Fitting Models to Data — I can fit a model to data with...

SECTION 1

Historical Context & Motivation

Humans have always looked for patterns in numbers. Long before graphing calculators and spreadsheets, scientists and mathematicians struggled with a fundamental question: given a set of observations, what equation best describes the pattern? The story of curve fitting — finding a mathematical model that closely matches collected data — stretches back centuries and is one of the most practically useful ideas in all of mathematics.

1805

Legendre's Least Squares

French mathematician Adrien-Marie Legendre published the method of least squares, a systematic way to find the line that minimizes the total squared error between predicted and actual data points.

1809

Gauss Refines the Theory

Carl Friedrich Gauss independently developed and extended least-squares fitting, using it to predict the orbit of the asteroid Ceres from limited observations — one of the first triumphs of data modeling.

1886

Galton and Regression

Sir Francis Galton coined the term regression when studying the heights of parents and children, establishing the foundation of modern statistical modeling.

1970s

Graphing Calculators Arrive

Handheld technology made regression analysis accessible to students and professionals, allowing anyone to fit models to data without doing tedious hand calculations.

Today

Big Data & Machine Learning

Modern tools like Desmos, Excel, Python, and AI-powered software fit complex models to massive datasets in seconds, driving decisions in medicine, finance, climate science, and more.

The core question this lesson addresses is deceptively simple: when you have a table of data points, how do you find the equation that best describes the pattern, and — just as importantly — what do the numbers in that equation actually mean in real life? Technology handles the computational heavy lifting, but interpreting the result is where your mathematical reasoning comes in.

SECTION 2

Core Principles & Definitions

Before you use any technology to fit a model, you need a clear vocabulary and a set of guiding ideas. The concepts below form the foundation of data modeling and will appear repeatedly throughout this lesson.

Model

A model is a mathematical equation (linear, quadratic, exponential, etc.) chosen to represent the relationship between variables in a dataset.

Parameters

Parameters are the constants in a model equation — like the slope and y-intercept of a line — that technology calculates to make the curve fit the data as closely as possible.

Residuals

A residual is the difference between an actual data value and the value predicted by the model: residual = actual − predicted. Smaller residuals mean a better fit.

Correlation Coefficient (r or R²)

The correlation coefficient (r) measures the strength and direction of a linear relationship. R² tells you what fraction of the data's variability is explained by the model (closer to 1 = better fit).

Constraints / Domain

Constraints are real-world limitations on a model's domain or range. A model for population can't produce negative people, and time can't be negative if you started collecting data at t = 0.

✦ KEY TAKEAWAY

Think of fitting a model to data like adjusting the zoom, angle, and focus on a camera until the picture is as sharp as possible. The technology adjusts the parameters (zoom, angle, focus) of your chosen model (camera type) until the residuals (blur) are as small as possible. But even the sharpest photo has limits — those are your constraints.

SECTION 3

Visual Explanation — Scatter Plot & Best-Fit Line

The diagram below shows a scatter plot of data points along with a best-fit line. The vertical dashed segments between each data point and the line represent the residuals. When technology runs a regression, it finds the line that makes the sum of the squared residuals as small as possible.

Each purple dot is a data point. The cyan line is the best-fit (regression) line ŷ = 2.5x + 0.5. The pink dashed segments show the residuals — the vertical distances between observed data and the line's prediction. Technology minimizes the sum of the squares of these residuals.

Notice how the data points don't all sit perfectly on the line — real data is rarely perfect. The best-fit line represents the overall trend. Points above the line have positive residuals, and points below the line have negative residuals. A good model keeps these residuals small and randomly scattered, with no obvious pattern left over.

SECTION 4

Mathematical Framework

Different datasets call for different types of models. Here are the three most common model families you will encounter, along with their equations and parameter meanings.

LINEAR MODEL

y = mx + b

m = slope (rate of change per unit increase in x); b = y-intercept (value of y when x = 0). Use when data shows a constant rate of increase or decrease.

QUADRATIC MODEL

y = ax² + bx + c

a = determines direction (opens up if a > 0, down if a < 0) and how wide/narrow the parabola is; b and c shift the parabola horizontally and vertically. Use when data rises then falls (or vice versa).

EXPONENTIAL MODEL

y = a · bˣ

a = initial value (when x = 0); b = growth factor (b > 1 means growth; 0 < b < 1 means decay). Use when data increases or decreases by a constant percentage.

COEFFICIENT OF DETERMINATION

R² = 1 − (SS_res / SS_tot)

SS_res = sum of squared residuals; SS_tot = total sum of squares. R² ranges from 0 to 1. Values closer to 1 indicate the model explains more of the data's variability.

When you use a graphing calculator, Desmos, or a spreadsheet to perform a regression, the technology applies the least-squares method behind the scenes. It tests millions of possible parameter values and returns the combination that produces the smallest total squared error. Your job is to choose the right model family by looking at the shape of the scatter plot, then interpret the parameters the technology gives you.

SECTION 5

Choosing the Right Model — A Visual Guide

One of the most important skills in data modeling is selecting the correct type of equation before running a regression. The shape of the scatter plot is your primary clue. The diagram below shows the three major scatter plot shapes and the model type each suggests.

Three scatter plot shapes and their corresponding model types. A linear pattern follows a straight line. A quadratic pattern forms a U-shape or arch. An exponential pattern shows rapid increase (or decrease) that curves steeply.

After choosing a model family, you use technology to perform the regression. In Desmos, you type your data as a table, then type an equation like y₁ ~ mx₁ + b, and the tool fills in the optimal values of m and b automatically. In a TI-84 calculator, you enter data into lists, then run LinReg, QuadReg, or ExpReg from the STAT → CALC menu. Spreadsheet tools like Google Sheets or Excel offer the LINEST function or trendline features. Regardless of the tool, the workflow is the same: enter data, choose a model type, and let the technology compute the best-fit parameters.

💻 Technology Tip

In Desmos, after entering your data table, type y₁ ~ a · b^x₁ for exponential regression. Desmos will display the values of a and b along with the R² value. You can also type y₁ ~ ax₁² + bx₁ + c for quadratic regression.

SECTION 6

Worked Example — Coffee Cooling

A student measures the temperature of a cup of coffee every 5 minutes after pouring it. The data is shown below. We will fit an exponential model, interpret the parameters, and discuss the constraints.

Coffee temperature data over 30 minutes

Time (min)	Temperature (°F)
0	180
5	156
10	137
15	121
20	109
25	99
30	92

Fitting an Exponential Model to Coffee Cooling Data

Step 1 — Plot the DataEnter the data into Desmos (or a graphing calculator) as a table. The scatter plot curves downward, dropping quickly at first and then leveling off — a classic exponential decay shape. This tells us an exponential model y = a · bˣ is appropriate.

Step 2 — Run Exponential RegressionIn Desmos, type y₁ ~ a · b^x₁. The tool returns the best-fit parameters.

a ≈ 179.1, b ≈ 0.977, R² ≈ 0.998

Step 3 — Write the ModelSubstituting the parameters, the model equation is:

T = 179.1 × (0.977)^t, where T is temperature in °F and t is time in minutes.

Step 4 — Interpret the Parameters in ContextThe parameter a ≈ 179.1 represents the model's predicted initial temperature (at t = 0), which is close to the actual starting temperature of 180 °F. The parameter b ≈ 0.977 is the decay factor: each minute, the temperature is multiplied by 0.977, meaning it drops by about 2.3% per minute. The R² ≈ 0.998 tells us that 99.8% of the variation in temperature is explained by the model — an excellent fit.

Step 5 — Identify ConstraintsThe model has physical constraints. Time must be t ≥ 0 (we can't go back before the coffee was poured). Temperature can never drop below room temperature (roughly 70 °F), but our exponential model will eventually predict values approaching 0 °F, which is unrealistic. Therefore, the model is only valid for roughly 0 ≤ t ≤ 50 minutes, beyond which the prediction becomes physically meaningless.

Domain constraint: 0 ≤ t ≤ ≈50 min; Range constraint: T ≥ ≈70 °F (room temperature)

SECTION 7

Strengths & Limitations of Fitted Models

Fitted models are powerful tools, but they are not perfect mirrors of reality. Understanding what a model can and cannot do is just as important as being able to compute one. The table below summarizes the key strengths and limitations.

Strengths vs. Limitations of Fitted Models

Strengths	Limitations
Provides a concise equation to summarize the trend in a large dataset.	A high R² does not prove causation — correlation and causation are different.
Allows predictions (interpolation) within the range of the data.	Extrapolation (predicting far beyond the data range) can give wildly inaccurate results.
Parameters have interpretable real-world meanings (rate of change, initial value, etc.).	Choosing the wrong model type (e.g., linear when data is exponential) produces misleading parameters.
Technology makes computation fast and accessible to anyone.	Models ignore outliers, noise, and context unless the user applies appropriate constraints.

✦ KEY TAKEAWAY

A fitted model is like a GPS route: it gives you the best path based on current data, but it can't predict a road closure two miles ahead. Interpolation (predictions within the data range) is generally reliable, while extrapolation (predictions beyond the data range) should be treated with caution. Always ask: does my prediction make sense in the real world?

SECTION 8

Connection to Advanced Modeling

The regression techniques you are learning now form the foundation for much more advanced modeling that you may encounter in college-level statistics, data science, or calculus-based courses. The table below shows how the concepts in this lesson connect to their advanced counterparts.

Current vs. Advanced Modeling Concepts

This Lesson	Advanced Version
Fitting y = mx + b to a scatter plot	Multiple linear regression (several predictor variables simultaneously)
Choosing between linear, quadratic, and exponential models	Model selection criteria like AIC, BIC, and cross-validation in machine learning
R² as a measure of fit	Adjusted R², residual analysis, and hypothesis testing for significance
Interpreting parameters in context	Confidence intervals and p-values for each parameter in inferential statistics
Domain constraints on models	Piecewise models and logistic growth functions that incorporate natural limits

For now, the key skills to master are recognizing the appropriate model type, using technology to compute the parameters, interpreting those parameters in the context of the problem, and identifying reasonable constraints on the model's domain. These skills transfer directly to AP Statistics, college-level data science, and real-world problem solving in virtually every STEM field.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

A classmate runs a linear regression and gets R² = 0.42. She concludes that her linear model is an excellent fit for the data. Is she correct? Explain your reasoning.

PROBLEM 2 — BASIC CALCULATION

A linear regression on sales data yields the model y = 12.5x + 200, where x represents months since the store opened and y represents monthly revenue in dollars. What does the slope of 12.5 mean in context? What does the y-intercept of 200 mean in context?

PROBLEM 3 — INTERMEDIATE

A biologist tracks a bacteria population and fits an exponential model: P = 50 × 1.35ᵗ, where P is the population in thousands and t is time in hours. (a) What was the initial population? (b) What is the hourly growth rate as a percentage? (c) Predict the population after 6 hours.

PROBLEM 4 — APPLIED

A basketball is launched from a height of 6 feet with an initial upward velocity. A motion sensor records the ball's height (in feet) at various times (in seconds), and a quadratic regression yields h = −16t² + 24t + 6. (a) Interpret each parameter in context. (b) At what time does the ball reach its maximum height, and what is that height? (c) What is a reasonable domain constraint for this model?

PROBLEM 5 — CRITICAL THINKING

A student collects data on the number of subscribers to a new social media platform over its first 12 months. A linear model gives R² = 0.91 and an exponential model gives R² = 0.97. The student decides to use the exponential model to predict the number of subscribers 5 years from now. Discuss: (a) Was the choice of exponential over linear model justified? (b) Is the 5-year prediction likely to be reliable? Why or why not? (c) What alternative model might be more realistic for long-term growth, and why?

SUMMARY

Lesson Summary

Fitting a model to data means choosing an appropriate equation type — linear (y = mx + b), quadratic (y = ax² + bx + c), or exponential (y = a · bˣ) — based on the shape of the scatter plot, then using technology to calculate the optimal parameters via least-squares regression. Tools like Desmos, graphing calculators, and spreadsheets handle the computation, returning the best-fit equation and R² to measure how well the model explains the data.

Interpreting the result is the most important skill. Each parameter has a real-world meaning: slopes represent rates of change, y-intercepts represent starting values, and growth factors describe percentage change per unit. Every model also has constraints — limits on the domain and range that reflect physical or logical reality. Predictions within the data range (interpolation) are generally reliable, while predictions beyond the data (extrapolation) must be treated with caution. Always ask: does my model's prediction make sense in the real world?