Linear Relationship in Statistics

< Probability and statistics definitions < Linear relationship

A linear relationship, also known as a linear association, is any relationship between two variables that creates a straight line when graphed in an x-y (Cartesian) plane. The straight line happens because one variable increases by approximately the same rate as the other variable changes by one unit.

When two variables have no linear relationship, there may be a non–linear relationship.

Ways to represent a linear relationship

Correlation is a common way to describe linear relationships; it describes the strength and direction of how one variable changes relative to changes in another variable. For example, there is a strong positive correlation with the linear relationship of “amount of items you purchase” vs. “how much money you spend.”

Linear regression can be used to extrapolate past events and make forecasts for the future. It attempts to model the relationship between two variables by fitting a linear equation to observed data [2], producing a line of best fit.

If a relationship exists between variables, not all of them are linear. Some relationships describe curves, such as S-shaped or J-shaped relationships, while other data is random and cannot be described by statistical relationships.

Here are several different ways to represent the same linear relationship between the same two data sets [4]:

1. Visual or Graphical Representations such as a scatterplot or line of best fit.
2. Verbal Representations: This is simply a statement in English concerning the variables of interest. For example, for pairs of numbers, the second number is always two more than three times the first number.
3. Numeric Representations: Ordered pairs, for example: {(1, 5), (2, 8), (3, 11), (4, 14), (5, 17)}.
4. Symbolic Representations: The equation y = 3x + 2, where x represents the first number and y represents the second number.
5. Table of values: A table of values (or a graph) may best reveal one characteristic of linear relationships: constant growth or decay.

x	y
1	5
2	8
3	11
4	14
5	17
Table of values.

The same relationship holds whether we represent it verbally, numerically, visually, or symbolically. In the relationship illustrated above, the constant change is 3 units. For each change of 1 unit in the first variable, the second variable changes 3 units. This characteristic of linear relationships is called slope. If we know two ordered pairs (x₁, y₁) and (x₂, y₂) that are part of a linear relationship, we have enough information to determine the slope of the relationship. The strongest linear relationship happens when the slope is 1 [5].

We can formally define linear relationships with linear functions.

Linear functions

To qualify as linear, a function must meet three criteria:

• It should have two variables,
• All variables must be to the first power (a variable to the second power or higher would create a polynomial relationship, not a linear one),
• The graph of the equation must be approximately a straight line — not curved, U-shaped or other.

Linear functions generate a straight line when graphed. The equation for a linear function is typically written as y = mx + b, where y represents the output, x represents the input, m is the slope of the line, and b is the y-intercept.

 

The equation y = mx + b is commonly known as the slope-intercept form, although it can also be expressed as y = ax + b or y = a + bx. These different representations will result in the same graph.

The domain and range of a linear function generally spans the set of real numbers, except in cases where the function is constant. When a linear function is constant, the range is restricted to the value of that constant, for instance, f(x) = 2.

Linear relationship examples

Many real life situations are linearly related. for example, height and weight usually has a strong linear relationship. In other words, increases in height (x) tend to result in increases in weight (y). 

On the other hand, there is no linear relationship between height and income.

Other examples of linear relationships which can be depicted by a straight line:

• The time it takes to drive to work and the distance of the commute. The longer the distance, the more time it will require to reach the destination.
• Amount of money spent and the number of items purchased. As the spending increases, the capacity to buy more items also increases.
• Height of a plant and the amount of sunlight it receives. With an increase in sunlight, the plant tends to grow taller.
• Effect of temperature on the volume of a substance. As the temperature rises, the volume also expands correspondingly.
• Number of hours spent studying and the grade achieved on an exam. The more hours dedicated to studying, the better the performance on the exam.
• The more practice put into mastering a skill, the higher the ability to perform that skill.

References

1. Dwendland, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons
2. Yale University. Linear Regression
3. Nicholas Longo, CC BY-SA 2.5 https://creativecommons.org/licenses/by-sa/2.5, via Wikimedia Commons
4. Day, R. MAT 312: Probability and Statistics for Middle School Teachers. Review: Linear Relationships. Retrieved July 28, 2023 from: https://math.illinoisstate.edu/day/courses/old/312/notes/twovar/twovar02.html
5. Scatterplots and Correlation