The percentage error it is the manifestation of a relative error in percentage terms. In other words, it is a numerical error expressed by the value that yields a relative error, subsequently multiplied by 100.
To understand what is a percentage error, it is first essential to understand what is a numerical error, an absolute error, and a relative error, since the percentage error is derived from these two terms.
A numerical error is one that appears when a measurement is taken in an equivocal way when using a device (direct measurement), or when a mathematical formula is applied incorrectly (indirect measurement).
All numerical errors can be expressed in absolute or percentage terms. For its part, the absolute error is that which is derived when making an approximation to represent a mathematical quantity resulting from the measurement of an element or from the erroneous application of a formula.
In this way, the exact mathematical value is altered by the approximation. The calculation of the absolute error is done by subtracting the approximation from the exact mathematical value, like this:
Absolute Error = Exact Result - Approximation.
The units of measurement used to express the relative error are the same as those used to speak of the numerical error. Similarly, this error can give a positive or negative value.
The relative error is the quotient obtained by dividing the absolute error by the exact mathematical value.
In this way, the percentage error is the one obtained by multiplying the result of the relative error by 100. In other words, the percentage error is the expression in percentage (%) of the relative error.
Relative Error = (Absolute Error / Exact Result)
A percentage value that can be negative or positive, that is, it can be a value represented by excess or by default. This value, unlike the absolute error, does not present units, beyond the percentage (%).
Relative Error = (Absolute Error / Exact Result) x 100%
The mission of the relative and percentage errors is to indicate the quality of something, or to provide a comparative value.
When measuring two lots or plots, it is said that there is approximately 1 m of error in the measurement. One plot is 300 meters and another is 2000.
In this case, the relative error of the first measurement will be greater than that of the second, since in proportion 1 m represents a higher percentage in this case..
300 m lot:
Ep = (1/300) x 100%
Ep = 0.33%
2000 m lot:
Ep = (1/2000) x 100%
Ep = 0.05%
An aluminum block is delivered in a laboratory. By measuring the dimensions of the block and calculating its mass and volume, the density of the block is determined (2.68 g / cm3).
However, when reviewing the number table for the material, it indicates that the density of aluminum is 2.7 g / cm3. In this way, the absolute and percentage error would be calculated as follows:
Ea = 2.7 - 2.68
Ea = 0.02 g / cm3.
Ep = (0.02 / 2.7) x 100%
Ep = 0.74%
It was assumed that 1,000,000 people would go to a certain event. However, the exact number of people who attended the event was 88,000. The absolute and percentage error would be the following:
Ea = 1,000,000 - 88,000
Ea = 912,000
Ep = (912,000 / 1,000,000) x 100
Ep = 91.2%
The estimated time it should take for a ball to reach the ground after being thrown from a distance of 4 meters, is 3 seconds.
However, at the time of experimentation, it is found that it took 2.1 seconds for the ball to reach the ground.
Ea = 3 - 2.1
Ea = 0.9 seconds
Ep = (0.9 / 2.1) x 100
Ep = 42.8%
It is approximated that if a car goes 60 km, it will reach its destination in 1 hour. However, in real life, the car took 1.2 hours to reach its destination. The percentage error of this time calculation would be expressed as follows:
Ea = 1 - 1.2
Ea = -0.2
Ep = (-0.2 / 1.2) x 100
Ep = -16%
Any length is measured by a value of 30 cm. When verifying the measurement of this length, it is evident that there was an error of 0.2 cm. The percentage error in this case would be manifested as follows:
Ep = (0.2 / 30) x 100
Ep = 0.67%
The calculation of the length of a bridge according to its plans is 100 m. However, when confirming this length once it is built, it is evident that it is actually 99.8 m long. The percentage error would be evidenced in this way.
Ea = 100 - 99.8
Ea = 0.2 m
Ep = (0.2 / 99.8) x 100
Ep = 0.2%
The head of a standard manufactured screw is given to be 1 cm in diameter.
However, when measuring this diameter, it is observed that the screw head is actually 0.85 cm. The percentage error would be the following:
Ea = 1 - 0.85
Ea = 0.15 cm
Ep = (0.15 / 0.85) x 100
Ep = 17.64%
According to its volume and materials, the weight of a given object is calculated to be 30 kilos. Once the object is analyzed, it is observed that its real weight is 32 kilos.
In this case, the value of the percentage error is described as follows:
Ea = 30 - 32
Ea = -2 kilos
Ep = (2/32) x 100
Ep = 6.25%
In a laboratory a steel sheet is studied. By measuring the dimensions of the sheet and calculating its mass and volume, the density of the sheet is determined (3.51 g / cm3).
However, when reviewing the number table of the material, it indicates that the density of the steel is 2.85 g / cm3. In this way, the absolute and percentage error would be calculated as follows:
Ea = 3.51 - 2.85
Ea = 0.66 g / cm3.
Ep = (0.66 / 2.85) x 100%
Ep = 23.15%
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