With any measurements, rounding off the results of calculations, performing rather complex calculations, this or that deviation inevitably arises. To assess such inaccuracy, it is customary to use two indicators - these are absolute and relative errors.
If we subtract the result from the exact value of the number, we will get the absolute deviation (moreover, when counting, the smaller number is subtracted from the larger number). For example, if you round 1370 to 1400, then the absolute error will be 1400-1382=18. If you round to 1380, the absolute deviation will be 1382-1380=2. The absolute error formula is:
Δx=|x – x|, here
x - true value, x is an approximation.
However, this indicator alone is clearly not enough to characterize the accuracy. Judge for yourself, if the weight error is 0.2 grams, then when weighing chemicals for microsynthesis it will be a lot, when weighing 200 grams of sausage it is quite normal, and when measuring the weight of a railway car, it may not be noticed at all. Sooften, along with the absolute error, the relative error is also indicated or calculated. The formula for this indicator looks like this:
δx=Δx/|x|.
Let's consider an example. Let the total number of students in the school be 196. Round this number up 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 quantity 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, it is impossible to calculate the exact value of the error. However, it is always possible to define some number that will always be slightly larger than the maximum absolute or relative error.
For example, a salesman is weighing a melon on a pan balance. In this case, the smallest weight is 50 grams. The scales showed 2000 grams. This is an approximate value. 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 of weight measurement does not exceed 50/2000=2.5%.
The value that is initially greater than the absolute error, or in the worst case equal to it, is usually called the limiting absolute error or the limit of the absoluteerrors. In the previous example, this figure is 50 grams. The limiting relative error is determined in a similar way, which in the above example was 2.5%.
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 marginal error.
It happens that the absolute marginal error is not specified. Then it should be considered that it is equal to half the unit of the last specified digit (if it is a number) or the minimum division unit (if it is an instrument). For example, for a millimeter ruler, this parameter is 0.5 mm, and for an approximate number of 3.65, the absolute limit deviation is 0.005.
