect negative relationshipo Relationships for r values in between are given in the following table

PEARSON’S PRODUCT CORRELATION COEFFICIENTIf you need more help with this, view the following video tutorial http://www.vcefurthermaths.com/2011/02/tutorial-18-pearsons-correlation-coefficient/

PEARSON’S PRODUCT CORRELATION COEFFICIENT Finding the Correlation Coefficient ‘r’ using the

calculator eg. Consider the problem given earlier – Results on a Maths test were compared to the time spent studying the day before the test. The following data was obtained from a sample of 10 students.

This data gave the scatterplot:

Study (hrs)

1 4 4 1 0 3 3.5 4.5 5 2

Result (%)

53 82 100 44 40 72 81 90 94 63

Study Time

Resu

lt

Using this data, we can use the calculator to find r.

First, using the Statistics function on your calculator, place the data into List 1 and List 2

Using what we know about ‘r’, view the scatterplot and predict – what approximate value will r have?

PEARSON’S PRODUCT CORRELATION COEFFICIENT Finding the Correlation Coefficient ‘r’ using the

calculatorStudy (hrs)

1 4 4 1 0 3 3.5 4.5 5 2

Result (%)

53 82 100 44 40 72 81 90 94 63

Choose: CalcLinear RegXlist: list with Independent VariablesYlist: list with Dependent Variables

PEARSON’S PRODUCT CORRELATION COEFFICIENT Finding the Correlation Coefficient ‘r’ using the

calculatorStudy (hrs)

1 4 4 1 0 3 3.5 4.5 5 2

Result (%)

53 82 100 44 40 72 81 90 94 63

Click OK.The following information is revealed.

Reading r off this gives the Correlation Coefficient r = 0.964This confirms that there is a strong positive relationship between the amount of study done and the score achieved on the maths test.

THE COEFFICIENT OF DETERMINATION Finding the Coefficient of determination ‘’ using the

calculatoro The coefficient of determination, r 2,  tells us how much the variation in the dependent variable can be explained by the variation in the independent variableo It is a measure that allows us to determine how certain one can be in making predictions from a certain model/graph.Reading off this gives the Coefficient of

Determination = 0.929From this, we can say that 92.9% of the total variation in y (test result) can be explained by variation in x (the amount of time spent studying)

We will cover more on this later…..

NOW TRY WORKBOOK SECTION TWO

THE COEFFICIENT OF DETERMINATION

o Is a measure that allows us to determine how certain one can be in making predictions just by looking at a certain model/graph. o The coefficient of determination, r 2,  tells us how much the variation in the dependent variable can be explained by the variation in the independent variable (note: not whether one causes the other). o We find r 2 simply by squaring our r value, ALSO our calculator also generates this value for uso Because r values have to be between -1 and +1, these r² values always fall between 0 and 1.o When answering questions about the Coefficient of Determination, the way we word our response is important – Variations to the dependent variable are not caused by variations of the independent variable, rather we can say,

The Variations of the Dependent Variable can be explained by variations of the independent variable.

THE COEFFICIENT OF DETERMINATION

If you need more help with this, view the following video tutorial

http://www.vcefurthermaths.com/2011/02/tutorial-19-coefficient-of-determination/

THE COEFFICIENT OF DETERMINATIONUsing the calculator to find Click

STATISTICS.Insert Data into

lists

Choose CALC, Linear

Reg

Xlist: Choose list which has the INDEPENDENT

variable

Ylist: Choose list which has the DEPENDENT

variable

is found!

THE COEFFICIENT OF DETERMINATION Finding the Coefficient of determination ‘’ using the

calculator

Reading off this gives the Coefficient of Determination = 0.929From this, we can say that 92.9% of the total variation in y (test result) can be explained by variation in x (the amount of time spent studying)

Study (hrs)

1 4 4 1 0 3 3.5 4.5 5 2

Result (%)

53 82 100 44 40 72 81 90 94 63

Again, looking at our prior example: Results on a Maths test were compared to the time spent studying the day before the test. This data was obtained from a sample of 10 students.

THE COEFFICIENT OF DETERMINATION

Exercise (hrs)

5 10 0 3 3 6 3.5 4.5 1 2

Heart Rate 61 48 83 66 71 61 69 64 80 65

Lets try another example: The resting heart rate of 10 individuals were compared with the number of hours per week they spent exercising, to see if one influences the other.

Choose CALC, Linear

RegClick

STATISTICS.Insert Data into

lists

is found!

THE COEFFICIENT OF DETERMINATION

Reading off this gives the Coefficient of Determination = 0.873From this, we can say that 87.3% of the total variation in y (heart rate) can be explained by variation in x (the amount of time spent exercising)

Now look at the r value…..what else does this tell us about the relationship between the two variables?

Exercise (hrs)

5 10 0 3 3 6 3.5 4.5 1 2

Heart Rate 61 48 83 66 71 61 69 64 80 65

Lets try another example: The resting heart rate of 10 individuals were compared with the number of hours per week they spent exercising, to see if one influences the other.

is found!

NOW TRY WORKBOOK SECTION THREE

LEAST SQUARES REGRESSION LINE

o The least squares regression line is simply a line of best fit of your scatter-plotted data.o Our calculator can easily be used to plot this line and give us the equation that represents this line.

What does it look like??

Choosey=

𝑦=38.4+11.97𝑥

LEAST SQUARES REGRESSION LINEExercise (hrs)

5 10 0 3 3 6 3.5 4.5 1 2

Heart Rate 61 48 83 66 71 61 69 64 80 65

Lets find the linear regression line for our earlier example: The resting heart rate of 10 individuals were compared with the number of hours per week they spent exercising.

Choose CALC, Linear

RegClick

STATISTICS.Insert Data into

lists

Select drop down box to display

The value of and the value of

We can then put these values into our straight line equation to find our regression line.

We are used to seeing this in the form The equation we are using in this topic is the same as

this, except the “m” is replaced by “b” and the “c” is replaced with “a”.

Which for this example gives:

LEAST SQUARES REGRESSION LINEExercise (hrs)

5 10 0 3 3 6 3.5 4.5 1 2

Heart Rate 61 48 83 66 71 61 69 64 80 65

are found!

NOW TRY WORKBOOK SECTION FOUR