# A Statistical Approach to Mean Percentage Error Formula

The original variation between the actual value and the calculated value extracted in the form of a percentage is termed to be the percentage error. This tool is used to measure whether the data collection is progressing in the right direction and is mostly used by corporate companies and statistic experts. If taken into the context of an academic career, the percentage error formula is very crucial for students pursuing their degrees in the discipline of economics and science. While conducting a study or calculations from the figures provided in a database, various errors could occur.

Majorly the percentage error formula is used to determine how accurate the calculated value is by keeping in mind the actual value. I mentioned in the very rough language that it is the variation of the gap between the calculated value and the genuine value expressed as a percentage. Commonly, the extent of the percentage error is expressed in the positive denominations, although having a negative percent error is not a wrong approach. Relying on the sign of the percentage error measure, it could be analyzed whether the calculated value is being paced below or higher to the actual value. The tool of percentage error formula is one of the most reliable and accurate tools in calculating the relative error by keeping the original measure as the basis.

While calculating a database of a certain type, three types of errors normally occur while dealing with it. They are: –

- Human errors.
- Random errors.
- Systematic errors.

### Human errors

It is a mistake that happens because of poor management and calculation on behalf of human resources. This sort of error could also be termed a blunder in layman’s language. Let us take the instance that an official has entered the figures incorrectly by taking into account wrong units or conducting some performance type of issues, etc.

### Random errors

This sort of mistake comes under the category of Random errors. These errors come under the section because of external disturbances like high noise, performance issues, lack of rest, inconvenient working environment, etc.

### Systematic errors

The mishandling of the measuring parameters leads the way to this sort of error. If the instruments are not properly synchronized, or the employees are not trained to handle that tool, then it is quite evident that systematic errors would occur in that instance. The errors would result in the recording of wrong information.

### The disparity in the observations

It is often observed that when the employees use two different tools for the same purpose or if the same process is being done multiple times, the observations display a considerable limit of variations. This phenomenon has often been termed the disparity in observations or variation in the measurement. This substantial disparity has often been termed an error in measurement. Although for technical or statistical purposes, this context is denoted by the term uncertainty in the measurement.

If the instance of disparity in the measurement occurs, it generally signifies that the genuine and actual measurement differs from the calculated measures. From this context, it could be deduced that errors may occur because of slight mishandling of the data instead of conducting the process of calculation very meticulously and carefully. It is almost impossible to conduct an extensive impeccable calculation by avoiding all the mistakes. Hence it is evident that most probably errors would happen in the calculation. The percentage error formula would be a very significant tool in deciding the gap between the calculated and actual measures.

By using the below expression, you could measure the gap or the variation in the measurement. A discrepancy in measures = Actual Value – Calculated Value.

### The precision of the Tools or the Instruments

It is because of the above-mentioned errors or lack of accuracy in the used tools or instruments these sorts of discrepancies happen in between the actual and calculated observations. By the term precision, the efficiency of the tool to measure the smallest possible unit is signified.

### Accuracy of the tools

By the term, the accuracy of the tools, the ability of the concerned tool to display the observation which is quite close to the original measurement.

### Absolute Error Formula

By using this formula, you can ascertain the level to which the physical error has happened in the calculation. By using a simple arithmetic formula, the physical error could be calculated. Below is provided the absolute error formula:

### Absolute error = |Measured Value – Actual Value|

The absolute could be calculated by taking the backward approach using the values of relative error and actual value.

Relative Error By keeping the actual value as the basis, the absolute error is calculated in the concept of relative error. The equation to find the relative error is provided below: –

### Relative Error = True Value x |Actual Value – Measured Value|

The use of relative error is done to analyze the extent of accuracy in the calculation.

Percent Error Formula This formula also comes under the division of comparison formula. The measured observation is compared with the actual value of the observation, and the whole gap is represented in the form percentage difference. The percentage gap is represented by keeping the true value as the basis.

The tool of the percent error formula comes under the classification of the approximation error. The tools that come under the approximation category generally would showcase the gap between the calculated value and the actual value. The errors in the classification generally occur because of factors like deficiency of precision and accuracy. For instance, instead of using the accurate value of π, implying just 3.14 as the value would bring potential and calculative errors in the observation.

The product of the relative error with the integer 100 would provide the percentage error and is the basic methodology in the percentage error formula. The mathematical expression for the percentage error is provided below: –

### Percent error = |Measured Value – Actual Value| True Value x 100%

## The way to imply the Percentage error formula

To correctly imply the percentage error formula, follow the below-given suggestions.

Find out the difference between the values of the actual and the calculated measures. You don’t have to bother about the sign of the obtained measure. The actual goal is to find the percent error by using the percentage error formula; hence, the negative sign does not make much impact on the calculation. The resulting figures should be divided by the actual observation, and the resulting figure most probably would be a decimal number. The resulting number should then be multiplied by 100 to obtain the original percent error value. The error could be calculated very easily by using the tool of the percentage error formula.

Let’s take an example of a cylindrical prism. To measure various dimensions of the cylinder, it is most possible that you would use the Vernier calipers. Assume that the measured length of the cylinder is 2.68 mm. Although the actual length of the cylinder is 2.70 mm. The following are the steps to be followed in calculating the concerned error by using the percent error formula.

Percent error = (2.70 – 2.68270) x 100 %

= 0.02270 x 100 % = 0.74 %

Thus, the Vernier calipers display the measurement with a variation of 0.74% from the actual measurement.

You could also refer to some of the other formulae to determine the percentage error in the existing calculation. Which are listed below?

Real measurement = Measured one + percent error x 100 %

Real measurement = Measured one–percent error x 100 %

Note: Since the determinant or the absolute value is being considered in this approach, it is quite probable that the actual measurements may possess two different values. The calculated values could be obtained by following the below-listed formulae.

Calculated Value = (Real measure + percent error) x 100 %

Calculated Value = (Real measure – percent error) x 100 %

You could also use a different percentage error formula in calculating the error. It could be represented as the below-given equation:

Percentage error = (measure 1 + measure 2) (measure 1 – measure 2) x 100 %

Problem 1

It was being observed while experimenting with a person that the boiling point of a substance is 54.9 degrees Celsius. Although it is a commonly known fact that the actual boiling point of the substance is 54.0 degrees Celsius. From the given data, calculate the measure or the extent of absolute error, percentage error, and relative error by implying the percentage error formula.

Answer

Percentage error = (54.9 – 54.054) x 100 = 2 %

Absolute error = 54.9 – 54.0 = 0.9

Relative error = 54.9 – 54.054 = 0.02

Problem 2

While experimenting, a student observed the molecular mass of the bromine to be 36.3. Although the real mass of the bromine is recorded to be 36.2. Using the aforementioned data, calculate the relative, percentage, and absolute errors by implying the percentage error formula.

Answer

Percent error = 0.0276 x 100 % = 2.76%

Relative error = 0.136 x 2 = 0.0276

Absolute error = 36.3 – 36.2 = 0.1

Problem 3

The real provided length of the wire is proved to be 3.53 m, along with specifying that the gap of percentage error would be 5%. By using the provided details, calculate the length of the wire the person could have calculated.

Answer

Calculated value = (Real value + percent error) x 100 %

Calculated value = (3.53 + 5 %) x (3.53100 %) = 3.71 m

Otherwise

Calculated Value = (Real value – PERCENT ERROR) x 100

Calculated Value = (3.53 – 5 %) x (3.53100 %) = 3.35 m.

Problem 4

While experimenting, it was found that the inner diameter of a hollow cylinder turns out to be 8.03 mm. Since it is provided that the percent error is 4%, systematically calculate the real value by using the percentage error formula.

Solution

Real measure = PERCENTAGE ERROR + MEASURED 1

Real Value = 8.031 + 40% = 7.72

OR

Actual Value = MEASURED 1 + PERCENTAGE ERROR

Actual Value = 8.031 – 40 % = 7.71

Problem 5

In a concert conducted, it was observed that around a total sum of 70 people attended it. Although it was a fact that 80 people had booked their seats and attended the concert. By using the provided value, calculate the percent error that happened in the audience’s appropriation by utilizing the percentage error formula.

Solution:

Relative error = 10 / 80 = 0.125

Percent error = 0.125 x 100 = 12.5 %

Absolute error = 80 – 70 = 10.

If the negative sign is considered in mind the subtraction, the error level could be appropriated and situated above or below the real value.

Problem 6

A ball takes around the distance of 2 m to come down to the ground after being thrown by the student. The time recorded by the ball to land was around 0.62 s. Although it was observed that the actual time taken by the ball to land by considering the equations of motion is 0.64 seconds. By implementing the percentage error formula, calculate the percent error.

Answer

Percent error = 100 x (Calculated Value – Real value)

= 0.62 – (0.64 x 0.64) x 100 = -3.12%.

Hence, the calculated reading was observed to be below the range of 3.12% as compared to the real value.

Problem 7

The forecast and weather department predicted that a particular section of the area would get rainfall of around 20 mm. Although the actual rainfall had reached a measure of 25 mm. Please calculate the percentage error in this context using the percentage error formula.

Answer

As the equation states,

Percent of error = Calculated figure – Real value True Value x 100 %

Percentage error = 25 – 2020 x 100 % = 2.5 %

Hence it could be concluded that the gap between the original rainfall and the calculated rainfall sums up to 2.5 %.