Data Modeling and Prediction Analysis
Year 13 Mathematics Understanding relationships between variables Making predictions using mathematical models
Learning Intentions - WALT (We Are Learning To)
Use a mathematical model to make predictions Improve our model by removing unusual points and fitting curves Use residuals to decide between linear and curved models Make interpolation and extrapolation predictions
Success Criteria
Identify and explain unusual points in datasets Justify the removal of unusual points Calculate residuals for data points Create and analyze residuals graphs Decide which curve best fits the data
Opening Question
Is there a relationship between two variables in a dataset? How can we tell if variables are connected? What tools can help us identify these relationships?
What is Data Modeling?
Creating mathematical representations of real-world relationships Using equations to describe patterns in data Building tools for prediction and understanding Connecting mathematics to practical applications
Types of Relationships
{"left":"Linear Relationships\nCurved (Non-linear) Relationships\nStrong positive correlation\nQuadratic patterns","right":"Weak or no correlation\nExponential growth\nNegative correlation\nLogarithmic patterns"}
Quick Data Exploration
Examine the height vs. shoe size dataset Plot the points on graph paper Describe the pattern you observe Predict: Is this linear or curved?
Scatter Plots - Our Starting Point
Visual representation of bivariate data Each point represents one observation Helps identify overall patterns and trends Shows the strength and direction of relationships
Linear Models - The Straight Line Approach
Equation form: y = mx + c Line of best fit through data points Useful for constant rate relationships Easy to interpret and calculate
When Linear Models Fall Short
What happens when data doesn't follow a straight line? How do we know if a linear model is appropriate? What are the signs of a poor linear fit?
Introducing Residuals
Residual = Observed value - Predicted value Measures how far each point is from our model Positive residuals: point above the line Negative residuals: point below the line
Calculate Residuals Practice
Given: Linear model y = 2x + 1 Data points: (1,4), (2,6), (3,7), (4,11) Calculate the residual for each point Which point has the largest residual?
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