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# Error Measurement

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Last updated date: 13th Jul 2024
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Measurement Error is the difference between the approximated value and the actual value of an object/quantity. The relative error can be computed in the percent with the help of relative error formula.

Errors in measuring instruments occur due to inescapable faults in the measuring instrument and limitations of the human eye. Errors come in all sizes, and sometimes we are required to decide if the error is so big that it makes the measurement meaningless. The smaller the error the more we are close to the actual value.

## Types of Errors

There are mainly 3 types of measurement errors:

### A. Absolute Error

The absolute error is the difference between the measured value and the real value. The expression for absolute error is:

Eabsolute=|xmeasured−xaccepted|

For example, if you know a length is 4.635 m + 0.007 m, then 0.007 m is an absolute error.

### B. Relative Error

It represents the ratio of the absolute error of the measurement to the accepted measurement. Relative Error can be represented as;

Erelative = Eabsolute/xaccepted

Relative Error formula =|measured value−actual value| / actual value

### C. Percentage Error

Percentage error is similar to relative error besides that the error here is converted to a percent value. The expression for percent error is

Percent of Error=|measured value−actual value| / actual value∗100 %

### What is the Relative Error Formula?

Percent error informs us about to what extent these inescapable errors affect our experimental outcomes.

The formula used to find percent relative error is:

Percent Relative Error = ∣ Actual Value - Estimated Value/Actual Value∣×100

That being said, Absolute value can also be sometimes referred to as theoretical value or true value. Mostly, the percentage error is represented as a positive value. The value of the absolute error is divided by an accepted value and presented as a percent.

### Relative Error and Percent Error

Relative Error = $\frac{丨measured - real丨}{real}$

Percent Error = $\frac{丨measured - real丨}{real}$ x 100%

### Define Absolute Error

In mathematics, Absolute error is the difference between the measured or the inferred value and the actual value of a quantity. The absolute error is insufficient because of the fact that it does not provide any information with respect to the significance of the error. That said, while measuring distances between cities kilometres apart, an error of some centimetres is trivial and is immaterial.

### What is Absolute Error Formula

Suppose that x is the actual value of a quantity or an instrument and x0 is the measured value of the quantity, then the absolute error value could be computed with the below-stated formula

Δx = x0-x.

Here, Δx is termed as an absolute error.

If we take into account multiple measurements, then the arithmetic mean of absolute error of individual measurements must be the ultimate absolute error.

### How To Find Cumulative Error

cumulative error calculation is done by finding the error of the real equation and multiplying that error by the number of times the error has been repeated. Find out the number of times the error has been made and multiply that by the actual error to calculate your cumulative error.

### Example:

Jessica measured his height and found it to be 5 feet. However, after she carefully measured her height a second time, she found her actual height to be 4.5 feet. What is the percent error Jessica made when she measured the first time?

### Solution:

Before solving the problem, let us identify the information:

Actual value: 4.5 ft and Estimated value: 5 ft

Now,

• Step1: Subtract one value from others in order to obtain the absolute value of error. Error = |4.5−5|=0.5| = 0.5

• Step-2: Then, Divide the error by the actual value, you will get; 0.5/4.5=0.1111(up to four decimal places)

• Step-3: Multiply that outcome by 100 and attach a% sign to express the answer in the form of a percentage. 0.111×100=11.11

Thus: Percentage error = 11.11%

### Example:

Police penalized Alex for speeding up the car. Alex got a traffic penalty notice for travelling 90 mph in a 70 mph zone. Alex insisted his speedometer was 70 mph, not 90 mph. What could Alex profess as his percent error?

Solution:

Let us arrive at % error in the below given three steps:

Absolute Error: |90−70| = 20

Relative percent error: 20/60 = 0.3333

= 0.3333 × 100

=33.33%

Thus: Alex can claim 33.33% as his percent error.

## FAQs on Error Measurement

Q1. What is Meant by Percent Error?

Answer: Percent error is the difference between the actual value and the approximated value compared to the actual value and is expressed in the form of a percentage.

Percent errors imply how huge our errors are when we measure something in an evaluative process. Smaller percent errors suggest that we are close to the accepted or the real value. For example, a 3% error suggests that we got very close to the accepted value, while an error of 46% indicates that we are quite far off from the true value. This is to say, you take the difference between the actual answer and the estimated answer, divide it by the actual answer, and express it in the percentage format.

Q2. What Makes Errors in Measurement Unavoidable?

Answer: Most of the time measurement errors are unavoidable because of certain reasons like shaking hands, imprecise material, or incapability of the instruments of estimating exactly. Percent error formula also notifies us how seriously these inevitable errors impact our results, and thus accuracy of measurement is determined.

Q3. What is the Purpose of Calculating Measurement Error?

Answer: The purpose of calculating the measurement error or percent error is to evaluate how close the measured value is to the original value. It is part of a comprehensive error evaluation. In most of the domains, percent error is always demonstrated as a positive number whereas, in others, it is appropriate to have either a positive or a negative value. The idea of keeping the sign is to identify whether recorded values consistently fall below or above expected values.