# What Is a Learning Curve? Formula, Calculation, and Example

## What Is a Learning Curve?

A learning curve is a mathematical concept that graphically depicts how a process is improved over time due to learning and increased proficiency. The learning curve theory is that tasks will require less time and resources the more they are performed because of proficiencies gained as the process is learned. The learning curve was first described by psychologist Hermann Ebbinghaus in 1885 and is used as a way to measure production efficiency and to forecast costs.

A learning curve is typically described with a percentage that identifies the rate of improvement. In the visual representation of a learning curve, a steeper slope indicates initial learning that translates into higher cost savings, and subsequent learnings result in increasingly slower, more difficult cost savings.

### Key Takeaways

• The learning curve is a visual representation of how long it takes to acquire new skills or knowledge.
• In business, the slope of the learning curve represents the rate in which learning new skills translates into cost savings for a company.
• A learning curve is usually described with a percentage that identifies the rate of improvement.
• The steeper the slope of the learning curve, the higher the cost savings per unit of output.

## Understanding a Learning Curve

The learning curve also is referred to as the experience curve, the cost curve, the efficiency curve, or the productivity curve. This is because the learning curve provides cost-benefit measurements and insight into all the above aspects of a company.

The idea behind this is that any employee, regardless of position, takes time to learn how to carry out a specific task or duty. The amount of time needed to produce the associated output is high. Then, as the task is repeated, the employee learns how to complete it quickly, and that reduces the amount of time needed for a unit of output.

That is why the learning curve is downward sloping in the beginning with a flat slope toward the end, with the cost per unit depicted on the Y-axis and total output on the X-axis. As learning increases, it decreases the cost per unit of output initially before flattening out, as it becomes harder to increase the efficiencies gained through learning.

Learning curves are often associated with percentages that identify the rate of improvement. For example, a 90% learning curve means that for every time the cumulative quantity is doubled, there is a 10% efficiency gained in the cumulative average production time per unit. The percentage states the percentage of time that will carry over to future iterations of the task when production is doubled.

## Learning Curve Formula

The learning curve has a formula to identify a target cumulative average time per unit or batch. The formula for the learning curve is:

\begin{aligned}&{\bf Y = aX^b}\\&\textbf{where:}\\&\textbf{Y} = \text{Cumulative average time per unit or batch}\\&\textbf{a} = \text{Time taken to produce initial quantity}\\&\textbf{X} = \text{The cumulative units of production or the}\\&\qquad\ \text{cumulative number of batches}\\&\textbf{b} = \text{The slope or learning curve index, calculated}\\&\qquad\text{as the log of the learning curve percentage}\\&\qquad\text{divided by the log of 2}\end{aligned}

## Learning Curve Calculation

Let's use an 80% learning curve as an example. This means that every time we double the cumulative quantity, the process becomes 20% more efficient. In addition, the first task we complete took 1,000 hours.

\begin{aligned}Y &= 1000\times1^{\frac{\text{log}\, .80}{\text{log}\, 2}} \\&= 1000\times 1\\& = \text{an average of 1,000 hours per task}\\&\quad\ \,\text{to complete one task}\end{aligned}

Now let's double our manufacturing output. The initial time spent on the first task will stay 1,000 hours. However, our value for X will now change from one to two:

\begin{aligned}Y&= 1000\times2 ^{\frac{\text{log}\, .80}{\text{log}\, 2}}\\&= 1000\times .8\\& = \text{an average of 800 hours per task}\\&\quad\,\,\,\text{to complete two tasks}\end{aligned}

This means that the total cumulative amount of time needed to perform the task twice was 1,600. Since we know the total amount of time taken for one task was 1,000 hours, we can infer that the incremental time to perform the second task was only 600 hours. This diminishing average theoretically continues as you advance along the learning curve. For example, the next doubling of tasks will occur at four tasks completed:

\begin{aligned}Y &= 1000 \times 4^{\frac{\text{log}\, .8}{\text{log}\, 2}}\\& = 1000 \times .64\\& = \text{an average of 640 hours per task}\\&\quad\,\,\,\text{to complete four tasks}\end{aligned}
In this final example, it took a total of 2,560 hours to produce 4 tasks. Knowing it took 1,600 hours to produce the first two tasks, the learning curve indicates it will only take a total of 960 hours to produce the third and fourth task.

In theory, the third and fourth task in the example above would have taken different amounts of time due to the fourth task being theoretically more efficient than the third task. However, the time taken for each of these two units is often shown as the average (i.e. 960 hours / two units = 480 hours for each of the third and fourth unit).

## Learning Curve Table

The learning curve can become complicated when trying to distinguish between the cumulative quantity, the cumulative production time, the cumulative average production time, and the incremental production time. Therefore, it is common to see a learning curve table that summarizes and neatly organizes each value. This type of information is very useful in cost accounting. The example above would have table as follows:

Note that the cumulative quantity must double between rows—to continue the table, the next row must be calculated using a quantity of eight. In addition, note that the incremental time is a cumulation of more and more units as the table is extended. For example, the 600 hours of incremental time for task No. 2 is the time it took to yield one additional task. However, the 960 hours in the next row is the time it took to yield two additional tasks.

## Learning Curve Graphs

Because learning curve data easily creates trend lines, it's fairly common to see learning curve data depicted graphically. There are several data points to choose from, one of which is the total cumulative time needed to produce a given number of tasks or units. In the graph below, the learning curve shows that more time is needed to generate more tasks.

However, the graph above fails to demonstrate how the process is becoming more efficient. Because of the graph's upward slowing curve, it appears it takes incrementally more time to perform more tasks. However, due to the nature of the learning curve, the x-axis is doubling and incrementally taking less time per unit. For example, consider the graph below that demonstrates the approximate average time needed to perform a given number of tasks.

## Shapes/Types of Learning Curves

Learning curves can be depicted visually in different ways. They can be represented in a chart, with linear coordinates, like the charts above in which the shape is an actual curve. A learning curve can also be depicted between axis points in a chart as a straight line or a band of points.

Lower learning curve percentages mean higher degrees of improvement. As a result, the lower the learning curve percentages, the steeper the slope of graphs.

## Benefits of Using the Learning Curve

Companies know how much an employee earns per hour and can derive the cost of producing a single unit of output based on the number of hours needed. A well-placed employee who is set up for success should decrease the company's costs per unit of output over time. Businesses can use the learning curve to inform production planning, cost forecasting, and logistics schedules.

The slope of the learning curve represents the rate in which learning translates into cost savings for a company. The steeper the slope, the higher the cost savings per unit of output. This standard learning curve is known as the 80% learning curve. It shows that for every doubling of a company's output, the cost of the new output is 80% of the prior output. As output increases, it becomes harder and harder to double a company's previous output, depicted using the slope of the curve, which means cost savings slow over time.

## Learning Curve Example

The learning curve has many applications within the realm of business. It can help assess the true cost of undertaking a project.

For example, the learning curve can play a fundamental part in understanding production costs and cost per unit. Consider a new hire who is placed on a manufacturing line. As the employee becomes more proficient at their job, they will be able to manufacture more goods in a smaller amount of time (all else being equal). In this example, a 90% learning curve would mean there is a 10% improvement every time the number of repetitions doubles. In the long run, a company can use this information to plan financial forecasts, price goods, and anticipate whether it will meet customer demand.

## Why Is a Learning Curve Important?

A learning curve is important because it can be used as a planning tool to understand when operational efficiencies may occur. The learning curve identifies how quickly a task can be performed over time as the performer of that task gains proficiency. This is useful for a company to know when allocating employee's time, dedicating training for new procedures, or allocating costs across new products.

## What Does a High Learning Curve Mean?

A high or steep learning curve indicates that it takes a substantial amount of resources to perform an initial task. However, it also signifies that subsequent performance of the same task will take less time due to the task being relatively easier to learn. A high learning curve indicates to a business that something might require intensive training, but that an employee will quickly become more proficient over time.

## How Is a Learning Curve Measured and Calculated?

A learning curve is measured and calculated by determining the amount of time it will take to perform a task. Then, a learning curve assigns an improvement value to identify the rate of efficiency the task performer will incur as they learn and become more proficient at the task.

The formal calculation to identify the cumulate time is Y = aX^b, where Y is the total amount of time taken, a is the time to produce the first task, X is the total number of tasks performed, and b is the slope of the learning curve.

## What Does a 90% Learning Curve Mean?

When a learning curve has a given percentage, this indicates the rate at which learning and improvement occur. Most often, the percentage given is the amount of time it will take to perform double the amount of repetitions. In the example of a 90% learning curve, this means there is a corresponding 10% improvement every time the number of repetitions doubles.

## The Bottom Line

Most people usually get better at doing something the more they do it. The time and resources spent to do something the first time is probably higher than the time and resources spent on performing the same task for the 100th time. This idea of continual improvement is measured through the learning curve. The learning curve graphically or mathematically depicts how time spent on completing tasks often decreases over time as proficiency is gained.

Article Sources
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1. Harvard Business Review. “Profit from the Learning Curve.”

2. Oxford Reference. “Learning Curve.”

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