Scatterplots and Correlation: Describing the Relation Between Two Variables

Scatterplots & Correlation

Interpreting Scatterplots

Scatterplots are graphical representations used to display the relationship between two numerical variables. Typically, one variable is considered independent (x-axis) and the other dependent (y-axis). The pattern of points can reveal the type and strength of the relationship between the variables.

• Scatterplot: A graph of paired numerical data, with each point representing a pair of values (x, y).

• Correlation: Two variables are correlated if their data points form a discernible pattern.

• Linear Correlation: The trend in the data forms a straight line.

• Positive Correlation: As x increases, y increases; the slope is positive.

• Negative Correlation: As x increases, y decreases; the slope is negative.

• No Correlation: No apparent relationship between x and y.

• Important: Correlation does NOT imply causation. Two variables may be correlated without one causing the other.

Example: A teacher surveys students to determine factors affecting test scores, such as time spent studying, number of pins on a backpack, time sleeping, and number of siblings. By plotting these variables against test scores, one can visually assess the type of correlation present (positive, negative, or none).

Practice: Interpreting Scatterplots

Given a table of mean driving speed and number of speeding tickets, plotting the data can help determine if there is a relationship between speed and tickets received. Typically, a positive correlation would indicate that higher speeds are associated with more tickets.

Matching Data Sets to Scatterplots

Students may be asked to match data sets to their corresponding scatterplots and describe the type of correlation (positive, negative, nonlinear, or none).

Creating Scatterplots

Using a Graphing Calculator

Scatterplots can be efficiently created using a graphing calculator by entering data into lists and using the calculator's plotting functions.

• Enter data into L1 (x-values) and L2 (y-values).

• Turn on STATPLOT and select PLOT1.

• Set the appropriate window for the axes (Xmin, Xmax, etc.).

• Graph the data to visualize the relationship.

Example: Engineers studying how cargo weight affects drone flight duration can plot cargo weight (x-axis) against flight time (y-axis) to determine if a correlation exists.

Additional info: The calculator image is included to reinforce the process of using a calculator for scatterplot creation, as described in the paragraph.

Correlation Coefficient

Introduction to the Correlation Coefficient

The correlation coefficient (denoted as r) is a numerical measure of the direction and strength of the linear relationship between two variables.

• Direction: The sign of r indicates the direction (positive or negative) of the correlation.

• Strength: The magnitude of r (how close it is to 1 or -1) indicates the strength of the correlation.

• Range: $-1 \leq r \leq 1$

• Strong correlation: r is close to 1 (strong positive) or -1 (strong negative).

• Weak correlation: r is close to 0.

• No correlation: r is approximately 0.

Important: The slope of the best-fit line does not affect the value of r; only the clustering of points around the line matters.

Example: Matching correlation coefficients to scatterplots helps students understand the visual meaning of different r values (e.g., r = 0.13 is weak, r = 0.64 is moderate, r = -0.96 is strong negative).

Practice: Interpreting r

Given a correlation coefficient (e.g., r = -0.92), students can identify which scatterplot best represents a strong negative linear relationship. Similarly, they can estimate r values for real-world scenarios, such as advertising budget vs. sales revenue.

Calculating the Correlation Coefficient

Using a Calculator to Find r

The correlation coefficient can be calculated using a graphing calculator's regression functions.

• Turn on diagnostics (if needed) to display r.

• Enter data into L1 and L2.

• Navigate to the CALC menu and select LinReg(ax+b).

• The calculator will output the correlation coefficient r.

Additional info: The calculator image is included to reinforce the process of using a calculator for correlation coefficient calculation, as described in the paragraph.

Example: Calculating r for test scores vs. time studying, or for speed of sound at different altitudes, allows students to quantitatively assess the strength and direction of the relationship.

Formula for the Correlation Coefficient

The formula for the sample correlation coefficient is:

$r = \frac{\sum (x_i - \overline{x})(y_i - \overline{y})}{\sqrt{\sum (x_i - \overline{x})^2 \sum (y_i - \overline{y})^2}}$

• Where $x_i$ and $y_i$ are the individual sample values, and $\overline{x}$ and $\overline{y}$ are the sample means of x and y, respectively.

Summary Table: Types of Correlation

Type of Correlation	Description	r Value
Strong Positive	As x increases, y increases; points closely follow a straight line	r close to +1
Weak Positive	As x increases, y increases; points loosely follow a straight line	r between 0 and +1
No Correlation	No apparent relationship between x and y	r ≈ 0
Weak Negative	As x increases, y decreases; points loosely follow a straight line	r between 0 and -1
Strong Negative	As x increases, y decreases; points closely follow a straight line	r close to -1