How to compute the formula for percent error.

We all make mistakes. However, if you handle them correctly, they can turn into valuable lessons. It's called 'human error' for a reason; even the most careful among us occasionally miss a 't' or forget to dot an 'i.' That's simply part of being human.

Before fixing a blunder, slip-up, or mix-up (by the way, did you know a thesaurus makes a great gift?), it's generally a good idea to first figure out where things went wrong.

The extent of the error is a crucial aspect. How far did you deviate from the intended result? Was it a minor misstep or a significant mistake?

Imagine a violinist in a symphony orchestra. On the evening of a major performance, he misses a key cue and plays a few notes late. If he was just half a second off, it might not matter too much. But if he missed the cue by thirty seconds, the situation would be much more problematic.

When there is a discrepancy between the expected value and the actual result, and you express that difference as a percentage, it's called a percent error or percentage error. Calculating this error involves comparing the expected value to the actual value to assess how much reality has strayed from the theoretical prediction.

Today, we're going to demystify the process of reporting percent error accurately and demonstrate how to apply it in everyday scenarios.

What Is the Percent Error Formula?

The equation you're about to see is remarkably straightforward. Here's how it looks:

Percent Error = | Experimental Value – Actual Value | / Actual Value x 100%

The value you first estimated goes by several names, such as exact value, accepted value, theoretical value, or experimental value, based on the situation. For example, in physics, a student calculating speed will reference an accepted value derived from the velocity formula, while the measured or approximate value of speed in their experiment might be slightly different. Similarly, a business owner might use an estimated value when predicting future revenue.

Likewise, your actual result can have different names, like actual value, measured value, or known value. Regardless of the term you use, the essential meaning behind the number remains unchanged.

Some people prefer written directions over mathematical expressions. If you fall into that category, here's a detailed guide to help you calculate percent error:

Examples of How to Compute Percent Error

Now we are ready to put the percentage error formula into action and see how it works in practice.

Example 1

Imagine you’re a book lover gearing up for a long vacation. You head to the library thinking you’ll check out three books. However, when you finally leave, you only manage to take home two. What is the percentage error in your estimate?

In this example, the experimental value is 3, and the actual value is 2. When you substitute these values, you get the following equation:

Percent Error = (3 – 2) / 2 x 100

Assuming you're familiar with basic math, we know that 3 minus 2 equals 1. So we’re left with:

Percent Error = 1 / 2 x 100

Now, dividing 1 by 2 gives us the following result:

To calculate Percent Error, we multiply the error by 0.5 and then by 100.

Next, we find that 0.5 multiplied by 100 equals 50. But remember, our final answer must be in percentage form. Therefore, we can conclude that the original estimate had a percent error of 50%.

This particular example is focused on quantity (such as the number of library books). However, you can apply the percent error formula to a wide range of other measurements like speed, mass, distance, and time.

With this in mind, let's revisit the formula once more.

Let's take a look at Example 2.

Imagine a college athlete estimating that he will need 45 seconds to finish a tough workout challenge. However, when he starts the routine at the gym, it takes him 60 seconds to complete. What is the percent error in his initial estimate of 45 seconds?

The formula for percent error is: (45 – 60) ÷ 60 x 100.

Immediately, we run into an issue. Subtracting 60 from 45 gives us a negative result (-15).

Next, divide -15 by 60 to get -0.25, another negative number. To proceed, we multiply -0.25 by 100, which gives us -25. Does this mean the percent error is -25%?

Percent error cannot be negative. It is always presented as a positive value, regardless of whether the initial estimate was too high or too low.

This is where we revisit the familiar concepts of 'absolute error' and 'relative error.' The number -15 represents only the relative error. Before proceeding with the calculation, you need to convert this to its absolute value, which is 15. Once you have the absolute error, divide it by 60, then multiply by 100 to get a percent error of 25%.