Absolute and relative error

At any measurements, rounding of the results of calculations, the performance of sufficiently complex calculations inevitably there is one or another deviation. To assess this inaccuracy, it is customary to use two indicators: absolute and relative error.

If we subtract the result from the exact value of the number, then we get an absolute deviation (and in calculating from a larger number , we subtract less). For example, if you round 1370 to 1400, the absolute error will be 1400-1382 = 18. When rounded to 1380, the absolute deviation will be 1382-1380 = 2. The formula for the absolute error is:

Δx = | x * - x |, here

X * is the true value,

X is an approximate value.

However, it is not enough to characterize the accuracy of this indicator alone. Judge for yourself, if the error in weight is 0.2 grams, then when weighing chemical reagents for microsynthesis it will be very much, when weighing 200 grams of sausage is quite normal, and when measuring the weight of a railway car, it may not be noticed at all. Therefore, often together with absolute, the relative error is indicated or calculated. The formula for this indicator is as follows:

Δx = Δx / | x * |.

Let's consider an example. Let the total number of pupils in the school be 196. We round this value to 200.

The absolute deviation will be 200 - 196 = 4. The relative error is 4/196 or rounded, 4/196 = 2%.

Thus, if the true value of a certain quantity is known, the relative error of the approximate value adopted is the ratio of the absolute deviation of the approximate value to the exact value. However, in most cases it is very problematic to reveal the true exact value, and sometimes it is completely impossible. And, therefore, it is impossible to calculate the exact value of the error. 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, the seller weighs a melon on the balance pan. 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 can not be more than 50 grams. Then the relative error in measuring the 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 marginal absolute error or the boundary of the absolute error. In the previous example, this figure is 50 grams. Similarly, the limiting relative error is determined, which in the example considered above was 2.5%.

The value of the limit error is not strictly specified. So, instead of 50 grams, we could easily 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 determined accurately, then it will simultaneously serve as the marginal error.

It happens that the absolute marginal error is not specified. Then it should be assumed that it is equal to half the unit of the last indicated digit (if this number) or the minimum division unit (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 limit deviation is 0.005.