Scatterplots & Correlation

Interpreting Scatterplots

Scatterplots are graphical representations of paired numerical data, where one variable is considered independent (x-axis) and the other dependent (y-axis). They are used to visually assess the relationship between two variables.

• Scatterplot: A graph of points representing paired values of two variables.

• Independent variable (x): The variable that is presumed to influence the other.

• Dependent variable (y): The variable that is measured as a response.

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

• Linear correlation: The trend in the scatterplot forms a straight line.

• Correlation does NOT imply causation: Just because two variables are correlated does not mean one causes the other.

Example: A teacher surveys students to determine factors affecting test scores. Data is plotted for variables such as time spent studying, number of pins on a backpack, time sleeping, and number of siblings. Each scatterplot is analyzed for the type of correlation (positive, negative, or none).

Types of Correlation

• 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 discernible pattern between x and y.

• Nonlinear correlation: The relationship is not a straight line.

Example: Test scores vs. time studying typically show positive correlation, while test scores vs. number of siblings may show no correlation.

Practice: Analyzing Scatterplots

Given a table of mean driving speed and number of speeding tickets, students are asked to plot the data and determine the relationship. Matching data sets to scatterplots helps describe the correlation type for each set.

Creating Scatterplots Using a Graphing Calculator

Steps to Create a Scatterplot on TI-84

Graphing calculators can be used to plot data efficiently. The process involves entering data into lists and activating the appropriate statistical plot.

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

• Turn on STATPLOT and select Plot1.

• Set Xlist to L1 and Ylist to L2.

• Adjust window settings (Xmin, Xmax, etc.) as needed.

Example: Engineers study how cargo weight affects flight duration for delivery drones. Data is entered and a scatterplot is generated to assess correlation.

Correlation Coefficient

Intro to Correlation Coefficients

The linear correlation coefficient, denoted as r, measures the strength and direction of the linear relationship between two variables.

• Strength: How closely the data points cluster around a trend line.

• Direction: Whether the relationship is positive or negative.

• Range:

• Strong correlation: Points are tightly clustered; r is close to 1 or -1.

• Weak correlation: Points are loosely clustered; r is closer to 0.

Example: Matching correlation coefficients to graphs: r = 0 (no correlation), r = 1 (strong positive), r = -1 (strong negative).

Properties of the Correlation Coefficient

• The slope of the best-fit line does not affect the value of r.

• Correlation does not imply causation.

Practice: Interpreting r Values

Given a data set with r = -0.92, students are asked to identify the graph that best represents this relationship. In another example, a marketing researcher finds a positive but imperfect correlation between advertising budget and sales revenue, suggesting r is positive but less than 1.

Finding the Correlation Coefficient Using a Calculator

Steps to Find r on TI-84

Calculators can compute the correlation coefficient using the LinReg function in the STAT menu.

• Turn on DiagnosticOn (only needed once).

• Enter data in L1 and L2.

• Navigate to CALC and select 4:LinReg(ax+b).

• Set XList and YList as needed.

Example: Test scores vs. time studying data is entered, and the calculator computes r. Another example involves measuring the speed of sound at different altitudes and finding the correlation coefficient.

Formula for the Correlation Coefficient

The formula for the sample correlation coefficient is:

• xi: Individual x-value

• yi: Individual y-value

• \bar{x}: Mean of x-values

• \bar{y}: Mean of y-values

Summary Table: Types of Correlation

Additional info: Academic context was added to clarify definitions, steps for calculator use, and the formula for the correlation coefficient. Examples were expanded for clarity and completeness.