Inverse Relationships: Inverse Relationship Graph and Formula

Inverse relationships, or negative relationships, are between two variables that have a negative correlation. When one variable increases, the other decreases, and vice versa. These are inverse correlations; when one variable in the data set trends high, the other will trend low. Thus, their ratios result in a negative number.

Inverse relationships illuminate cause and effect and help businesses predict trends. Understanding how one variable influences another can prove valuable in laying out more efficient and cost-effective strategies.

In an inverse relationship, when one variable increases, the other variable decreases. The formula to calculate an inverse relationship is: y = k ÷ x.

Mathematically, x and y represent the two variables, and k is a constant. As x increases, y decreases, and vice versa. For example, in a half marathon race, y represents time, k represents the distance, and x represents the runner’s speed. If speed increases, the time to complete the race decreases.

A direct or proportional relationship is the opposite of an inverse relationship. In a direct relationship, one variable’s increase results in another variable’s similar increase. On a graph, that would mean the x-values and y-values move up or down together. For example, the amount of money a car costs to manufacture and its overall cost to the consumer might be, in part, reflect a causal relationship: a higher manufacturing cost leads to a higher price tag for the buyer.

In inverse relations, variables move in opposite directions, or, as one moves up, the other moves down, and vice versa. Take plane travel time and customer comfort: the longer a flight is, the less comfortable the flier might be: as one variable (flight time) increases, the other variable (customer comfort) decreases.

See below for examples of inverse relationships you might encounter in real life:

1. 1. Hot weather and soup sales: People tend to eat soup in the cooler months, so soup companies will see a regression in sales as temperatures rise.
2. 2. Interest rates and home purchases: Here, the first variable is interest rates (x); the second is the rate of home purchases (y). Low interest rates entice homebuyers, so when x increases, y decreases.
3. 3. Unemployment rates and consumer spending: Higher unemployment rates mean less consumer spending. People with less money or lower financial stability are more likely to conserve, spend less, and rely on coupons and discounts for purchases. In the opposite scenario, this would be a positive correlation: when employment rates rise, so might consumer spending.

An inverse relationship always has a negative slope on a graph. Generally, you need at least three data points to reveal a negative relationship on a graph. Create a graph on which your x-axis represents one variable, and the y-axis represents the other. In this example, the x variable is soup sale revenue in thousands of dollars on a given day, and the y variable is the daily mean temperature on those days. Your variables might look like this:

X (1, 2, 3)
Y (50, 40, 30)

On a fifty-degree day, soup sales were at $1,000, but on a thirty-degree day, sales were at $3,000. On your chart, you will note a downward slope, indicating an inverse relationship: when temperatures go down, soup sales go up.

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