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# Lecture 12. Theme: Repeated measurement on an order scale. Bases of the theory of .. 413

**Theme: Repeated measurement on an order scale. Bases of the theory of selective statistical control.**

Repeated measurement on the graduated scales. Repeated measurement with ravnotochny values of counting. Repeated measurement with neravnotochny values of counting. Processing of several series of measurements.

Plan:

1. Repeated measurement on an order scale

2. Bases of the theory of selective statistical control

3. Repeated measurement on the graduated scales

4. Repeated measurement with ravnotochny values of counting

5. Repeated measurement with neravnotochny values of counting

6. Processing of several series of measurements

*Repeated measurement with ravnotochny values of counting*

Fundamental idea of repeated measurement. Sequence of performance of repeated measurement on the graduated scales of intervals and the relations. Formation of the massif of experimental data. Amending. Exception of mistakes. Promotion and check of hypotheses about the law of distribution of probability of result of measurement. The solution of a return task at various laws of distribution of probability of result of measurement. Ensuring demanded accuracy of measurements.

*Repeated measurement with neravnotochny values of counting*

Average weighed. Dispersion of an average weighed. Solution of a return task.

*Processing of results of several series of measurements*

Homogeneous and non-uniform series of measurements. Check of a normality of results of measurements in each series. Check of the importance of distinction between average arithmetic values of result of measurement in two series. Check of a ravnorasseyannost of results of measurements in two series. Processing of results of homogeneous and non-uniform series of measurements.

Repeated ravnotochny measurements

Need for repeated supervision of some physical size arises at existence in the course of measurements of considerable casual errors.

Direct repeated measurements share on ravnotochny and neravnotochny measurements. Measurements which are led by gages of identical accuracy by the same technique under invariable external conditions are called as Ravnotochnymi. At ravnotochny measurements of SKO of results of all ranks of measurements are equal among themselves.

The problem of processing of results of measurements consists in finding of an assessment of the measured size and a confidential interval in which there is its true value. Processing should be carried out according to GOST 8.207-76 «GSI.

Direct measurements with repeated supervision. Methods of processing of results of supervision. General provisions».Initial information is a row from n (n> 4) results of measurements

X1, the X-th 2, x1 x2 from which known systematic errors - sample are excluded. The number of "n" depends on requirements to accuracy of received result and from real possibility to carry out repeated measurements.

The sequence of processing of results of direct repeated measurements consists of a number of stages:

I stage: Definition of dot estimates of the law of distribution of results of measurements.

Are defined:

- average arithmetic value x of the measured size;

- SKO of result of measurement of Sx;

- SKO of average arithmetic value S*.*

According to criteria misses are excluded, then repeated calculation of estimates of average arithmetic and its SKO is carried out. For more reliable calculation can other dot estimates will be defined: factor of an assimetriya of v, эксцесc ε and its counterexcess To;

П stage: Definition of the law of distribution of results of measurements or casual errors of measurements.

From sample of results of measurements *х1.х2 хз..., x„ *pass to sample of deviations from average arithmetic *Akh1, Akh2, Akh3.... Akhp* where *Ah, *= *x, *- *x.*

At identification of the law of distribution determine construction by the corrected results of measurements х1*, *where *by i=I, 2..., n, *a variation row (the ordered sample), and also Vi where *y1=min (x1* and *yn=max (*In a variation row results of measurements or their deviation have *x1* in ascending order; a row break into optimum number *of t *of identical intervals of grouping in length *of h = (y1 + unitary enterprise)/t. Leah *of practical application it is expedient to use *mmin* expression = *0,55п04* and *ттах* = *1,25п ° *received for often meeting distributions with an excess in limits from 1,8 to 6, those from uniform Laplas before distribution. Required value *of t *in an interval *ттin* and *ттах* should be odd as at even value *of t *the middle of a curve of distribution it is artificial уплощается.

Define intervals of grouping of experimental data in a look

^{}^{ }also count up number of hits of n, (frequency) of results of measurements in each interval of grouping. The sum of these numbers should equal to number of measurements. On the received values count probabilities of hit of results of measurement to frequency in each of grouping intervals on a formula:

*рk _{= }n) / п, *where

*к=1,2 … t.*

These calculations allow to construct the histogram, the range and a cumulative curve.

Drawing 7.1 - the Histogram, the range (and) and a cumulative curve

For creation of the histogram on an axis of results of supervision *x *∆t intervals in ascending order numbers are postponed and on each interval is under construction *in p1* rectangle in height*. *The area concluded under the schedule, is proportional to number of supervision of "n" Sometimes height of a rectangle postpone equal empirical density of probability *р1 = pt/∆k* = n k/(∆n k) which is an assessment of average density of n “n” interval In this case the area under the histogram is equal 1. At increase in number of intervals and respectively reduction of their length the histogram more and more comes nearer to a smooth curve - to graphics of density of distribution of probability.

The range represents the broken curve connecting the middle of the top bases of each column of the histogram; it reflects a form of a curve of distribution. Outside of the histogram on the right and at the left there are empty intervals in which the points corresponding to their middle, lie on an axis of abscissae. These points at creation of the range connect among themselves pieces; as a result together with an axis *x *the closed figure, which area according to rationing rules is formed, should be equal to 1 (unit) (or to number of supervision when using by frequency).

The cumulative curve is a schedule of statistical function of distribution. For its construction on an axis of results of supervision *x *postpone intervals Д4 in ascending order numbers and on each interval build a rectangle

Fk value is called as a cumulative particular, and n sum - cumulative frequency. By the form constructed dependences the law of distribution of results of measurements can be estimated.

III stage: An assessment of the law of distribution by statistical criteria.

At number of supervision of n> 50 for identification of the law of distribution Pearson is used criteria!' - *х2* (хи - a square) or Mizesa-Smirnov's criterion (ω2). At *50> п> 15 *the compound criterion is applied *to *check of a normality of the law of distribution (by d - criterion), given in Gost8.207-76. At *п <the 15th *belonging of experimental distribution to the normal is not checked.

IV stage: Definition of confidential borders of a casual error.

If it was possible to define the law of distribution of results of measurements, with its use find a kvantilny multiplier of Zp at a preset value of confidential probability of R; in this case confidential borders of a casual error д = ± z p Sk

V stage: Delimitation of not excluded systematic error *in *results of measurements.

Borders of not excluded systematic error are accepted equal to limits of allowed main and additional errors of measuring instruments if their casual components are negligible. The confidential probability at delimitation *0 *is accepted the equal confidential probability used at finding of borders of a casual error.

**Control questions:**

1. According to what state standard specification processing of results of measurements is carried out?

2. Ravnotochnye and neravnotochny measurements.

3. Call sequence of stages of processing of results of measurements.

4. What dot estimates are applied to more reliable calculation?

5. How intervals of grouping of experimental data are defined?

6. How the probability of hit of results of measurements (frequency) in grouping intervals pays off?

7. How the histogram, the range and a cumulative curve on the basis of the carried-out calculations is under construction?

8. How confidential borders of a casual error are defined?

9. Delimitation of not excluded systematic error.