Indirect measurement error estimation rules

INDIRECT MEASUREMENT ERRORS evaluation rules can be obtained only by means of the errors mathematical theory and differential calculus methods. The error of indirect measurement depends on the velocity of function change while the argument varies i.e.

, ,

where y and x are absolute indirect and direct measurement errors.

Therefore, if the function is presented as

,

then the measurement error of quantity y is defined by the expression

,

where ,… are partial derivatives of function in respect to independent arguments.

Fractional error is defined as follows

.

We get from the previous expressions

.

According to differentiation rules, we have

; … .

Thus, we obtain expression for fractional error of indirect measurement. It looks as follows:

.

Let’s apply this expression to the case where the required quantity is connected with the results of direct measurements by dependence occurring in physics.

, where A and k are constant values, which can be integer or fraction, negative or positive. Take the logarithm of the function.

.

Calculate partial derivatives using the table of derivatives

; ;

Using the expression for a fractional error, one can obtain the formula for fractional error calculation of the function.

.

Laboratory work 1.1

Physical measurement errors calculation and a solid body density determination

The purpose of the work: to study measurement and error types, errors calculation technique; to determine a body density experimentally.

Theoretical information

The main purpose of each laboratory work is to measure some physical value employing special technical means. The result of measurement shows how the measured value differs from those accepted for a unit of measurement.

As an example, we consider the expression for determination of a solid body density of the cylindrical shape:

,

where m, d and h are mass, diameter and height of the cylinder respectively.

Work procedure and data processing

To carry out this work, students should use the data given in table 5.1. The variant number is chosen in compliance with the teacher’s directive. The data given in the table are a cylinder height (has been measured with the help of slide-clipper) and diameter (has been measured with the help of micrometer). Measurement of each parameter has been repeated five times.

Table 5.1

Var. No.	Exp. No.	Height h, mm	Diam. d, mm	Mass m, g	Var. No.	Exp. No.	Height h, mm	Diam. d, mm	Mass m, g
1	1	20.1	40.12	26.1	6	1	29.5	20.04	70.6
	2	20.3	40.03	25.9		2	29.3	20.08	70.4
	3	19.9	39.98	26.4		3	29.9	20.10	70.2
	4	20.5	40.09	26.5		4	29.1	20.12	70.9
	5	19.8	40.04	25.7		5	29.6	20.05	70.3
2	1	40.1	40.17	127.1	7	1	25.5	40.11	250.5
	2	40.7	40.09	125.9		2	25.3	40.17	250.3
	3	39.8	39.98	125.4		3	25.9	39.18	250.9
	4	40.1	40.06	125.5		4	25.1	40.14	250.1
	5	39.9	40.04	126.7		5	25.8	39.19	250.8
3	1	50.6	20.04	137.1	8	1	26.1	20.31	24.3
	2	50.4	20.08	135.9		2	25.9	20.33	24.8
	3	50.2	20.10	135.4		3	26.4	19.39	24.1
	4	50.9	20.12	135.5		4	26.5	20.35	24.9
	5	50.3	20.05	136.7		5	25.7	19.38	24.5
4	1	24.3	60.04	23.5	9	1	38.5	43.12	324.3
	2	24.8	60.08	23.3		2	38.3	43.03	324.8
	3	24.1	60.11	23.9		3	38.8	42.98	324.1
	4	24.9	60.02	23.1		4	38.2	43.09	324.9
	5	24.5	60.05	23.8		5	38.9	43.04	324.5
5	1	10.6	20.11	32.3	10	1	59.5	20.04	15.2
	2	10.4	20.17	32.8		2	59.3	20.08	15.4
	3	10.2	19.18	32.1		3	59.9	20.11	15.7
	4	10.9	20.14	32.9		4	59.1	20.02	15.1
	5	10.3	19.19	32.5		5	59.6	20.05	15.9

The student should do the following:

1. Enter the data of his/her variant into table 5.2.

2. Calculate the average values, random deviation, root-mean-square error, random and total error of diameter, height, and mass using formulas (5.1…5.6). Apply the following values of instrumental error: d_in= 0.01 mm; h_in= 0.05 mm; m_in= 0.1 g.

3. Calculate fractional and absolute errors of a solid body density measurement according to the rules of an indirect measurement error determination. Therefore to calculate an indirect measurement error of a solid body density, we use the expression

.

Take the logarithm of the function.

.

Calculate partial derivatives using the table of derivatives

; ;

Finally, one can obtain the formula for fractional error of the density.

or

where Δρ, Δm, Δd, Δh are absolute measurement errors of the cylinder mass, diameter and height; are average values of the cylinder density, mass, diameter and height.

Knowing the fractional error, get the absolute error of the density indirect measurement:

.

The final result of measurement is expressed as follows:

.