We assume that a certain quantity has been independently observed n times with equal precision, and the observed values are l 1, l2, l3, …, ln. Its arithmetic mean is
Architectural engineering survey
We believe that the arithmetic mean is the most reliable value among a group of observations with the same accuracy. Why? It can be proved by the characteristics of accidental error.
Let the observed truth value be x, then the observed true error is
Architectural engineering survey
The sum of the two ends in formula (5-8) is obtained by dividing by n.
Architectural engineering survey
From formula (5-7), we know that x=
, enter the type and move the project, get
Architectural engineering survey
When the number of observations n increases indefinitely, according to the accidental error characteristics, there are
Architectural engineering survey
therefore
Architectural engineering survey
So when n increases infinitely, the arithmetic mean is close to the true value. If n is finite.
It is also a very small quantity, and the arithmetic mean value x is still closer to the real value than the observed value. We call the approximation closest to the true value "the most probable value" (or "the most reliable value").
Second, the observation correction number
The difference between the most likely observed value and the observed value is called "observation correction number". When observing with equal precision, the difference between the arithmetic mean X and the observed value L is the correction number V of the observed value. have
Architectural engineering survey
Add the two ends of the above types to get it.
[V]=nx-[l]
From the formula (5-7), we can see that NX = [L]. Substituting the above formula, we get
[V]=0 (5- 10)
Equation (5- 10) shows an important feature of observation correction, that is, in equal-precision observation, the sum of observation correction is zero, which can be used as a check in calculation. If there is a rounding error in the calculation of the arithmetic mean, the sum of the correction numbers is less than or equal to 0.5n, that is, ∑ V is less than or equal to 0.5n, and n is the number of observed values.
3. Calculate the error of the observed value through the corrected number of the observed value.
In practical work, the true value x of the observed value is often unknown. In the observation with equal precision, generally only the arithmetic mean value X and the correction number V of the observed values are known, so the formula (5-4) cannot be used to calculate the mean error. In this case, the true error can be replaced by v, and the average error of the observed value can be calculated by the following formula.
Architectural engineering survey
The proof of the above formula is as follows:
From (5-8) and (5-9), we can get
Architectural engineering survey
Add the squares at both ends of the above categories.
[δδ]=[VV]+n(X-X)2-2(X-X)[V](b)
Because [v] = 0, (x-X) is the true error of the arithmetic mean. Let δ=(x-x) and substitute it into formula (b)
[δδ]=[VV]+nδ2(c)
Divide both ends by n.
Architectural engineering survey
Add all the items in (a) to get
Architectural engineering survey
Square both ends of formula (e)
Architectural engineering survey
δ1δ 2, δ 2 δ 3, ... are the products of accidental errors. When the number of observations increases indefinitely, these products also have the characteristics of accidental error, so there are
Architectural engineering survey
According to formula (5-4a)
Substitute this formula and formula (g) into (d) to get it.
Architectural engineering survey
After finishing, you get it.
Architectural engineering survey
Complete the certificate.
Fourth, the arithmetic mean error
The average error m of the arithmetic mean x can be calculated by the following formula.
Architectural engineering survey
or
Architectural engineering survey
Simply prove it.
Equation (5- 12) shows that the mean error m of the arithmetic mean is only the mean error m of any observation in the group.
In other words, its accuracy is improved. It can be seen that the accuracy can be improved by increasing the observation times of a quantity and taking its average value. However, when the number of increases is large, not only the workload is large, but also the increase of accuracy tends to be slow. For example, when n= 16, the accuracy is 1/4 times of the observation error. When n=36, the number of observations is 20 times higher than when n= 16, but the accuracy is only 2 times higher than the former. Therefore, when high precision is required, more sophisticated instruments and improved observation methods should be considered if possible.
Example 5- 1 has a certain distance. Under the same observation conditions, measure with 30m steel ruler for 4 times, and the results are shown in the second column of Table 5-2. Find the most possible value of this distance and its error.
Table 5-2
Solution: In order to eliminate the system error, the correction length is obtained by adding the correction numbers of scale length, temperature and inclination. The corrected length mainly contains accidental errors. Because it is an observation with equal precision, the arithmetic average is taken as the most possible value, and it is obtained.
Architectural engineering survey
See Table 5-2 for the calculation of correction number and average error of observed values. M = 5.8 mm, which is the average error of any observed value; M = 2.9 mm, which is the average error of arithmetic mean. The end result is
X = 89.574m m 2.9mm.
The relative average error is
Architectural engineering survey
Example 5-2 Using the same theodolite, a horizontal angle is observed by back measurement method, and back measurement is carried out * * * five times. The results are shown in Table 5-3. Find the most probable value of horizontal angle and its error.
Table 5-3
Solution: Because it is an observation with equal precision, its arithmetic mean is the most possible value. In order to make the calculation simple, take the initial value x0 = 64 2 1' 00 ",then
Architectural engineering survey
See Table 5-3 for the calculation of correction number and average error of observation value. The mean error of the observed values in the first survey is mm =19.5 ",and the mean error of the arithmetic mean is m = 8.7. So the final result is x = 6421'06 "8.7". Because the angle observation error has nothing to do with the angle, it is not necessary to calculate the relative median error.