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IB DP Maths AA SL Study Notes

4.2.2 Line of Best Fit

Introduction to the Line of Best Fit

The line of best fit is a straight line that best represents a set of data points on a scatter plot. For more details on how to create and interpret scatter plots, visit our scatter plots page. This line is determined using a mathematical method that minimises the sum of the squared differences between the observed values and the values predicted by the line.

  • Purpose: The main goal of the line of best fit is to provide a clear visual representation of the relationship between two variables. It allows us to predict the value of one variable based on the value of another.
  • Components: The line is typically represented by the equation y = mx + c, where:
    • y is the dependent variable.
    • x is the independent variable.
    • m is the slope of the line.
    • c is the y-intercept.

Calculation of the Line of Best Fit

The most common method to calculate the line of best fit is the least squares method. This method aims to minimise the sum of the squares of the vertical distances between the data points and the line. Understanding the correlation coefficient can also help in assessing the strength and direction of the relationship between variables.

The formulae for the slope m and the y-intercept c are:

m = (n(sum xy) - (sum x)(sum y)) / (n(sum x2) - (sum x)2)

c = (sum y - m(sum x)) / n

Where:

  • sum xy is the sum of the product of each pair of x and y values.
  • sum x is the sum of the x values.
  • sum y is the sum of the y values.
  • n is the number of data points.

Interpretation of the Line of Best Fit

  • Slope (m): The slope represents the rate of change of the dependent variable concerning the independent variable. A positive slope indicates a positive correlation, while a negative slope suggests a negative correlation.
  • Y-intercept (c): The y-intercept provides the value of the dependent variable when the independent variable is zero. It offers a starting point for the line.
  • Goodness of Fit: The coefficient of determination, denoted as R2, measures how well the line fits the data points. A value close to 1 indicates a good fit, while a value close to 0 suggests a poor fit. You can read more about the basics of probability to understand how probability theory underpins the concept of goodness of fit.

Practical Applications

The line of best fit has numerous applications across various fields:

  • Economics: Economists use it to predict future sales based on past data or to understand the relationship between variables like price and demand.
  • Biology: Biologists might use it to understand the relationship between variables like dosage of a drug and its effectiveness.
  • Engineering: Engineers might use it to predict the performance of a component based on its characteristics. Additionally, understanding the normal distribution can help in assessing the variation and spread of data around the line of best fit.

Example Questions

Question: Given the following data points, calculate the line of best fit:

Hours of Study: 1, 2, 3, 4, 5Exam Scores: 50, 55, 60, 65, 70

Answer: Using the formulae for slope and y-intercept, we can determine the line of best fit. After calculations, the line of best fit is given by the equation y = 5x + 45. This means that for every additional hour of study, the exam score increases by 5 points.

Question: Interpret the line of best fit given by the equation y = 3x + 20.

Answer: The slope of the line is 3, which means that for every unit increase in the independent variable x, the dependent variable y increases by 3 units. The y-intercept is 20, indicating the value of y when x is zero. The positive slope suggests a positive correlation between the two variables. For a deeper understanding of mathematical concepts, you can explore our section on direct and indirect proofs.

Advanced Insights

The line of best fit can also be represented in more complex forms, such as polynomial regression, where the relationship is represented by a curve rather than a straight line. This is particularly useful when the relationship between the variables is non-linear.

Moreover, while the least squares method is the most common approach to determine the line of best fit, other methods, such as robust regression, can be used when the data contains outliers.

FAQ

Outliers are data points that deviate significantly from the other data points in a dataset. The presence of outliers can significantly influence the slope and y-intercept of the line of best fit. Depending on their position, outliers can either increase or decrease the slope, leading to a potentially misleading representation of the relationship between the variables. It's essential to identify and, if necessary, remove outliers before calculating the line of best fit to ensure a more accurate representation of the data.

Some relationships between variables are non-linear, meaning they don't follow a straight line pattern. In such cases, a curved line, such as a polynomial or exponential curve, might better represent the data. Using a straight line for non-linear data can lead to inaccurate predictions and misinterpretations of the relationship between the variables. It's crucial to analyse the scatter plot and the nature of the relationship before deciding on the type of line or curve that best fits the data.

Theoretically, there can be multiple lines that fit a set of data, but the line of best fit is the one that minimises the sum of the squared differences between the observed values and the values predicted by the line. While different methods might produce slightly different lines, the least squares method is widely accepted as the standard for determining the line of best fit, ensuring consistency in its calculation.

The strength of the relationship between variables is often determined using the coefficient of determination, denoted as R2. This value measures how well the line of best fit explains the variation in the dependent variable. An R2 value close to 1 indicates a strong relationship, with the line of best fit accounting for a large proportion of the variation in the data. Conversely, an R2 value close to 0 suggests a weak relationship, indicating that the line of best fit does not explain much of the variation in the data.

A line of best fit is a straight line that best represents the data points on a scatter plot. It is determined using a method, such as the least squares method, that minimises the sum of the squared differences between the observed values and the values predicted by the line. On the other hand, a line of perfect fit is a line where every data point lies exactly on the line, indicating a perfect linear relationship between the two variables. In real-world scenarios, a line of perfect fit is rare due to the presence of random variations and other factors affecting the data.

Practice Questions

A maths teacher recorded the number of hours her students studied for a test and their respective scores.

The data is as follows:

Hours of Study: 2, 4, 5, 7, 9Test Scores: 50, 55, 63, 70, 80

Using the least squares method, determine the equation of the line of best fit for the given data.

To determine the equation of the line of best fit, we first need to calculate the slope (m) and the y-intercept (c) using the given formulas. After performing the calculations, we find that the slope m is approximately 3.5 and the y-intercept c is approximately 44. Thus, the equation of the line of best fit is y = 3.5x + 44. This equation suggests that for every additional hour of study, the test score increases by approximately 3.5 points.

A company analysed the relationship between the number of advertisements they aired on television and the number of products sold. The equation of the line of best fit they obtained was y = 4x + 10, where y is the number of products sold and x is the number of advertisements aired. Interpret the slope and y-intercept of this line in the context of the company's data.

The equation of the line of best fit is y = 4x + 10. The slope of this line is 4, which means that for every additional advertisement aired on television, the number of products sold increases by 4 units. This suggests a positive correlation between the number of advertisements and product sales. The y-intercept is 10, which indicates the number of products sold when no advertisements are aired. In other words, even without any advertisements, the company expects to sell 10 products. This could be due to other marketing strategies or existing brand recognition.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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