# Basic Principle of Lot Sampling (Metrology)

19.21.
Sampling inspection techniques are used to estimate the lot quality for lot acceptance purposes. As already pointed out, 100% inspection (or screening) is not at all economical and in some cases it is impossible ; whereas the sampling inspection is generally more practical, quick, economical, and provides incentive to the producer to improve his quality as the rejection of a sample means rejection of the whole lot.
Table 19.9. Summation of Terms of Poisson’s Exponential Bionomial Limit 1,000 x probability of c or less occurrences of event that has average number of occurrences equal to e’ or np’

 c -» c or np’ I 0 1 2 3 4 5 6 7 8 9 0.02 980 1000 0.04 961 999 1000 0.06 942 998 1000 0.08 923 997 1000 0.10 905 995 1000 0.15 861 990 999 1000 0.20 819 982 999 1000 0.25 779 974 998 1000 0.30 741 963 996 1000 0.35 705 951 994 1000 0.40 670 938 992 999 1000 0.45 638 925 989 999 1000 0.50 607 910 986 998 1000 0.55 577 894 982 998 1000 0.60 549 878 977 997 1000 0.65 522 861 972 996 999 1000 0.70 497 844 966 994 999 1000 0.75 472 827 959 993 999 1000 0.80 449 809 953 991 999 1000 0.85 424 791 945 989 998 1000 0.90 407 772 937 987 998 1000 0.95 387 754 929 984 997 1000 1.00 368 736 920 981 996 999 1000

Since the value is less than 2.583.
The pieces manufactured by first machine are not bigger than those manufactured by 2nd machine.
19.31.2.

### Paired t-Test.

This test is applied to determine whether or not there is a significant difference between the means of two related lots. Such a situation arises when two samples are so given that the observations of one sample correspond to the observations of the other.
For this test, the differences between observations in each pair are found out; (the null hypothesis for such a case is that the mean of the difference is zero); and the t-test applied on these differences as if testing of the mean of these against zero.
Thus in case of two machines set to produce cylindrical plugs at a specified diameter, if the observations of two lots are quite close to each other, then this test is applied as under :

If this value is less than the critical value of t found from Table 19.18 for critical values of ^-distribution for 5% significance level from two-sided or one-sided as the case may be with n – 1 degrees of freedom, then it is presumed that the means of the two lots are same or in other words, the nuLUiypothesis that the average of the two lots does not differ with each other is not questioned.

### Note.

It may be noted that the £-test is applied when the sample size is small and the information about the standard deviation is not available and, therefore, inferences are made

 c -» c or np’ I 0 1 2 3 4 5 6 7 8 9 1.1 333 699 900 974 995 999 1000 1.2 301 663 879 966 992 998 1000 1.3 273 627 857 957 989 998 1,000 1.4 247 592 833 946 986 997 999 1,000 1.5 223 558 809 934 981 996 999 1,000 1.6 203 525 783 921 976 994 999 1,000 - 1.7 183 493 757 907 970 992 998 1,000 1.8 165 463 731 891 964 990 997 999 1,000 1.9 150 434 704 875 956, 987 997 999 1,000 2.0 135 406 677 857 947 983 995 999 1,000 2.2 111 335 623 819 928 975 993 998 1,000 2.4 091 308 570 779 904 964 988 997 999 1,000 2.6 074 267 518 736 877 951 983 995 999 1,000 2.8 061 231 469 692 948 935 976 992 998 999 3.0 050 199 423 647 815 916 966 988 996 999 3.2 041 171 380 603 781 895 955 983 994 998 3.4 033 147 340 558 744 871 944 977 992 997 3.6 027 126 303 515 706 844 927 969 988 996 3.8 022 107 269 473. 668 816 909 960 984 994 4.0 018 092 238 433 629 785 889 949 979 992 4.2 015 078 210 395 590 753 867 936 972 989 4.4 012 066 185 359 551 720 844 921 964 985 4.6 010 056 163 326 513 686 818 905 955 980 4.8 008 048 143 294 476 651 791 887 944 975 5.0 007 040 125 265 440 616 762 867 932 968 5.2 006 034 109 238 406 581 732 845 918 960 5.4 005 029 095 213 373 546 702 822 903 951 5.6 004 024 082 191 342 512 670 797 886 941 5.8 003 021 072 170 313 478 638 771 867 929 6.0 002 017 062 151 285 446 606 744 847 916 C-» 10 11 12 13 14 15 16 2.8 1,000 3.0 1,000 3.2 1,000 3.4 999 1,000 3.6 999 1,000 3.8 998 999 1,000 4.0 997 999 1,000 4.2 996 999 1,000
 c -» c or rap’ 1 0 2 3 4 5 6 7 8 9 4.4 994 998 999 1,000 4.6 992 997 999 1,000 4.8 990 996 999 1,000 5.0 986 995 998 999 1,000 5.2 982 993 997 999 1,000 5.4 977 990 996 999 1,000 5.6 972 988 995 998 999 1,000 5.8 965 984 993 997 999 1,000 6.0 957 980 991 996 999 999 1,000 6.2 002 015 054 134 259 414 574 756 826 902 6.4 002 012 046 119 235 384 542 687 803 886 6.6 001 010 040 105 213 355 511 658 780 869 6.8 001 009 034 093 192 327 480 623 755 850 7.0 001 007 030 082 173 103 450 599 729 830 7.2 001 O06 025 072 156 276 420 569 703 810 7.4 001 005 022 063 140 253 392 539 679 788 7.6 001 004 019 055 125 231 365 510 648 765 7.8 000 004 016 048 112 210 338 481 620 741 8.0 000 003 014 042 100 191 313 453 593 717 8.5 000 002 009 030 074 150 256 386 523 653 9.0 000 001 006 021 055 116 207 324 456 587 9.5 000 001 004 015 040 089 165 269 392 522 10.0 000 000 003 010 029 067 130 220 333 458 c -> 10 11 12 13 14 15 16 17 18 19 20 21 22 6.2 949 975 989 995 998 999 1,000 6.4 939 969 986 994 997 999 1,000 6.6 927 963 982 992 997 999 999 1,000 6.8 915 955 978 990 996 998 999 1,000 7.0 901 947 974 987 994 998 999 1,000 7.2 887 937 967 984 993 997 999 999 1,000 7.4 871 926 961 980 991 996 998 999 1,000 7.6 854 915 954 976 989 995 998 999 1,000 7.8 835 902 945 971 986 993 997 999 1,000 8.0 816 888 936 966 983 992 996 998 999 1,000 8.5 763 849 909 949 973 989 993 997 999 999 1,000 9.0 706 803 876 926 959 978 989 995 998 999 1,000 9.5 645 752 836 838 940 967 982 991 996 999 999 1,000 10.0 583 697 792 864 917 951 973 986 993 997 998 999 1,000
 C -¥ c or np’ I 0 1 2 3 4 5 6 7 8 9 10.5 000 000 002 007 021 050 102 179 279 397 11.0 000 000 001 005 015 038 079 143 232 341 11.5 000 000 001 003 Oil 028 060 114 191 289 12.0 000 000 001 002 008 020 046 090 155 242 12.5 000 000 000 002 005 015 035 070 125 201 13.0 000 000 000 001 004 Oil 026 054 100 166 13.5 000 000 000 001 003 006 019 041 079 135 14.0 000 000 000 000 002 006 014 032 062 109 14.5 000 000 000 000 001 004 010 024 048 088 15.0 000 000 000 000 001 003 008 018 037 070 c -* ci 10 11 12 13 14 15 16 17 18 19 10.5 521 639 742 825 888 932 960 978 988 994 11.0 460 579 689 781 854 907 944 968 982 991 11.5 402 520 633 733 815 878 924 954 974 986 12.0 347 462 576 682 772 844 899 937 963 979 12.5 296 406 519 628 725 806 869 916 948 969 13.0 252 353 463 573 675 764 835 890 930 957 13.5 211 304 409 518 623 718 798 861 908 942 14.0 176 260 358 464 570 669 756 827 883 923 14.5 145 220 311 413 518 619 711 790 853 901 15.0 118 185 268 363 466 568 664 749 819 875 c -> cl 20 21 22 23 24 25 26 27 28 29 10.5 997 999 999 1,000 11.0 995 998 999 1,000 11.5 992 996 998 999 1,000 12.0 988 994 997 999 999 1,000 12.5 983 991 995 998 999 999 1,000 13.0 975 986 992 996 999 999 1,000 1,000 13.5 965 980 989 994 997 998 999 1,000 14.0 952 971 983 991 995 997 999 999 1,000 14.5 936 960 976 986 992 996 999 998 999 1,000 15.0 916 947 967 981 989 994 997 997 999 1,000
 c -> c or np’ I 0 1 2 3 4 5 CO 7 8 9 16 000 000 004 010 022 043 077 127 193 275 17 000 000 002 005 013 026 049 085 135 201 18 000 000 001 003 007 015 030 055 092 143 19 000 000 001 002 004 009 018 035 061 098 20 000 000 000 001 002 005 001 021 039 066 21 000 000 000 000 001 003 006 013 025 043 22 000 000 000 000 000 002 004 008 015 028 23 000 000 000 000 000 000 002 004 009 017 24 000 000 000 000 000 000 001 003 005 Oil 25 000 000 000 000 000 000 001 001 003 006 c -» ci 14 15 16 17 18 19 20 21 22 23 16 368 467 566 659 742 812 868 911 942 963 17 281 371 468 664 655 736 805 861 905 937 18 208 287 375 469 562 651 731 799 855 899 19 150 215 292 378 469 561 647 725 793 849 20 105 157 221 297 381 470 559 644 721 787 21 072 Oil 163 227 302 384 471 558 640 716 22 048 077 117 169 232 306 387 472 556 637 23 031 052 082 123 175 238 310 389 472 555 24 020 034 056 087 128 180 243 314 392 473 25 012 022 038 060 092 134 185 247 318 394 c -> c4- 24 25 26 27 28 29 30 31 32 33 16 978 987 993 996 998 999 999 1,000 17 959 975 985 991 995 997 999 999 1,000 18 932 955 972 983 990 994 997 998 999 1,000 19 895 927 951 969 980 988 993 996 998 999 20 843 888 922 948 966 978 987 992 995 997 21 782 838 883 917 944 963 976 985 991 994 22 712 777′ 832 877 913 940 959 973 983 989 23 635 708 772 827 873 908 936 956 971 981 24 554 632 704 768 823 868 904 932 953 969 25 472 553 629 700 763 818 863 900 929 950 c -> ci 34 35 36 37 38 39 40 41 42 43 19 999 1,000 20 999 999 1,000 21 997 998 999 999 1,000 22 994 995 998 999 999 1,000 23 988 993 996 997 999 999 1,000 24 979 987 992 995 997 998 999 999 1,000 25 966 977 985 991 994 997 998 999 999 1,000

According to the method of inspection, the quality (i.e., the characteristic of interest) of a product or material can be broadly classified as (i) attributes, (ii) count of defects, (Hi) variables. In the method of classification by attributes, each item inspected is classified as satisfying one particular characteristic or not, e.g., ininspection by gauges-either a component is satisfactory or not. In inspection by attributes, the object is to limit the number of items having one or more defects. In second method (count of defects), and number of defects on each inspected item are counted and it is carried out when the object is to limit the average number of defects in the items. In the method of classification by variables, the characteristic is measured on a continuous scale. For these three methods of quality classification, the lot quality is described respectively in terms of (i) percentage defective, (ii) number of defects per item or for 100 items, (Hi) arithmetic mean or standard deviation or both, or coefficient of variation which is the ratio of the standard deviation to the average.
19.21.1.

### Whether to go for attributes inspection, or inspection by variables.

Whenever there is a choice to decide which of the two methods of inspection should be followed, the following considerations may be looked for :

 Consideration Attributes Inspection Inspection by variables (i) Cost of inspection Since it involves use of GO or NO GO gauges, inspection cost is low. Measurement of any characteristic by measuring instruments involves more time and labour and thus cost. It also involves more record keeping and calculations for computation of mean, standard deviation etc. (ii) Items to be inspected for the same degree of efficiency in drawing inference about the lot. More Less. This method is, therefore, suited in case were destructive or costly testing is involved. (iii) Method of estimating the sample quality from the item quality. Less complicated More complicated. (iv) Information furnished It does not give much information It gives much more information about the quality of item lot. (v) Assumptions No. Therefore, more applications Characteristics, under consideration should have normal distribution. (vi) Limitations No. Many characteristics can be considered at a time to classify an item as defective or non-defective. Only one characteristics can be considered at a time. (vii) General Inspection is slightly subjective, i.e. items of border-line quality may be classified as defective or non-defective depending on the sense of the operator. Inspection is more objective. As it involves taking actual measurements, it minimises the possibilities of inspection bias and error.

19.21.2.

### Principles of Sample Selection.

It has been realised in most of the industrial and trade applications that 100% inspection (checking) and testing of components is not feasible and the sampling procedures have become indispensable tools because of their economy, reliability and practicability. However, the reliability of the conclusions drawn (e.g.,
estimating the quality of a lot or ascertaining its conformity to the requirements of a specification) on the basis of the sample depends on its representativeness and the method of its selection. The method of sampling adopted should be such that it ensures a truly random and representative sample leading to sound and satisfactory estimation of lot quality. Bureau of Indian Standards has prepared a specification (IS : 4905-1968 Methods for Random Sampling) to achieve this objective.
It may be understood that in the lot sampling, the inference about the lot is drawn on the basis of information given by a sample, and there is every possibility that the sample may not be true representative of the lot because out of any lot many samples can be made and all the samples cannot be identical. Thus there are chances that a bad lot may be accepted or good lot may be rejected ; the error on this account is known as the sampling error. Similarly when lot sampling is being carried out to estimate the lot quality then due to sampling error the sample estimate may be different from the actual value of the lot quality.

### The sampling error depends upon :

(i) the degree of homogeneity of the lot, (ii) the size of the sample, (iii) method of sample selection.
It may be stressed here that the method of sampling adopted should be such that it avoids bias of any form (i.e., giving preference to certain positions in the lot, intentionally choosing either defectives or non-defectives only, unplanned sampling). Method of random sampling (in which the chance for inclusion of any items in the sample is predetermined and is independent of the quality of the item and is also independent of the quality of other items selected for the sample) avoids all types of biases.
For random selection, if possible, the lot should be thoroughly mixed, sample units selected blindly from all parts of the lot, the various parts so arranged that sampler has access to all, the position of selection of sample unit in each part being changed.
The method of simple random sampling can be followed if the lot consists of a number of items such that each item is easily identifiable and, apart from the lot size, all other information about the composition of the lot is available. According to this method, once an item has been drawn, it will not be put back into the lot till complete sampling is over. IS : 4950-1968 gives a table of random numbers containing 3000 digits (from 0 to 9), arranged randomly in 15 pages. For convenience of reading, the digits have been grouped into sets of 5 and arranged in 40 rows and 50 columns at each page. All the items in a lot are numbered at random and then the item numbers to be selected from the lot to form a random sample are found out from this table. The random numbers are selected by choosing any page and starting at random from any place vertically downwards. The digits greater than the lot size may be ignored.
19.21.3.

### Methods of Random Sampling. Other methods of sampling include :

(i) Systematic sampling.
(ii) Stratified sampling. (iii) Cluster sampling. (iv) Sampling in stages.
Method of systematic sampling is applicable to cases where the items are presented in an orderly manner and the sample units are chosen at regular intervals. Here it is assumed that there is no deliberate attempt to manipulate the sequence of the items in the lot in any desired manner while the lot is presented for inspection.
In the method of systematic sampling, first a single sample item is selected from the lot of N items at random and then the items are selected at regular predetermined intervals (of N/n) to make up the desired sample of size n.
The method of stratified sampling is possible where the lot can be sub-divided into a certain number or more homogeneous groups or strata and then each stratum sampled separately. As this method draws items from each stratum of the lot, the sample drawn by this method is considered to be more representative of the lot.
The division of a lot into the strata may be on the basis of the homogeneity of the items within a lot, or convenience of sampling, or such other considerations which would make the item within each stratum as much alike as possible. The number of items selected from each stratum are proportional to the size of the stratum, ensuring that at least two items are drawn from each stratum.
When the lot submitted for inspection consists of certain groups of clusters of items, {e.g., when a lot consists of items packed in cartons and it is either impracticable or costly to repack the cartons opened for selecting sample items) it is advisable to go in for cluster sampling method. In this method, first a few clusters are selected and then all the items are inspected in the selected clusters. The results of this method will be satisfactory only if the items within a cluster are quite heterogeneous.
In cases where the lot is in the form of packages and it is impossible to take sample from each package after opening it, then first some packages are selected at random and then from each package a random sample of items proportional to the size of the package is chosen. This method is known as sampling in stages. In this method, in the first stage, a desired number of primary units are selected at random and in the second stage, the required number of items are chosen at random from the selected primary units.
19.21.4.

### Planning sample selection.

Proper planning of the sample selection is very important to satisfy the assumptions made in sample theory and to achieve the maximum possible sampling efficiency. For this purpose it is desirable that the maximum possible information be gathered about the lot, the nature of items and the quality characteristics etc. Some of the important points requiring due considerations are discussed below :
(i) We should be clear about the purpose for which the sample has to be selected, i.e. whether for assessing the quality of the lot, or for lot acceptance. It may be noted that though the method of sampling will be same for both the purposes, the size of the sample will be different in two cases.
(ii) The next important step is the formation of lots, for which due consideration has to be given to the nature of the lot, i.e. stationary lot or moving lot, item lot or bulk lot, homogeneity of lot, formation of sub-lots etc.
It may be emphasised that for inspection purposes, a lot is viewed as an aggregate of items which may be in the form of a batch, a group, a continuous stream or a bulk of products of raw materials.
A stationary lot is one in which all the items are simultaneously available for inspection. Its advantages are that the planning is easier, the various items can be easily identified and it is possible to have re-sampling and sequential sampling. However, to arrange all the items of the lot such that each item is accessible to the inspector to avoid bias is a difficult problem. In the case of moving lot, the various items flow past the point of inspection and as designed,
a few items are selected at a time from the moving batch. Sampling from moving lot is very convenient in the case of bulk materials like coal, grains, etc.
An item-lot consists of an aggregate of discrete items, or an aggregate of items of continuous or bulk materials which have been rendered discrete in the form of specimens or increments. A bulk lot is presented in the form of continuous or bulk material. As stated above if the continuous or bulk material is presented in the form of increments, (e.g., continuous length of cloth presented in the form of pieces of 1 metre, or oil tinned into containers of 1 kg capacity) then it will be treated as item lot.
A homogeneous lot is one which comes out from a process under statistical control. In order to have high efficiency of sampling it is desirable to maintain high degree of homogeneity of a lot, and for this purpose the lot should be confined to the items or products originating from essentially similar conditions. Under the conditions where a lot is formed by combining various sub-lots coming from different sources then, as far possible, the identity of the sub-lots should be preserved so that efficient stratification can be achieved.
When the lot size is very big, then for considerations of economy in inspection or efficiency of sampling, it may be considered desirable to divide the lot into sub-lots. Even if the sub-lot is very bulky then the division into sub-lots may be carried on two or more stages as well. An example of such a case will be to find the quality of coal coming from a mine. Say the coal is coming in the truck-load by the rails. First it may be decided to sample the rails, i.e. sampling be done once in a week. After a particular train is selected, then a small number of wagons are selected and further from each wagon equal quality samples are taken and inference drawn. It may be stressed here that inference will finally be made about the lot as a whole and not about each sub-lot. Such a method of sampling is also known as sampling in stages.
(iii) In the formation of lot, the decision about the size of the lot is also very important item. As a general rule the lot size should be as large as possible keeping two things in view, viz.,
(a) a reasonable degree of homogeneity should be maintained, i.e., the items in the lot should not differ widely in quality, and
(b) we should be able to arrange the various items of the lot such that the inspector will have easy access to all the items for selection of a sample. This calls for proper storage and handling facilities.
The advantage in having a bigger lot size is that, for any given degree of efficiency of sampling and for the same degree of homogeneity in the lot, the sample size will not increase as rapidly as the lot size and will not increase after a certain size for a lot. In other words, having a bigger lot size means the inspection cost is low.
On the other hand, if a large lot is rejected, then 100% inspection means cost of inspection will increase enormously and therefore a suitable compromise is essential in deciding the lot size.
As it is very important condition in the formation of a lot that the items must be of a single type, grade, class, size etc., produced under relatively uniform conditions of a manufacture ; it is proposed that the items for a lot be taken from a single batch of raw material or from component parts obtained from a single source, or manufactured by a single production method, or obtained from a single production line with the same dies and fixtures, or manufactured during single production shift, or produced from one setting of the machine and so on.
(iv) Determination of sample size : It may be again stressed that the method of determining the sample size for the purpose of assessing the lot quality is different than the method of
determining the sample size for the purpose of lot acceptance. Both these methods are discussed in details later.
Here we are broadly outlining the factors which require due consideration in deciding the size of the sample :
(a) We should be clear about the extent of error (risk element) that we can tolerate due to sampling.
(6) Degree of homogeneity in the lot.
(c) Lot size: As already stated a smaller lot size means that the sample size will be bigger in percentage (relatively), whereas the percentage of items in a sample for a bigger lot will be comparatively very less.
(d) The inspection cost should be minimum.
(v) Determination of item size or sample-unit size : This problem arises in case of bulk or continuous materials. (It is obvious that items in the sample can be either in the discrete form, i.e. single integral unit of product which can’t be further sub-divided, or in the continuous or bulk materials form). Under such situation, item is defined as a specimen of specified length, or as a specimen of a specified area, or as a relatively small portion of specified weight or volume of the bulk or the increment.
The determination of size of the item in such cases is very important factor in sampling techniques.
In the case of bulk-lots, usually larger the size (area, volume or weight) of the item, greater would be sampling efficiency, but beyond a certain size for the item the rate of gain in the sampling efficiency may be negligible. Besides from practical considerations, the large size of the increment for bulk products like iron ore, coal etc., may also pose problems like limitations of manual sampling, introduction or errors due to crushing, etc.
However, if sampling efficiency is the only criterion for determining the item size, then that size, beyond which the rate of gain in efficiency would be very small, may be taken as the appropriate item size.
For determining the item size following this principle, statistically designed experiments should be conducted covering different sizes for the item with respect to the item quality characteristics of interest. Corresponding to each size considered, from the values of the item quality, a measure of variability in item quality should be calculated. These values plotted against different sizes for the item would give a curve. In general, this curve would slope downwards in the direction of increase in the item size and would be more or less flat after a certain point. The size corresponding to this point would be an appropriate item size.
In an investigation conducted on coal of sizes 38 to 60 mm, seven series of 35 increments (to be considered as items) each were collected. The increments in each series were of approximately equal size which were analysed for ash content, individually. The average size of the increments in each series and the corresponding measure of variability (co-efficient of variation, v) of ash content were as follows :

 Series Average size of increments (weight) Co-efficient of variation v of ash content 1 94 g 61.1 2 345 g 50.7 CO 978 g 41 4 1.7 kg 33.5 5 2.4 kg 29 6 3.6 kg 23.4 7 3.9 kg 22

The co-efficient of variation of ash content between the increments has been plotted against the average size of the increments in Fig. 19.27.
On the basis of this investigation about 3.5 kg could be taken as a suitable size of the increment as increments of larger size do not decrease the co-efficient of variation appreciably.
(vi) General conditions for gross sample, i.e. mixture of all items (increments) selected from single bulk-lot or a sub-lot.
In order to gather maximum information regarding item quality in

Fig. 19.27. Determination of Item Size.
case of bulk material, it is advisable to keep each increment in a separate container so that its identity is maintained. However, in cases, where the inspection cost of bulk material is high ; the increments selected in accordance with the principles of the sample selection may be mixed up thoroughly to form a gross sample (it may be noted that the samples being of smaller size, compared to the whole bulk, can be easily mixed up thoroughly so that a small sample taken out of the gross sample will be representative of the full sample ; this process is sometimes known as reduction) and inspection done on it. This method provides an estimate of the average quality as efficiently as the one based on inspection of each individual increment within the gross sample. However, it may be appreciated that a single gross sample is not sufficient to give any information about the variability in the lot, or to provide a measure of the sampling error involved and of the reliability of the assessment made of the average quality level.

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