What is a Percent?
Before discussing percent error, let’s see what a percent is.
A percentage is a number or ratio that is written as a fraction of 100. Percentage refers to the number of parts per hundred. The word originally comes from the Latin term per centum, which means ‘per hundred’. Percentage is denoted by the sign ‘%’ in mathematics.
What Causes Errors?
Whenever we perform an experiment or calculation, we obtain results that are either equal to, or, differ from the actual value. The difference between the estimated and actual value is called an error. Errors may happen due to human inattention or the limitations of the devices used in the experiment. Errors come in all sizes and it is essential to calculate how large an error is, as the result of the experiment depends on it.
This is where percent error comes in handy, which assists us in finding the accuracy of the result.
What is the Percent Error?
Percent error indicates the difference between the actual value and the estimated value. This estimated value when compared to the actual value, is expressed in the form of a percentage value. This reveals the extent of the error. Smaller errors tell us that we’re pretty close to the estimated or actual value.
How do we Compute Percent Error?
We can calculate the percent error by applying the following formulas:
- If the actual value is greater than the estimated value, then,
Percent Error = \(\frac{Actual~value~-~Estimated~value}{Actual~value}~\times~100 \)
- If the estimated value is greater than the actual value, then,
Percent Error = \(\frac{Estiamted~value~-~Actual~value}{Actual~value}~\times~100 \)
Here, the result obtained from (Actual value – Estimated value) or ((Estimated value- Actual value) is called the absolute error.
Also, the result after solving \(\frac{Actual~value~-~Estimated~value}{Actual~value} \) or \(\frac{Estimated~value~-~Actual~value}{Actual~value} \) gives us the value for relative error.
Rapid Recall
Solved Percent Error Examples
Example 1:
There are 40 candies in a small container box. But we miscounted it as 32 candies. What would be the percent error?
Solution:
The actual number of candies = 40
The candies miscounted = 32
Here, the actual value is greater than the estimated value. So, the formula to calculate percent error is:
Percent Error = \(\frac{Actual~value~-~Estimated~value}{Actual~value}~\times~100 \).
Percent error = \(\frac{40~-~32}{40}~\times~100 \) (Substitute the value)
= \(\frac{8}{40}~\times~100 \) (Determining the relative error)
= \(0.2~\times~100 \) (Multiply with 100)
= 20
The percent error is 20%.
Example 2:
Harry measured himself and found he is 5.5 feet tall. However, after careful observation, he realized that his actual height is 5 ft. Calculate the percent error in Harry’s height measurement.
Solution:
The actual height of Harry = 5 ft (This is the actual value)
Height measured = 5.5 ft (This will be the estimated value)
Since, the estimated value is greater than the actual value, so the formula to find the percent error is:
Percent Error = \(\frac{Estimated~value~-~Actual~value}{Actual~value}~\times~100 \)
= \(\frac{5.5~-~5}{5}~\times~100 \) (Substitute the values)
= \(\frac{0.5}{5}~\times~100 \) (Determining relative error)
= \(0.1~\times~100 \) (Multiply with 100)
= 10
Therefore, the percent error in Harry’s height is 10%.
Example 3:
There were 30 students in Emma’s science class yesterday. But, she mistakenly recorded that the total number of students who attended the class was 24. What is Emma’s percent error?
Solution:
The actual number of students attended the class = 30
But, the recorded number of students = 24
Here, the actual value is greater than the estimated value. Hence, the formula for percent error would be,
Percent Error = \(\frac{Actual~value~-~Estimated~value}{Actual~value}~\times~100 \).
= \(\frac{30~-~24}{30}~\times~100 \) (Substitute the values)
= \(\frac{6}{30}~\times~100 \) (Determining relative error)
= \(0.2\times~100 \) (Multiply with 100)
= 20
Hence, Emma’s percent error is 20%.
Example 4:
Jenny went to a grocery store and bought 2 pounds of apples for 50$. But after seeing the bill, she realized that the original cost of the apples was only 45$. Find the absolute and percent error of the apples?
Solution:
The actual cost of the apples = 45 $
Cost paid by Jenny = 50 $
As we can see, the actual cost of the apples is less than the cost paid by Jenny. Then, the formulas for absolute and percent error are:
Absolute error = (Estimated value – Actual value)
= (50 – 45) (Substitute the values)
= 5
Percent error = \(\frac{Estimated~value~-~Actual~value}{Actual~value}~\times~100 \)
= \(\frac{5}{45}~\times~100 \) (Substitute absolute error directly)
= \(0.1111~\times~100 \) (Multiply with 100)
= 11.11
Thus, the absolute error and percent error in the cost of apples are 5$ and 11.11%.
Percent error tells us how closely the measured value matches the actual value.
No, having a high percent error of sixty or seventy percent indicates that the measured value is far away from the actual value. The smaller the percent error, the closer we are to the actual value.
If the estimated error is greater than the actual value, the formula for absolute error is (Estimated value- Actual value). Or else the formula will be (Actual value – Estimated value)