Graphs, Models, and Experimental Uncertainty

Chapter 5 Overview

A table of numbers from an experiment tells you what was measured. A graph of those numbers reveals the relationship. A model tells you why. And an uncertainty estimate tells you how much to trust it all. These four tools — graph, table, model, uncertainty — are how physics converts raw data into knowledge. This chapter builds the analytical habits you will use in every experiment and every data-driven problem for the rest of this book.

On This Page

1. Why Physicists Use Graphs and Models
2. Independent and Dependent Variables
3. Reading Scientific Graphs
4. Constructing Proper Graphs
5. Interpreting Slope and Rate of Change
6. Interpreting Area Under a Curve
7. Linear vs. Nonlinear Relationships
8. Physical Models and Idealizations
9. Assumptions in Scientific Modeling
10. Experimental Error and Uncertainty
11. Random Error vs. Systematic Error
12. Propagation of Uncertainty
13. Accuracy, Precision, and Reliability
14. Common Mistakes in Data Interpretation
15. Chapter Summary
16. Key Terms
17. Practice and Review
18. Chapter 5 Checkpoint

5.1 Why Physicists Use Graphs and Models

A physicist rarely stops at measuring. The measurement is the beginning, not the end. Once data is collected, the physicist asks: What pattern do these numbers reveal? What relationship connects the variables? Does that relationship match what theory predicts? Could the agreement be accidental? These questions require tools beyond arithmetic.

Graphs are among the most powerful of those tools. A well-constructed graph makes a pattern visible at a glance that would take minutes to detect by scanning a data table. A curve's shape tells you immediately whether a relationship is linear, quadratic, inverse, or something else. The slope of a line on a graph can represent a physical quantity — velocity on a position-time graph, acceleration on a velocity-time graph, spring constant on a force-extension graph. The area under a curve on a graph can represent another physical quantity — distance on a velocity-time graph, work done on a force-displacement graph.

Models are simplifications. No experiment perfectly matches the idealized equations of a textbook, because the real world has air resistance, friction, temperature fluctuations, and imperfect instruments. A model selects the most important factors, ignores the less important ones, and produces a mathematical description that is tractable without being false. Learning to work with models — to know when they apply and when they fail — is one of the most important intellectual skills in physics.

“All models are wrong, but some are useful.” — George Box, statistician

5.2 Independent and Dependent Variables

Every experiment involves at least two variables: the quantity the experimenter controls or varies, and the quantity that responds. Understanding which is which is essential before constructing a table or graph.

The independent variable is the quantity the experimenter deliberately changes or controls. It is the input to the experiment. The dependent variable is the quantity that responds to those changes; it is the output or result. In a well-designed experiment, all other relevant quantities, called controlled variables or controlled conditions, are held constant while the independent variable is varied.

The Rule: The independent variable is plotted on the horizontal axis, the x-axis. The dependent variable is plotted on the vertical axis, the y-axis.

5.3 Reading Scientific Graphs

A well-constructed scientific graph contains several standard elements. Learning to read each element quickly is a skill used in every chapter of this book and in every science course you will take.

When reading a graph, extract specific values by reading coordinates at points of interest. Estimate intermediate values by interpolation. Avoid extrapolation beyond the range of the data unless you have strong theoretical reasons to believe the relationship continues.

Common Mistake: Extrapolating beyond the range of your data is unreliable. Always state clearly when an answer depends on extrapolation.

5.4 Constructing Proper Graphs

Constructing a graph is not merely drawing data points on paper. A scientific graph must be built to communicate clearly and honestly.

Steps for Constructing a Scientific Graph

1. Identify which variable is independent and which is dependent.
2. Choose a scale that spreads the data across at least 75% of the graph area.
3. Label both axes with variable name and unit in parentheses.
4. Plot data points as small, clear symbols. Do not connect points with zigzag lines.
5. Draw a best-fit line or curve that follows the trend of the data.
6. Add a title that identifies the graph.

Tip: The best-fit line should pass among the data points, with roughly equal numbers of points above and below. Do not force the line through the origin unless the theory specifically requires zero-zero.

Common Mistake: Do not connect data points with straight line segments between each pair. Draw a single smooth best-fit curve instead.

5.5 Interpreting Slope and Rate of Change

The slope of a line on a graph tells you how rapidly the dependent variable changes with respect to the independent variable. Slope is one of the most important physical quantities you can read from a graph.

slope = rise / run = Δy / Δx = (y₂ − y₁) / (x₂ − x₁)

Choose two widely separated points on the best-fit line, not individual data points, to calculate slope. The physical meaning of the slope depends on what is plotted.

Common Mistake: Calculate slope using two points on the best-fit line, not two raw data points.

5.6 Interpreting Area Under a Curve

While slope gives a rate of change, the area enclosed between a graph and the horizontal axis gives an accumulated quantity. This is the graphical version of integration, but it can be understood geometrically without calculus.

To estimate the area under a curve when the relationship is not simple, count grid squares under the curve, or break the region into triangles and rectangles and add their areas.

Example: A car accelerates from 0 to 20 m/s in 5.0 s. On a velocity-time graph, this is a right triangle with base 5.0 s and height 20 m/s. Area = ½ × 5.0 s × 20 m/s = 50 m.

Bridge Note: Slope and area under a curve are the graphical versions of differentiation and integration.

5.7 Linear vs. Nonlinear Relationships

Not all physical relationships produce straight lines on a standard graph. Recognizing the shape of a curve gives you important information about the mathematical relationship between the variables.

Linearizing a Nonlinear Relationship

A powerful technique is to replot the data in a way that produces a straight line even when the original relationship is nonlinear. This is called linearization.

Example: If you plot y vs. x and get a parabola, try plotting y vs. x². If the relationship is truly y = kx², the new graph will be a straight line through the origin with slope k.

Tip: Whenever a graph produces a curve, ask what power of x, or what function of x, would make this a straight line.

5.8 Physical Models and Idealizations

A physical model is a simplified representation of a real system that captures the essential physics while deliberately ignoring secondary complications. Models are not failures of precision; they are strategic decisions about which factors dominate.

A useful test is to estimate the size of the neglected effect. If it is 10 times smaller than the measurement uncertainty, ignoring it is justified. If it is comparable to or larger than the uncertainty, the model must be refined.

Historical Note: Newton's laws with no air resistance predict that a cannonball's range is maximized at 45°. With significant air resistance, the optimal angle can drop to around 30°–35°.

5.9 Assumptions in Scientific Modeling

Every model rests on explicit and implicit assumptions. Making assumptions explicit before doing the calculation is a hallmark of careful scientific reasoning.

How to State Assumptions

1. Identify quantities being held constant that might vary.
2. Identify forces or effects being neglected.
3. Identify the range of conditions within which the model is expected to be valid.
4. Identify whether the system is open or closed, isolated or in contact with an environment.

An assumption is justified if the quantity being neglected produces a correction smaller than the measurement uncertainty. Estimate the correction and compare it to the uncertainty.

Key Point: Writing “assume frictionless” or “neglect air resistance” is careful physics because it signals exactly what you are and are not accounting for.

5.10 Experimental Error and Uncertainty

No measurement is perfectly precise. Every measured value differs from the true value by some amount due to instrument limits, experimental technique, environmental variability, and the fundamental limits of measurement. This unavoidable imperfection is called experimental error, though the word error does not imply a mistake.

Uncertainty is the quantitative expression of this spread. A measurement reported as 4.75 ± 0.05 m tells you that the true value is believed to lie between 4.70 m and 4.80 m.

measurement = best estimate ± uncertainty (unit)

Example: m = 12.4 g ± 0.1 g means the true mass is believed to be between 12.3 g and 12.5 g.

The uncertainty is typically estimated in one of three ways. For a single measurement, use half the resolution of the instrument. For repeated measurements, use half the range or one standard deviation. For calculated results, use propagation of uncertainty.

5.11 Random Error vs. Systematic Error

Random errors are unpredictable, non-repeating fluctuations. They cause repeated measurements to scatter around the true value. Random error affects precision but not accuracy, and it is reduced by taking more measurements and averaging them.

Systematic errors are consistent, repeatable deviations in one direction. They shift every measurement by the same amount, so averaging does not help. Systematic errors affect accuracy and require calibration or redesign.

Example — Random error: Timing a pendulum ten times gives values scattered between 1.98 s and 2.04 s. Average them to get 2.01 s.

Example — Systematic error: A ruler whose first millimeter has worn away always reads 1 mm too small.

5.12 Propagation of Uncertainty (Introductory)

When a calculated quantity depends on two or more measured quantities, each with its own uncertainty, those uncertainties combine to produce an uncertainty in the result. This process is called propagation of uncertainty.

Rule 1: Addition and Subtraction

When adding or subtracting quantities, add the absolute uncertainties.

If z = x + y or z = x − y, then Δz = Δx + Δy

Example: L₁ = 3.50 ± 0.05 m; L₂ = 2.20 ± 0.03 m; L_total = 5.70 ± 0.08 m.

Rule 2: Multiplication and Division

When multiplying or dividing quantities, add the relative or percentage uncertainties.

If z = xy or z = x/y, then Δz/z = Δx/x + Δy/y

Example: L = 4.0 ± 0.1 m (2.5%); W = 2.5 ± 0.1 m (4.0%); Area = 10 m² ± 6.5% = 10 ± 0.7 m².

Rule 3: Power

When raising a quantity to a power n, multiply the relative uncertainty by |n|.

If z = xⁿ, then Δz/z = |n| × Δx/x

Key Point: These rules give the maximum possible uncertainty in the result. They are safe and conservative.

Bridge Note: The advanced series uses the statistical quadrature method for propagating uncertainty.

5.13 Accuracy, Precision, and Reliability in Experiments

A good experiment must be accurate, precise, and reliable. Accuracy means close to the true value. Precision means reproducible. Reliability means robust under independent repetition.

percent error = |measured − accepted| / accepted × 100%

5.14 Common Mistakes in Data Interpretation

1. Connecting data points with a zigzag line. Draw a smooth best-fit line or curve instead.
2. Forcing the best-fit line through the origin. Do this only when theory requires a zero-zero relationship.
3. Reading slope from data points instead of the best-fit line. Use two widely separated points on the line.
4. Extrapolating beyond the data range. Label extrapolated values and justify them theoretically.
5. Ignoring units on graph axes. A graph without units is incomplete.
6. Reporting more significant figures than uncertainty supports. Round to the precision justified by the measurement.
7. Ignoring uncertainty entirely. Without uncertainty, you cannot compare results honestly.

Common Mistake: The most dangerous graph error is the “ruler through any two convenient points” method for finding slope. Choose two widely separated points on the best-fit line.

Chapter Summary

Graphs, models, and uncertainty analysis are the analytical infrastructure of experimental physics. Together, they allow physicists to extract relationships from data, simplify complex systems, and report results honestly.

1. The independent variable is plotted on the x-axis; the dependent variable is plotted on the y-axis.
2. A scientific graph requires a title, labeled axes with units, data points, and a best-fit line or curve.
3. Slope equals Δy/Δx and is calculated from two points on the best-fit line.
4. Area under a graph equals y-units times x-units and represents an accumulated quantity.
5. Common curve shapes include linear, parabolic, inverse, and exponential relationships.
6. A model is a deliberate simplification that retains essential physics and ignores secondary effects.
7. Random error causes scatter and is reduced by averaging. Systematic error shifts all readings and requires calibration or redesign.
8. Measurements are reported as value ± uncertainty.
9. Accuracy is closeness to truth, precision is reproducibility, and reliability requires independent replication.

Key Terms

independent variable: The variable the experimenter controls or varies; plotted on the horizontal axis.

dependent variable: The variable that responds to changes in the independent variable; plotted on the vertical axis.

controlled variable: A quantity held constant throughout an experiment.

best-fit line: A line drawn through a scatter of data points to represent the underlying trend.

slope: The ratio Δy/Δx for a straight line.

area under the curve: The area enclosed between a graph and the horizontal axis; represents an accumulated quantity.

interpolation: Estimating between measured data points.

extrapolation: Estimating outside the range of the data.

linearization: Replotting data to convert a nonlinear relationship into a straight line.

physical model: A simplified representation of a real system.

idealization: A simplifying assumption within a model.

experimental error: The unavoidable discrepancy between a measured value and the true value.

uncertainty: The quantitative estimate of the range within which the true value is believed to lie.

random error: Unpredictable fluctuations that cause scatter.

systematic error: A consistent deviation from the true value in one direction.

propagation of uncertainty: Calculating uncertainty in a derived quantity from uncertainties in measured quantities.

percent error: |measured − accepted| / accepted × 100%.

precision: Reproducibility of a measurement.

accuracy: Closeness of a measurement to the true or accepted value.

reliability: The property of an experimental result that holds up under independent replication.

Practice and Review

Concept Review Questions

1. What is the difference between the independent and dependent variables in an experiment? Give an example from daily life.
2. List six elements a properly constructed scientific graph should include. Why does each matter?
3. Explain why a best-fit line should not be drawn by connecting data points with straight line segments.
4. What physical quantity does the slope represent on a position-time graph, a velocity-time graph, and a force-extension graph?
5. What physical quantity does the area under a velocity-time graph represent? Why does unit analysis support this?
6. Describe two different curve shapes that appear in physics graphs and give one physical example of each.
7. What is a physical model? What is an idealization?
8. Why is stating assumptions explicitly considered good scientific practice?
9. What is the difference between random error and systematic error?
10. A student measures the period of a pendulum ten times and gets values from 1.96 s to 2.04 s. Estimate the uncertainty and report the result.
11. What does it mean for a result to be precise but not accurate?
12. Explain two common mistakes students make when reading slope from a graph.

Graph Interpretation Exercises

1. A position-time graph passes through (0 s, 2.0 m) and (5.0 s, 17.0 m). Find the slope, units, physical meaning, and y-intercept meaning.
2. A velocity-time graph rises from (0 s, 0 m/s) to (8.0 s, 24.0 m/s). Find slope, physical meaning, area, and area meaning.
3. A force-extension graph has slope 250 N/m. Identify the slope and calculate force for 0.12 m extension.
4. A graph of y vs. x² gives a straight line with slope 4.9 m/s². Explain the original relationship and why linearization was used.
5. A velocity-time graph has rest, acceleration, then constant velocity intervals. Find acceleration and total distance from the described intervals.

Graph Construction Practice

Construct proper graphs from the given spring and ramp datasets in the source chapter. Determine best-fit line equations and extract the relevant physical quantities from slope.

Experimental Uncertainty Problems

1. A block's length is measured five times: 8.23, 8.25, 8.22, 8.26, 8.24 cm. Find mean, range, half-range uncertainty, and report the result.
2. A force F = 25.0 ± 0.5 N acts over displacement d = 4.0 ± 0.2 m. Find work and propagated uncertainty.
3. The accepted value of g is 9.81 m/s². A student measures 9.65 m/s². Calculate percent error.
4. A radius r = 3.0 ± 0.1 cm is used to compute A = πr². Find area and uncertainty.
5. Two resistors R₁ = 120 ± 3 Ω and R₂ = 80 ± 2 Ω are in series. Find total resistance and uncertainty.
6. A pendulum period is measured as 1.41, 1.44, 1.42, 1.45, 1.43 s. Report the mean with uncertainty, calculate percent error, and decide whether the accepted value lies within the uncertainty range.

Lab Analysis Activity: Ramp and Roll

Purpose: To practice data collection, graph construction, slope extraction, uncertainty estimation, and model evaluation in a single integrated experiment.

Materials: A smooth ramp, a ball or cart, a ruler, a stopwatch, and graph paper.

1. Set the ramp at a fixed angle and measure travel times for five distances.
2. Construct a data table with distance, trial times, mean time, and uncertainty.
3. Plot distance vs. mean time and describe the curve shape.
4. Plot distance vs. mean time squared and check for a straight line.
5. Use d = ½at² to compare the slope to ½a and find acceleration down the ramp.

Chapter 5 Checkpoint

Before moving on to Chapter 6, confirm you can do each of the following without notes.

Continue to Chapter 6: Describing Motion →