1、UDC 519.2 :311.1/.2 :001.4 DEUTSCHE NORM March 1985 I I Statistical interpretation of data Countable (discrete) characteristics - DIN 53 804 Part 2 Statistische Auswertungen; zhlbare (diskrete) Merkmale In keeping with current practice in standards published by the international Organization for Sta
2、ndardization (/SO), a comma has been used throughout as the decimal marker. Contents Page 1 Scope and field of application . 1 2 Concepts . 1 3 Poisson distribution . 2 4 Characteristics of a sample consisting of n count values 2 5 Graphical representation of count values 2 6 Estimated value and con
3、fidence interval for the expectation of a Poisson distribution 3 6.1 With a single count . 3 6.2 With n counts 4 6.3 Conversion to a different counted unit 5 1 The properties of products and activities are differen- tiated by characteristics. Values of a suitable scale will be allocated to the value
4、s of a characteristic. The scale values are - any real numbers (as values of physical quantities) Scope and field of application when measurable (continuous) characteristics are concerned; - whole numbers (integers) when countable (discrete) characteristics are concerned; - property categories which
5、 follow a ranking (e.g. smooth, somewhat creased, heavily creased) when ordinal characteristics are concerned; - attributes (e.g. presenthot present or red/yellow/blue) when attribute characteristics are concerned. Measureable or countable characteristics are designated as quantitative, whilst ordin
6、al characteristics and attribute characteristics are designated as qualitative (assessable). These types of characteristic correspond to the fundamental concepts in metrology: measuring, counting, sorting and classifying (see DIN 1319 Part 1 ). It is generally not reasonable to determine characteris
7、tic values from all the units of a population and therefore samples are taken and the characteristic values of the samples determined. Parameters of the probability distribution, which describe the behaviour of the characteristic in the population, are estimated from the characteristic values of the
8、 sample. These estimated values are subject to a definable uncer- tainty. Hypotheses concerned with a population investi- gated by way of a sample can be checked by means of statistical tests. Page 7 Testing of the expectation of a Poisson distribution . 5 7.1 Comparison of the expectation with a sp
9、ecified value 5 7.2 Comparison of two values of expectation 5 Appendix A: Examples from textile technology . 7 Appendix B: Key to symbols used 10 Standards and other documents referred to 11 Other relevant documents . 11 Explanatory notes 11 This standard describes statistical methods allowing chara
10、cteristic values to be processed and parameters of the underlying probability distribution to be estimated or tested. The statistical methods are governed by the kind of scale used. This series of standards therefore is issued in four Parts, DIN 53 804 Part 1 dealing with measurable characteristics,
11、 Part 3 with ordinal characteristics and Part 4 with attribute characteristics. This standard covers countable characteristics. It describes statistical methods, with which count values (number of events, e.g. accidents, thread breaks) can be processed and the parameters of the probability distribut
12、ion, in this case the parameters of the Poisson distribution, can be estimated and tested. 2 Concepts The statistical concepts used in this standard are to be found in Standards DIN 13 303 Part 1 and Part 2 and DIN 55 350 Part 12, Part 14 (at present at the stage of draft), Part 21, Part 22, Part 23
13、 and Part 24. In addition to these, the following concepts are used. Counted unit The counted unit (observed section) is the unit of observation in which the occurrence of particular events is being counted. The counted unit (or counted units if several are being counted) forms the sample obtained f
14、rom the population. Countable Characteristic The countable characteristic is the number of events in one counted unit. Continued on pages 2 to 11 Beuth Verlag GmbH. Berlin 30. has exclusive sale rights for German Standards (DIN-Normen) DIN 53 804 Part 2 Engl. Price group Sales No. O109 04.86 Page 2
15、DIN 53 804 Part 2 One hour Count value The count value xi is a particular value of the countable characteristic. Note. Where measurable characteristics are concerned (see DIN 53 804 Part 11, the individual value xi corresponds to the count value xi. Count values ranked by size are designated X(U. In
16、 the case of the countable (discrete) characteristics referred to in this standard, the values of the charac- teristics are represented in a discrete scale 6. Number of customers using the counter in one hour Note. Uncertainties with regard to the delimitation of the counted unit can affect the resu
17、lt of the count; quantitative determination of these uncertainties does not however form part of the subject matter of this standard. One minute 24 hours 1000 rn of cable 3 Poisson distribution It is assumed in this standard, for calculating confidence intervals and testing hypotheses, that the coun
18、table characteristic follows a Poisson distribution ). The probability function of the Poisson distribution shows the probability that the event will occur zero times, once, twice, . . ., m times,. . . in one counted unit. The only parameter of the Poisson distribution is the expectation p; this val
19、ue shows how frequently the event occurs on average in the counting interval concerned. The expected value p is proportional to the size of the counted unit, .e. if the counted is increased u times, the expected value will be a Xp. The variance and the expectation of the Poisson distribu- tion are a
20、lways equal. u* =p (1) This relationship can be used for testing for a Poisson distribution. For statistical tests for a Poisson distribu- tion, see i. Number of electrons emitted from a heated cathode in one minute Number of vehicles passing a toll checkpoint in 24 hours Number of insulation faults
21、 per 1000 m of cable Examples showing counted units and countable charac- teristics Value of the countable characteristic Population Number of equal count values Counted unit O 1 2 3 4 5 6 7 8 13 7 5 3 4 2 O 1 O Countable Number of accidents per year One year Interval from 1960 to 1980 Opening times
22、 of a Post Off ice counter The first two hours of the switched-on time Year 1980 Cable pro- duction in the month of May 1981 4 Mean : Characteristics of a sample consisting of n count values %=-I: 1 xi ni31 (2) See DIN 53 804 Part 1 with regard to other characteristic values. 5 Graphical representat
23、ion of count values It may be advisable to classify the R count values into groups of the same numerical value. For this purpose, 100000m 1 Number of thread breaks of yarn per 100 O00 m of yarn Yarn delivery Pieces of one length of warp One piece of fabric Number of defects in the piece of fabric Ta
24、ble 1. Sorted count values Cumulative number Number of erythrocytes in the counting area Counting area of specified size I cm3 of suspension Four Paw Blood sample ce1 I culture Technical booklet III? I cl= i= E O ni 13 20 25 28 32 34 34 35 35 Number of yeast cells per cubic centimeter of suspension
25、Number of printing errors in four pages Daily production of table tops rhree table tops Number of surface discontinuities on three table tops Oven batch of currant bread One currant bread loaf Number of currants in one currant bread loaf k 15 o n,=35 For ), see page 3. 1 I DIN 53 804 Part 2 Page 3 5
26、1 the number of equal count values with the value 1 is designated by nl (I = O, 1,2, . . . , k) : i n,=n (3) I= o Table 1 shows an example of this (see also example A.4). If 1 bars of heights n1 are plotted against the numerical values, the result is a bar diagram for the n count values (see figure
27、1 ). - - - - - Figure 1. Bar diagram for the sorted count values given in table 1 The bar diagram provides easily readable information on the properties of the distribution (symmetry, outliers etc.) Apart from a bar diagram, a representation in the form of a cumulative step chart, the horizontal inc
28、rements of which are at the numerical values 1 and the step heights are equal to nl (see figure 2). From the cumulative step chart, it is possible to read how many count values are less than or equal to a specified value. ) The Poisson distribution is also described as a distri- bution of rare event
29、s. The meaning of “rare“ in this description can be taken for example from 3. Instead of the numbers nl it is also possible to use the relative frequencies nl/n for the bar diagram and the cumulative step chart. When this is done, the number n of count values is to be specified. 6 Estimated value an
30、d confidence interval for the expectation of a Poisson distribution 6.1 With a single count The estimated valuep for the expectation of p of the number of events is The confidence limits pun and pob of the two-sided confidence interval for p; /lunsps pob (5) or the one-sided confidence interval, can
31、 be taken from table 2 as a function of the count value x for the confidence level 1 - a= 0,95 (see example A.1). Exact equations for calculating pun and ouni talue X - O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 120 150 200 500 lo00
32、 Confidence level 1 - a= 0,95 Two-sided limitation Lower 1 Upper confidence limit O O ,025 O 242 0,619 1P9 1 82 220 2 81 3,45 4.12 5,49 620 6 92 7,65 8,40 9,15 9 90 1 0.67 11.44 1222 13.79 1538 1698 18.61 2024 2438 28 38 3282 37,ll 41,43 45,779 50,17 5467 58 $9 63.44 6790 7237 7686 81 36 99,49 12696
33、 17324 457,13 938,97 480 kb 3.69 5857 7 22 8.77 1024 1 1,67 13p6 14,42 15,76 17P8 1 8 39 19.68 2096 2223 23,49 24,74 25 98 27 22 28,45 29,67 30 89 3331 35.71 38.10 40,47 4283 48,68 54,47 6021 6592 71 39 7723 8285 88,44 94,Ol 99,57 105.10 110.63 116.13 121.63 143,49 176 ,O2 229,72 545,81 1 064,OO One
34、-sided limitation Lower I Upper confidence limit Hin O 0,051 0,355 0,818 1 37 197 2,61 329 398 4,70 5,43 6,17 692 7.69 8,46 925 10,04 1083 11,63 1294 1325 1489 16,55 1822 1990 21.59 2587 3020 34,56 38,96 43.40 47,85 52,33 56,83 61,35 65,88 70,42 7498 7966 84,14 102.57 130,44 177,32 463m 948,56 pob 3
35、PO 4J4 633 775 9,15 1061 11,84 13.15 14,43 15.71 1636 1821 1 9.44 20.67 21 89 23,lO 2430 2530 26,69 27 88 29 ,O6 31,41 33,75 36 ,O8 3839 40,69 46.40 52 ,O7 57 JO 6329 68 85 74 39 7991 85,40 90 89 96 35 101 #80 107 24 112,66 1 18,08 1 39,64 171,76 224.87 5383 1 053,60 (11) it is possible to calculate
36、 the confidence limits from 3 and (121 and in the case of onesided limitation, using the auxiliary value (131 from or (14) See 5. Table 3. Tabulated values of u1-=/2 and of the standardized normal distribution 1,645 2,576 In the case of n counts (count values xi) in counted units of the same type an
37、d of size Ai, it is necessary to group the individual counted units into one enlarged counted unit of size B = xi into the count value obtained for this, x = 2 xi. The estimated value for the expectation p of the number of events in the enlarged counted unit of size B is i = x, the two-sided confide
38、nce interval for pug is obtained from (5) and in the case of one-sided limitation from (61, see example A.2. Conversion to another counted unit is to be carried out as described in subclause 6.3. The estimated value PA for the expectation pA of the number of events, in the case of n-times counting o
39、n counted units of the same type and of size A is n A i by calculation and the count values i-1 n i=l p* =x. (15) To calculate the confidence limits for the expectation PA, using the count value x = n i=l xi, the confidence DIN 53 804 Part 2 Page 5 Typeof test . Null hypothesisHo Alternativehypothes
40、isH1. HO is not rejected, if Two-sided p=K PfK /2x1;1a/2 of the Fdistribution which is obtained from table 5 for a significance level a = 0.05. The null hypothesis is to be rejected, if F F2(x2+ 1 ); 2x, ; 1 - an (23) (see example A.7). If the alternative hypothesis is Pi p2 Hl: - - Bl a, this const
41、itutes a test with one-sided formulation of the question, see l. The above considerations are collated in table 6. Table 4. Summary of test instructions Page 6 DIN 53 804 Part 2 12 39,41 8.75 5.37 4,20 3,62 328 3.05 2.89 2.77 2,68 2,41 2.17 2.05 2,Ol 1,97 Table 5. Tabulated values Ffl;f2;1-a12 of th
42、e F distribution for the two-sided test with a = 0.05 14 39,43 8.68 5.30 4,13 3,55 321 2.98 2.82 2,70 2,60 2.34 2.09 1,98 1.93 1,89 f2 6 3933 9,20 5,82 4.65 4,07 3,73 3.50 334 322 3,13 2,87 1 2,63 2,52 2.47 2.43 - 16 8 3937 898 5,60 4,43 3,85 3,51 3.29 3.12 3,Ol 2,91 265 2,41 2.30 226 2.22 39,OO 10,
43、65 726 6,06 5,46 3925 9.60 623 5,05 4,47 5,lO 4,86 4,69 4.56 4,46 4,12 3.89 3,73 3,61 3,51 Type of test Null hypothesis Ho Alternative hypothesis Hl Ho is not rejected, if Ho is rejected, if Two-sided One-sided Pi k2 -+- FFf1,fz;1 -a12 FFf1,f2;l -a12 Pl - P2 Bl B2 Bl 82 Pl Pz Pi l -a FFf1.f2;i -Ca B
44、; B2 Bl B2 - - Pl = + Pi 19.00 6.94 5.14 4.46 4,lO 3,89 3.74 3.63 3,55 3.49 332 3.15 3.07 3.04 3,Ol 1925 19,33 19.37 639 6.16 6,04 4.53 4,28 4,15 3,84 3.58 3,44 3.48 322 3.07 3,26 3,OO 2,85 3,ll 2.85 2,70 3,Ol 2.74 2,59 2,93 2,66 2,51 2,87 2,60 2,45 2,69 2,42 227 2.53 2.25 2.10 2.45 2,18 2.02 2.42 2
45、.14 198 239 2.12 1.96 1.99 1,82 1.73 1,69 1.66 1.96 1.93 1.89 1.78 1.75 1,70 1,69 1.66 1,61 l,66 1.62 1,57 1,62 1.59 1.54 2,09 1,92 1.83 1.80 1.77 2,04 1.86 1.78 1,74 1.71 - 20 24 I 50 18 10 - 39.44 8.59 5,20 4,03 3,45 399 134 5,46 4,30 3,72 3994 8.63 5,24 4.08 3,50 39.45 8,56 5,17 4,OO 3 42 39,46 3
46、9,48 8,51 8.38 5.12 4,98 3.95 3,81 3,37 3,22 2 4 6 8 10 3,68 2,74 2,50 12 14 16 18 20 3.37 3.1 5 2.99 2,87 2,77 3,15 2.92 2,76 2.64 2,55 3 ,O7 2.84 2,68 2,56 2,46 3.1 1 2,88 2,72 2,60 2.50 2.23 1.98 1,87 1,82 1,78 2,28 2,03 1,92 1.87 1,83 - 2,20 1.94 1.82 1.78 1,74 - 2,51 2.16 2 ,O7 227 2,ll 200 500
47、 Example of reading: fOrf1 = 14 and f2 = 188, Ff,f2;l-Ca/;! = F14,188;0,g75 = 1,94 is obtained by interpolation. I Table 6. Summary of test instructions Withfl = 2 (x2 + 1) andf2 = 2xl. Table 7. Tabulated values Ff,f2;1-.of the Fdirtribution for a one-sided test with a- 0,05 1 I I I 500 19,5 - 5,64
48、3.68 2,94 2,55 2,31 2,14 2 ,o2 1,93 1,86 - 16 1 18 I 20 I 24 10 1 9,40 5,96 4 ,O6 3.35 2.98 2,75 2,60 2,49 2,41 2,35 2,16 1.99 1.91 1 -88 135 50 1 9.48 5,70 3.75 3 .o2 2,64 2 4 6 8 10 12 14 16 18 20 3.20 2,74 2,40 2.24 2,12 2 ,o4 1,97 1,76 1.56 1,46 1,41 1,38 - 30 60 120 200 500 1.64 1.41 1.28 1,22
49、1,16 - Example of reading: for fi = 46 and f2 = 12, Ffl,f2;l-Ca = F46,12;0,995 = 2,41 is obtained by interpolation. DIN 53 804 Part 2 Page 7 Length Ai of the piece of fabric, in m Appendix A Examples from textile technology Number xi of yarn thick places Example A.l Re subclause 6.1 For assessing the appearance of pieces of fabric and for determining specified values for mending, among other factors, the detection of thick places in the fabric itself is of importance. 22 yarn thick places were found in a piece of fabric, 25 m long and 1,50 m wide, in a singl
