Calculating The Gradient of Line of Best Fit

This is part of the NSW HSC science curriculum part of the Working Scientifically skills.

How to Calculate and Use the Gradient in HSC Physics and Chemistry

This video explains the use of gradient calculation in a scientific setting including physics and chemistry specific examples. This video also discusses helpful tips and common mistakes surrounding this topic.

What is the Gradient?

The gradient of a line describes how steep the line is. It represents the rate of change of one variable (y) with respect to another (x). Mathematically, the gradient (m) is given by:

$$m = \frac{\Delta y}{\Delta x}$$

Where:

  • `\Delta y` is the change in the y-values (vertical change).

  • `\Delta x` is the change in the x-values (horizontal change).

 

 

The gradient of a graph represents the trend or relationship between the independent and dependent variables of an experiment. For example, if the gradient is constant (straight line graph), the trend is then constant or remains unchanged between the two variables. When the gradient is not constant, the trend also changes.

 

 

Positive gradient values suggest positive relationships between variables, i.e. as the independent variable increases, the dependent variable also increases. Negative gradient values suggestive negative relationships between variables.

 

 

In scientific terms, the gradient often corresponds to a physical quantity, such as speed (Physics) or reaction rate (Chemistry). 

Use of Gradient in HSC Physics:

  • Kepler's third law of planetary motion
  • Circular motion and centripetal force
  • Torque
  • Electromagnetic forces e.g. force acting on a moving charge in magnetic field
  • Faraday's Law
  • Malus' Law
  • Photoelectric effect

 

Use of Gradient in HSC Chemistry:

  • Analysis of reaction rate
  • Gas laws
  • Calorimetry
  • Titration curves
  • Calibration graphs in colourimetry, AAS and spectrophotometry

Step-by-Step Guide to Calculating the Gradient

Step 1: Draw the Line of Best Fit

  • Plot your experimental data on a graph.

  • Use a ruler to draw a straight line that best represents the data points. This line should pass as close to as many points as possible, balancing the scatter above and below the line.

  • The line of best fit should also be drawn between the first and last data points only. Extending the line beyond these points assumes the trend is valid outside the data set tested in the experiment.

Step 2: Choose Two Pairs of Data on the Line

  • Identify two clear points on the line of best fit. These points do not have to be data points from your experiment, but they must lie directly on the line.

  • Compute the differences in the y-values and x-values between the two chosen points:

Step 3: Calculate the Gradient

  • Use the formula to calculate the gradient.

  • Simplify the result and include appropriate units (if applicable).

Example: Velocity in Physics

In a Physics experiment, you might measure the displacement of an object (y-axis) over time (x-axis). The gradient of the displacement-time graph gives the object’s velocity. This relationship is given by the simple formula for velocity of an object:

$$v = \frac{\Delta s}{\Delta t}$$

Since displacement and time are the y and x variables respectively, velocity is represented by the gradient of the following graph.

 

For instance, let’s say you have a line of best fit with the following points:

and

  1. Calculate the change in y value: `= 10.0-8.0 = 2.0`

  2. Calculate the change in x value: `= 2.7 - 2.0 = 0.7`

  3. Compute the gradient: `m = \frac{2.0}{0.7} = 2.9 = 3 \text{ m/s (1 s.f.)`

The gradient is , which represents the velocity of the object in units of displacement per unit time (e.g., m/s).

Example: Reaction Rate in Chemistry

An example of a variable gradient is its use in analysis of reaction rate in chemical reactions. For a gas producing reaction, the volume of gas produced can be monitored over time as shown in the following graph.

 


The gradient of this graph represents the reaction rate and it decreases over time. This means the reaction slows down over time. The instantaneous reaction rate can be determined by calculating the gradient at a specific time point.

For example, at t = 0, `m = \frac{20-0}{10-0} = 2.0 \text{ mL/s}`

At t = 46 s, `m = \frac{39-23}{74-16} = 0.28 \text{ mL/s}`

By studying the gradient, it is also easy to identify when the reaction comes to a stop. This occurs at t = 88 s when the gradient becomes zero.

Note in this case, the unit of the reaction rate (gradient) depends on the units of the independent and dependent variables. If the unit of volume was in litres, the gradient unit's would be L/s instead of mL/s. 

Why Use the Gradient Over a Single Data Point?

 

Using the gradient of a line of best fit is more accurate than relying on a single data point because:

  1. Minimises Errors: Individual data points may have random errors or deviations due to experimental inaccuracies. The line of best fit averages these variations, providing a more reliable representation of the trend.

  2. Considers the Full Dataset: The gradient uses multiple points along the line, incorporating more data into the calculation. This approach ensures the result reflects the overall pattern, not just one measurement.

Other Uses of Gradient

As explained above, the gradient provides insight into the relationship between variables. 

In addition, the line of best fit and its gradient can also be used to predict values outside your data range (extrapolation) or within it (interpolation).

 

 

The gradient value can also be used to determine the validity and accuracy of experimental data. This is done by comparing the gradient’s value to theoretical predictions or expected trends. This is discussed in more detail here.

 

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