Scatterplots and Relationships

Bivariate data involves two variables measured on the same set of items. A scatterplot displays each data pair as a point on a coordinate plane, allowing us to visually assess the relationship.

Visual: Types of Relationships

Strong Positive

r close to +1

Strong Negative

r close to -1

No Correlation

r close to 0

Pearson's Correlation Coefficient (r)

The correlation coefficient r measures the strength and direction of a linear relationship between two variables. It ranges from -1 to +1.

Interpreting r values:

r = 1: Perfect positive linear correlation
0.8 ≤ r < 1: Strong positive
0.5 ≤ r < 0.8: Moderate positive
0 < r < 0.5: Weak positive
r = 0: No linear correlation
-0.5 < r < 0: Weak negative
-0.8 < r ≤ -0.5: Moderate negative
-1 < r ≤ -0.8: Strong negative
r = -1: Perfect negative linear correlation

Key reminders:

r measures linear relationships only.
r has no units and is unaffected by changes in scale.
Correlation does not imply causation.
Outliers can strongly affect the value of r.

Describing Bivariate Relationships

When describing a bivariate relationship from a scatterplot, always comment on these four features:

Direction

Positive (upward trend) or negative (downward trend)?

Form

Linear or non-linear? Could a straight line model the data?

Strength

Strong, moderate, or weak? How tightly clustered are the points?

Outliers

Are there any points that do not follow the general pattern?

Key Vocabulary

Bivariate Data

Data that involves two variables measured on the same set of observations.

Scatterplot

A graph displaying paired data points on a coordinate plane to reveal patterns and relationships.

Correlation Coefficient (r)

A numerical measure from -1 to +1 indicating the strength and direction of a linear relationship.

Outlier

A data point that lies far from the general pattern of the other data, which may unduly influence r.

Worked Examples

1

A dataset has r = 0.87. Describe the correlation.

Step 1: r = 0.87 is positive, so the direction is positive (as x increases, y tends to increase).

Step 2: Since 0.8 ≤ 0.87 < 1, the strength is strong.

Answer: There is a strong positive linear correlation between the two variables.

2

Hours studied: 2, 4, 5, 7, 8. Marks: 45, 60, 62, 78, 85. Find r using technology and interpret.

Step 1: Enter the data into a calculator or spreadsheet and compute Pearson's r.

Step 2: r = 0.993 (using technology).

Step 3: Since r = 0.993 is very close to 1 and positive, this indicates a very strong positive linear relationship.

Answer: r = 0.993. There is a very strong positive linear correlation between hours studied and marks achieved.

3

A scatterplot shows points curving upward. The r value is 0.62. Should we use a linear model?

Step 1: The scatterplot shows a curved (non-linear) pattern.

Step 2: Although r = 0.62 suggests moderate correlation, r only measures linear association.

Step 3: A linear model would be inappropriate because the relationship is non-linear. A transformation or different model (e.g., exponential) may be more suitable.

Answer: No. The non-linear pattern means a linear model is inappropriate despite the moderate r value.

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

If r = -0.72, what type of correlation exists?

Strong positive Weak negative Moderate negative Strong negative

Question 2

Which value of r indicates the strongest linear relationship?

r = 0.65 r = -0.45 r = 0.88 r = -0.91

Question 3

A study finds a strong positive correlation (r = 0.92) between ice cream sales and drowning incidents. What can we conclude?

There is a strong association but correlation does not prove causation Drowning causes ice cream sales Ice cream causes drowning The data must be incorrect

Question 4

The coefficient of determination r² = 0.81. What percentage of the variation in y is explained by the linear relationship with x?

90% 81% 19% 9%

Question 5

When describing a scatterplot, which of the following is NOT one of the four features you should comment on?

Direction Strength Sample size Form

Bivariate Data Analysis