For any measurements, rounding off the results of calculations, and performing rather complicated calculations, one or another deviation inevitably arises. To assess such inaccuracy, it is customary to use two indicators - this is the absolute and relative error.
If we subtract the result from the exact value of the number, then we get the absolute deviation (moreover, when counting from the
larger number, the smaller is subtracted). For example, if you round 1370 to 1400, then the absolute error will be equal to 1400-1382 = 18. When rounding to 1380, the absolute deviation will be 1382-1380 = 2. The formula for the absolute error has the form:
Δx = | x * - x |, here
x * is the true value,
x is an approximate value.
However, to characterize the accuracy of this indicator alone is clearly not enough. Judge for yourself if the weight error is 0.2 grams, then when weighing chemicals for microsynthesis, this will be very much, when weighing 200 grams of sausage, it is quite normal, and when measuring the weight of a railway carriage, it may not be noticed at all. Therefore, relative error is often indicated or calculated along with the absolute. The formula for this indicator is as follows:
δx = Δx / | x * |.
Consider an example. Let the total number of students in the school be 196. Round this value to 200.
The absolute deviation will be 200 - 196 = 4. The relative error will be 4/196 or rounded, 4/196 = 2%.
Thus, if the true value of a certain value is known, then the relative error of the accepted approximate value is the ratio of the absolute deviation of the approximate value to the exact value. However, in most cases, revealing the true exact value is very problematic, and sometimes even impossible. And therefore, the exact value of the error cannot be calculated . Nevertheless, it is always possible to determine a certain number, which will always be slightly larger than the maximum absolute or relative error.
For example, a seller weighs melon on a cup scale. In this case, the smallest weight is 50 grams. Scales showed 2000 grams. This is an estimate. The exact weight of the melon is unknown. However, we know that the absolute error cannot be more than 50 grams. Then the relative error in measuring weight does not exceed 50/2000 = 2.5%.
A value that is initially greater than the absolute error or, in the worst case, equal to it, is usually called the ultimate absolute error or the boundary of the absolute error. In the previous example, this figure is 50 grams. The marginal relative error, which was 2.5% in the above example, is determined in a similar way.
The value of the marginal error is not strictly specified. So, instead of 50 grams, we could well take any number greater than the weight of the smallest weight, say 100 g or 150 g. However, in practice, the minimum value is chosen. And if it can be accurately determined, then it will simultaneously serve as the ultimate error.
It happens that the absolute marginal error is not indicated. Then it should be assumed that it is equal to half the unit of the last indicated discharge (if it is a number) or the minimum unit of division (if the tool). For example, for a millimeter ruler, this parameter is 0.5 mm, and for an approximate number of 3.65, the absolute maximum deviation is 0.005.