Guide to Linear vs. Nonlinear Analyses (With Example Uses)

By Indeed Editorial Team

Published June 17, 2022

The Indeed Editorial Team comprises a diverse and talented team of writers, researchers and subject matter experts equipped with Indeed's data and insights to deliver useful tips to help guide your career journey.

Statistics and mathematics have far-reaching applications, involving everything from health care to technology. Identifying patterns, both linear and non-linear, can help analysts make predictions and understand situations. Learning what these terms mean and how to apply them can help you thrive in your career. In this article, we explore linear vs. nonlinear analyses by defining each and providing examples of practical applications for each of these mathematical analyses.

Linear vs. nonlinear analyses

An understanding of linear vs. nonlinear analyses ensures that you interpret data in the correct way. The gathering, mapping, and studying of numbers on a Cartesian plane are the basis for this type of analysis. On a graph, you plot ordered pairs of values (x, y), where x is the horizontal plane and y is the vertical plane. These data points connect in different ways, ranging from a parabolic nonlinear pattern to a straight linear correlation.

When you perform an analysis, it's important to consider what type of pattern the data presents. To learn whether your results form a linear or non-linear pattern, plot the data points on a graph. You can also consider other factors, such as the number and behaviour of the degrees that apply to the x or y variables. Once you determine which type of equation you have, then you can analyze it accordingly.

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What is a linear equation?

A linear equation is y = mx + b. The value of y is the vertical location on the graph. X is the variable that the slope, m, affects. The slope is the line's incline, measured by the difference between your highest and lowest y values, called the rise. You divide this number by the difference between your highest and lowest x values, called the run. The result is your slope, m. Finally, b is the value of the y coordinate when the line intersects with the x-axis. To calculate it, set x to zero and calculate for b.

To identify a linear equation, the primary criteria is that both variables have a degree of one, no less or greater. The line requires a consistent linear pattern that applies in both directions, negative and positive, and follows the same slope throughout. Some examples of linear equations include:

  • 45x + 67 = 92y

  • 5y - 4 = x

  • 0 = 8 + x + 2y

Read more: Example of Positive Correlation (And How to Calculate It)

Types of linear analytics categories

Linear analyses are common in daily life and understanding how to recognize them can offer ample value both at work and for personal applications. Some useful categories where you can use linear analytics include:

Predicting simple costs

Linear equations occur whenever you aim to predict an outcome. For instance, if a lemonade stand spends 20 dollars to get supplies and earns 10 dollars in sales in one week, you can express it as y = 10x +20 in equation form. To predict the income results after five weeks, you execute the function y = 10(5) + 20 for a result of 30 dollars. This assumes that the costs and sales remain static throughout the entire period.

Making budgets

Budgets are amongst the most common linear applications, especially when there is a fixed amount. Consider a wedding, where the cost of the venue is 500 dollars, and the price per attendee is 10 dollars. The equation y = 10x + 500 allows you to do two things. First, you can input the number of attendees you want as the value x and see the total costs. Then, you can set a maximum budget by inputting it as a y value and solving for x to determine your maximum number of attendees.

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Analyzing rates

When receiving a job offer, it includes the annual salary. For example, consider a job that offers 50,000 dollars per year. The first linear equation calculates your weekly earnings by inputting 50,000 as the y value and the x as the weeks, so 50,000 = 52x gives you a weekly earning of approximately 962 dollars per week. You can continue this pattern by then inputting your weekly hours and calculating their value by changing y to 962 and x to the number of hours. In this case, the equation is 962 = 40x for an hourly rate of 24 dollars.

What is a non-linear equation?

A non-linear equation is anything that has either an x or y degree that is greater than or lesser than zero. Most non-linear equations follow the pattern of ax² + by² = c. In this equation, a and b are each numbers, also called coefficients, that affect the variables x and y. The c value represents the y-intercept point of the pattern. While a linear pattern has a sequential form, denoted by the variable degree of one, non-linear data follows a different pattern. When you plot the data graph, it forms a curve whose pattern depends on the variables' degrees.

Some examples of non-linear equations include:

  • 4x³ + 5y² = 987

  • a² + 5ab² + b = 0

  • 3x³ +2y = 943

Important non-linear analyses

Non-linear equations are more common than linear ones, mostly because of the chaotic nature of the world. Consider a runner moving at a consistent speed for 10 kilometres. That pattern follows a linear equation. A non-linear equation occurs if the runner accelerates, slows down, or stops for a short period. Because of this difference, it's more likely to encounter a non-linear situation. Some common examples include:

Infrastructure

When urban planners make decisions, non-linear function analysis is common. For example, consider a town that decides to build a fountain that comprises a square land plot and includes a path surrounding it. To determine the length of the trail, the urban planner considers the area of the square first, finding the length of the one side, denoted as S by taking the square root of the area using S = A^½. Because half is lower than one, it's a non-linear expression.

Disease prevention

Studying infectious diseases requires nonlinear differential calculations. This is common when creating models for population growth, both of people and of pathogens. The patterns of population growth and movement change regularly, meaning only a non-linear expression can represent them. With infections, the premise is that the function has an odd degree, such as three or five, and therefore moves in a wavelike curve that grows at different rates. Infectious disease models take regular information to determine the curve pattern and attempt to alter it using human intervention.

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Economics

In economics, the standard of practice is to account for all graphical patterns as curves due to how common they are in the field. Three functions that economists rely on include:

  • Exponential functions consider the growth or decay patterns that an investment, company value, or revenue follows over time.

  • Quadratic functions allow the economist to have the independent variable x change its effect on the dependent variable, y. It's essential to supply and demand calculations.

  • Logarithmic functions allow economists to convert non-linear functions into linear representations for ease of use. The final resulting linear representation is the log of the contributing variables.

Read more: Positive vs. Normative Economics (Definitions and Comparison)

Physics

Most physics involves chaotic patterns that follow a non-linear pattern. Depending on the situation, the number of variables increases drastically. The qualifying criteria for a non-linear pattern are the degrees of the variables, not the number. Some complex quantum physics equations move beyond the Cartesian plane and incorporate more dimensions.

The applications of physics in non-linear patterns can be as simple as acceleration due to gravity or the parabola of throwing a ball into the air. These same patterns can represent neuron patterns, weather systems, and machine learning. If there is any unpredictability in the situation, you express it as a non-linear function.

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