How To Calculate the Average Rate of Change in 5 Steps

The average rate of change represents a measurement that can provide insight into a variety of applications. From finance and accounting to engineering applications, you can calculate the average rate of change using the simple algebraic formula: (y1 - y2) / (x1 - x2).

Additionally, understanding how you can apply the average rate of change can be beneficial for different uses. 

In this article, we define the average rate of change, explore how it's used and explain how to calculate the average rate of change with examples to give you a deeper understanding of why this measurement is important.

What is the average rate of change?

In mathematics, the average rate of change measures how much a function changes per unit interval, in other words, the movement between two points on a coordinate plane. Another term for the average rate of change is "slope," and you can calculate this value with the following algebraic formula:

y = (mx + b)

If you are working with two sets of coordinates, you can use this formula to find the average rate of change:

(y1-y2) / (x1-x2)

On a graph, the average rate of change either increases or decreases and represents a rise or fall in elevation, such as with land features. However, this value is a useful and often important measurement across many applications.

Related: What Is Internal Rate of Return (IRR)?

What is the average rate of change used for?

Essentially, the average rate of change, or slope, is an extremely useful way to measure the rate at which something changes over a specific period of time. Consider the following applications where the average rate of change is a useful way to measure change over time:

Construction, engineering and architecture

Slope is a mathematical concept that is extremely important in the construction industry. For instance, new building plans require accurate measurements and calculations about a location's topography and how the change in elevation can affect the construction of structures. This data then gives engineers and architects the information they need to develop structural plans for construction projects. The plans that engineers and architects develop take into account land slope and other features to ensure viable structures like buildings, roadways, bridges and other infrastructure.

Related: What Is Terminal Growth Rate? (With Definition and Example)

Finance and accounting

The average rate of change can provide valuable and crucial insight into things like investment expectations, projected outcomes in stocks and other applications where it's necessary to understand how various changes over time can affect investment returns. For example, investors in the stock market can use the rate of change to identify trends like security momentum and price points over time, which is extremely important to consider when investing in financial channels like mutual funds.

Related: What Is the Average Rate of Return? (And How To Calculate It)

Sales

In sales applications, the average rate of change can be useful for gauging the effect that different product costs, prices and margins have on a business's profit growth. For example, assume a business sells different products that all have varying costs to produce, price points and profit margins when sales occur. The business would use the average rate of change for costs, prices and margins to measure the effect these changing values have on its overall profit generation. This can be extremely useful for identifying where certain products are failing to generate expected revenue and where others exceed revenue expectations.

Related: A Guide to Profitability Ratios

How to calculate the average rate of change

In its simplest sense, the average rate of change uses the following formula when applying it to coordinates on a graph:

(y1 - y2) / (x1 - x2)

1. Identify your first set of coordinates

With two pairs of coordinates, determine which set to designate as "set 1." For example, assume you have a coordinate pair of (3, 4) and another coordinate pair of (1, 2). Designate the preceding coordinates (3, 4) as "set 1." This results in your x1 and y1 for the formula.

2. Identify your second set of coordinates

Designate the remaining pair of coordinates as "set 2." Using the previous coordinates (3, 4) and (1, 2), the coordinate pair (1, 2) becomes "set 2" and will be your x2 and y2 values in the formula. Therefore, x1 is 3, y1 is 4, x2 is 1 and y2 is 2. Now that you have your values, you can plug them into the formula.

3. Subtract your y values

Plug your "y" values into the corresponding place in the formula. Use the coordinate pairs (3, 4) and (1, 2) as an example:

(y1 - y2) / (x1 - x2) = [(4) - (2)] / (x1 - x2) = 2 / (x1 - x2)

4. Subtract your x values

Plug your "x" values into their appropriate place in the formula and subtract. With (3, 2) and (1, 4), this is what it looks like:

(y1 - y2) / [(3) - (1)] = (2) / (2)

5. Divide the differences

Once you have subtracted both your "x" and "y" values, you can divide the differences:

(2) / (2) = 1 so the average rate of change is 1.

You can convert the average rate of change to a percent by multiplying your final result by 100 which can tell you the average percent of change. Additionally, the rate of change can be positive or negative, where a positive slope indicates an increasing pattern and a negative slope points to a decreasing trend.

Related: How To Calculate Ratios

Examples of the average rate of change

The following examples can provide additional insight into how the average rate of change can be useful in real-world applications:

Average rate of change in finance

Assume an investor who invests in the stock market wants to understand how the rate of change affects the momentum of their securities. This means that a higher rate of change in an upward trend (increasing on a graph) can be beneficial for higher investment returns. However, a lower rate of change in a downward trend (decreasing on a graph) can indicate potential losses or a decrease in return value over time. To find the rate of change in this situation, the investor would use the formula:

(Closing price{p} - Closing price{p - n}) / (Closing price{p - n}) x 100

Closing price{p} represents the ETF's closing price for the most recent period and closing price{p - n} represents the closing price "n" periods prior to the most recent period. If the investor knows that the closing price for the most recent period is $10 and it was $8 four days prior, they will use these values in the formula:

[($10) - ($8)] / ($8) = (2) / (8) = 0.25 x 100 = 25, where 25 represents the rate of change the ETF's price point experiences between the two time periods. This value is also useful as a percentage, which indicates the percent of change the price points exhibit between two time periods.

In this example, the percent of change between days one and four is 25%, which means the price point of the investor's ETF increased by 25% from day one to day four. Because this value is positive, it indicates that the securities are experiencing upward buying momentum. If the rate of change results in a negative value below zero, then it indicates downward buying momentum.

Related: How To Calculate Different Types of Rate

Average rate of change in temperature

You can easily calculate the average rate of change in temperature using the coordinate method. Assume you measure the temperature one day at 50 degrees, and five days later you measure the temperature at 57 degrees. Using a graph, you set the temperatures as your y-axis and the days you measure as your x-axis. Using days one and five as your "x" values and your temperature readings as your "y" values, you can calculate the slope like this:

Day one = x1 and day five = x2, with 50 degrees = y1 and 57 degrees = y2

Plug these values into the formula like this:

(50 - 57) / (1 - 5) = -7 / -4 = 1.75

The value 1.75 indicates that the average rate of change in temperature was 1.75 degrees each day between days one and five.