Bivariate Data - aceh.b-cdn.net

Chapter 6

Bivariate Data

6A Introduction to Bivariate Scatterplots (pg. 63) 6B Bivariate Data Relationships (pg. 65)

6C Pearson's Correlation Coefficient (pg. 68) 6D Line of Best Fit (pg. 69)

6E Interpolation and Extrapolation (pg. 71) 6F Statistical Investigation (pg. 72)

Written by Benjamin Odgers

Maths Teacher B Teaching / B Science

The following theory booklet lines up with the Cambridge Year 12 NSW Standard Mathematics 2 Textbook. This can be found using the following link:





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6A Introduction to Bivariate Scatterplots Bivariate data comprises of two variables that may or may not have a correlation. A good example of bivariate data was given by the Roman architect, Vitruvius in the first century BC. He claimed that a person's arm span is approximately the same as a person's height. We can attempt to prove Vitruvius' claim by using a bivariate scatter plot. We can measure a sample of recipients and plot them on a bivariate scatterplot. Each person's height and arm span represent the two variables for our scatterplot. If Vitruvius' claim is true we should see a strong correlation between people's height and arm span.

The image at right is called the Vitruvian Man and was drawn by Leonardo da Vinci. The image was obtained from the following site:



Example 1 The following table represents a sample of 15 people. Each person had their height and arm span measured and recorded.

Height (cm)

152 180 159 165 187 183 161 158 165 168 172 176 169 178 178

Arm Span (cm) 153 184 160 167 187 180 159 162 166 168 170 179 171 175 183

a) Construct a scatterplot by plotting the points on

200

the number plane at right.

190

b) What is the scale for the horizontal axis?

180

Arm span (cm)

c) According to this data, would you say there is a

170

strong correlation between a person's height and

arm spam? Why?

160

150 150 160 170 180 190 200 Height (cm)

d) What do you notice about the shape of the scatter plot?



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Example 2 The following table represents a sample of 15 students. Each student recorded the number of hours they studied for an exam as well as their marks.

Study Time (h) 0 18 5 1 2 11 15 18 0 2 7 16 13 10 4 Exam Mark (%) 5 95 80 10 20 80 85 80 35 30 60 80 85 85 40

a) Construct a scatterplot by

100

plotting the points on the

number plane at right.

80

Exam Mark (%)

60

b) What is the scale for the

vertical axis?

40

c) How many students got a mark greater than 60%?

20

0 0 246

d) How many students completed less than 8 hours of study in preparation for the exam?

8 10 12 14 16 18 20 Study Time (h)

e) According to this data, would you say there is a strong correlation between a person's study time and exam mark? Why?

f) What do you notice about the shape of the scatter plot?

g) Were there any students that seemed to lie outside the normal trend when comparing students study time to their exam mark?



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6B Bivariate Data Relationships When interpreting bivariate scatterplots, we often talk about variables having (or not having) a relationship. When talking about the relationship between variables we often use language such as variables having a correlation or an association with each other. We can tell if variables have a relationship by observing the position of the points. If there is an obvious pattern then we can see that a relationship exists between the variables.

Strength of the Relationship (or Association) By observing the scattering of points on a scatter plot we can find the strength of the relationship. When a relationship is strong it will have an obvious trend that can be easily graphed using a straight line or curve. As relationships become weaker it becomes harder to graph the trend.

Strong

Moderate

Weak

No Relationship

Sale Price ($) Persons IQ

Arm Span (cm) Wife's Age (years)

Height (cm)

Husband's Age (years)

Form of the Relationship (or Association) Linear Form

Size of Land (Acres)

Persons Height (cm)

Non-Linear Form

Company Profits ($)

Life Expectancy (years)

Person's Age (years)

Direction of the Relationship (or Association) Positive

Number of Employees

Negative

Life Expectancy (years)

Arm Span (cm)

Height (cm)

Person's Age (years)



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Example 1

Describe the (i) strength, (ii) form and (iii) direction for each scatterplot below

(a)

(b)

(c)

(d)

(e)

(f)

What is the difference between dependent and independent variables when referring to bivariate scatterplots?



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