Exploring Bivariate Numerical Data(KA/SAP/U5ExploringBivariateNumericalData)

Unit 5: Exploring Bivariate Numerical Data

Topics Covered:

• Introduction to scatterplots
• Correlation coefficients
• Introduction to trend lines
• Least-squares regression equations
• Assessing the fit in least-squares regression
• More on regression

Introduction to Scatterplots

A scatterplot is a graph that shows the relationship between two numerical variables. Each point represents an observation with two values (x, y). Scatterplots help us see patterns, trends, and possible relationships between variables.

Scatterplot Example

Quiz: What does each point on a scatterplot represent? Select An observation with two values A category A frequency count

Correlation Coefficients

The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables. It ranges from -1 (perfect negative) to +1 (perfect positive). A value near 0 means little or no linear relationship.

Correlation Visualizations

Quiz: What does a correlation coefficient of r = -0.9 indicate? Select A strong negative linear relationship A strong positive linear relationship No relationship

Introduction to Trend Lines

A trend line (or line of best fit) is a straight line drawn through the points on a scatterplot to show the general direction of the data. It helps us make predictions and see the overall pattern.

Scatterplot with Trend Line

Quiz: What is the main purpose of a trend line? Select To show the overall pattern in the data To show outliers To show categories

Least-Squares Regression Equations

The least-squares regression line is the line that best fits the data by minimizing the sum of the squared vertical distances from the points to the line. Its equation is usually written as y = a + bx, where a is the intercept and b is the slope.

Regression Line Visualization

Quiz: What does the slope (b) of the regression line represent? Select The change in y for each unit increase in x The value of y when x = 0 The correlation coefficient

Assessing the Fit in Least-Squares Regression

The fit of a regression line can be assessed using the coefficient of determination (R²), which tells us the proportion of the variance in y explained by x. Residual plots can also help us see if a linear model is appropriate.

Residual Plot Example

Quiz: What does a random scatter of points in a residual plot suggest? Select A linear model is appropriate A linear model is not appropriate A perfect fit

More on Regression

Regression can be used for prediction, but beware of extrapolation (predicting outside the data range). Outliers and influential points can strongly affect the regression line. Always check the data and context before making conclusions.

Regression with Outlier

Quiz: What is a risk of extrapolating with a regression line? Select Predictions may be unreliable Predictions are always accurate There is no effect