4.4 Bivariate data & Linear Correlation

Bivariate Data

This topic is all about looking for relationships between variables. Does life expectancy depend on GDP? By collecting 2 variables about the things we survey we can use scatter graphs to represent the data and then look for correlation and models that we can then use for prediction.

Key Concepts

In this unit you should learn to…

Collect, present and analyse 2 variable data looking for linear correlation
Calculate the Pearson's Product Moment Correlation Coefficient as a measure of degree of correlation
Calculate and use a regression line that models data that shows correlation.

Essentials

Slides Gallery

Use these slides to review the material and key points covered in the videos.

1. Bivariate data and Scatter diagrams

This video is an introduction to the concept of bivariate data and looking for relationships/correlation

2. Scattergraphs

This is about plotting and interpreting these diagrams

3. Correlation

This video is about judging correlation

4. Product moment correlation coefficient

This is about calculating this key measurement of correlation

5. Lines of best fit

This is about how lines of best fit become calculated 'Least squares regression lines' that are accurate measures of the pattern.

6. Making predictions

If we have correlation, and we have calculated a regression line, we can use it to make predictions.

Summary

Revision Card - Review this condensed 'key point' revision card to help you check and keep ideas fresh in your mind.

Test yourself

Self Checking Quiz

Practice your understanding on these quiz questions. Check your answers when you are done and read the hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above.

Question 1

Look at the scatter diagrams below.

Write the letter corresponding to the diagram next to the descriptions of the correlation

Write A, B, C, D, E in the boxes below (Use them all)

Strong Positive correlation

Moderate positive correlation

No correlation

Moderate negative correlation

Strong negative correlation

Linear Correlation is a measure of how close the points are to a straightline. The gradient of the line is not a consideration.

...

Question 2

Which of the following would it be possible to show on a scatter graph. Write either 'Y' or 'N' next to each.

1. 100 students had the heigth and armspan measured.

2. 200 Cars were surveyed for their make and engine size

3. 1000 trees were surveyed for their height and diameter at the base.

4. 75 Countries were surveyed to compare their GDP with infant mortality

5. 100 25 year olds were asked for their gender and had their 100 sprint time measured

For a scatter diagram, you need 2 pieces of numerical data about each of the things you survey.

1. Y - Height and Armspan are both numericam

2. N - Car make is categorical

3. Y - Both numerical measures about the trees

4. Y - Both numerical measures

5. N - Gender is categorical

Question 3

Thee table below shows the IB scores of students who studied Geography and History

a) Calculate the the Pearsons product moment correlation coefficient to 3sf

r =

b) The equation of the regression line for this data is y = mx + c. What are the values of m and c, rounded to 3sf.

m = , c =

c) Using the rounded values for the regression line, estimate the English score for a student who scored 1 in Geography?

English score = (note round to the nearest whole number)

d) Should we consider the answer to the above question valid? (Y or N)

a) b) Using lists on the stats menu of your calculator. NOTE TI users need to make sure you have turned catalogue on. See the calculator page for help.

c) Substitute x = 1 into y = 0.506x + 2.25

d) N - 1 is outside the range of data we had in the beginning.

Question 4

A student collected the ages and sprint times of 10 people to see if the was a relationshop between the two. The results are in the table below. the student thinks that people will get faster the older they are.

a) What kind of correlation was the student expecting? Negatine (N) of Positive (P)

b) Calculate the correlation coeficient for this data. r = (3sf)

c) The student stated that their hypothesis was correct. Was this a correct conclusion? (Y or N)

a) Student was expecting that times would get faster (so lower numbers) as age increased. This is negatove correlation.

b) From GDC and carefully rounded

c) N - In fact the result shows no signifcant correlation, even though it was negative.

Question 5

The following shows the results of an experiment to investigate what happens to the length of a spring as the weight attached to it is increased.

a) Calculate the PMCC for this data. r =

b) The equation of the regression line for this data is y = mx + c. What are the values of m and c, rounded to 3sf.

m = , c =

c) Using the rounded values for the regression line, estimate the length of the spring with a weight of 9 kg attached.

Length = (3sf)

d) Using the rounded values for the regression line, estimate the length of the spring with a weight of 17 kg attached.

Length = (3sf)

e) Using the rounded values for the regression line, estimate the length of the weight required to stretch the spring to a length of 17cm

Weight = (3sf)

a) From GDC

b) From GDC

c) d) For the next 2 parts it might be wise to enter y = 1.76x + 8.41 into the table function and read from the table when x is 9 and 17 respectively.

e) For this part, you could plot the function y = 1.76x + 8.41 and then solve for y = 17

For a given winter, a householder compared the average monthly temperature with the volume of wood that was used for heating. The results are as follows.

a) What type of correlation might the householder be expecting. Negative (N) or Positive (P)

b) Work out the PMCC for this data. r = (3sf)

c) The equation of the regression line for this data is y = mx + c. What are the values of m and c, rounded to 3sf.

m = , c =

d) The expected average temperature for the coming month is 7°. Use the rounded answers to the question abouve to predict the volume of wood that might get used. Volume = m³

e) What are the mean values of the two variables? (3sf)

Mean (Ave temp) = ° (3sf), Mean (Volume) = m³

a) It is reasonable to expect that as temperature falls, the volume of wood used increases - thus negative correlation.

b) c) From GDC

d) Using the function y = -0.115x + 1.24, read from the table or solve when x = 7.

e) From GDC. This point is needed to plot the regression line on paper. The other is the y - intercept. After which the line can drawn.

Exam Style Questions

The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.

Question 1

Video solution

Question 2

Video solution

Question 3

Video solution

Question 4

Mikel believes there is a linear relationship between time spent studying and performance in an IB exam.To test this, he collected data on ten of his classmates. He recorded the time spent studying, x, alongside the percentage they received in their most recent exam, y.

The following table shows the data he found.

Hours studying, x.	4	5	4	3	7	2	7	9	10	11
Percentage Score, y	58	60	43	40	65	35	70	75	75	80

a. Why does Pearson’s Product Moment Correlation Coefficient appear to be valid for this data?

b. Find the value of Pearson’s Product Moment Correlation Coefficient, r.

c. Comment on the strength of the correlation

Mikel decides to study for 8 hours for the test.

d. By using the equation of the regression line, what score could he expect to get?

Video Solution

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Self-assessment

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4.4 Bivariate data & Linear Correlation

Bivariate Data

Self Checking Quiz

IBDP Maths: Applications & Interpretations SL

Statistics and Probability Home Free Access Weekend! Start here Examination Questions Practice Questions Question Bank Your Graphical Display Calculator Number and Algebra Functions Geometry and Trigonometry Calculus Pre-IB Preparation Exams & Assessment For Teachers

4.4 Bivariate data & Linear Correlation

Bivariate Data

Self Checking Quiz

Statistics and Probability
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Free Access Weekend!
Start here
Examination Questions
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Question Bank
Your Graphical Display Calculator
Number and Algebra
Functions
Geometry and Trigonometry
Calculus
Pre-IB Preparation
Exams & Assessment
For Teachers