13.7 Statistical Quality Control with Sampling by Variables
As described in the previous section, sampling by attributes is based on a classification of items as good or defective. Many work and material attributes possess continuous properties, such as strength, density or length. With the sampling by attributes procedure, a particular level of a variable quantity must be defined as acceptable quality. More generally, two items classified as good might have quite different strengths or other attributes. Intuitively, it seems reasonable that some "credit" should be provided for exceptionally good items in a sample. Sampling by variables was developed for application to continuously measurable quantities of this type. The procedure uses measured values of an attribute in a sample to determine the overall acceptability of a batch or lot. Sampling by variables has the advantage of using more information from tests since it is based on actual measured values rather than a simple classification. As a result, acceptance sampling by variables can be more efficient than sampling by attributes in the sense that fewer samples are required to obtain a desired level of quality control.
In applying sampling by variables, an acceptable lot quality can be defined with respect to an upper limit U, a lower limit L, or both. With these boundary conditions, an acceptable quality level can be defined as a maximum allowable fraction of defective items, M. In Figure 13-2, the probability distribution of item attribute x is illustrated. With an upper limit U, the fraction of defective items is equal to the area under the distribution function to the right of U (so that x U). This fraction of defective items would be compared to the allowable fraction M to determine the acceptability of a lot. With both a lower and an upper limit on acceptable quality, the fraction defective would be the fraction of items greater than the upper limit or less than the lower limit. Alternatively, the limits could be imposed upon the acceptable average level of the variable
Figure 13-2 Variable Probability Distributions and Acceptance Regions
In sampling by variables, the fraction of defective items is estimated by using measured values from a sample of items. As with sampling by attributes, the procedure assumes a random sample of a give size is obtained from a lot or batch. In the application of sampling by variables plans, the measured characteristic is virtually always assumed to be normally distributed as illustrated in Figure 13-2. The normal distribution is likely to be a reasonably good assumption for many measured characteristics such as material density or degree of soil compaction. The Central Limit Theorem provides a general support for the assumption: if the source of variations is a large number of small and independent random effects, then the resulting distribution of values will approximate the normal distribution. If the distribution of measured values is not likely to be approximately normal, then sampling by attributes should be adopted. Deviations from normal distributions may appear as skewed or non-symmetric distributions, or as distributions with fixed upper and lower limits.
The fraction of defective items in a sample or the chance that the population average has different values is estimated from two statistics obtained from the sample: the sample mean and standard deviation. Mathematically, let n be the number of items in the sample and xi, i = 1,2,3,...,n, be the measured values of the variable characteristic x. Then an estimate of the overall population mean is the sample mean :
An estimate of the population standard deviation is s, the square root of the sample variance statistic:
Based on these two estimated parameters and the desired limits, the various fractions of interest for the population can be calculated.
The probability that the average value of a population is greater than a particular lower limit is calculated from the test statistic:
which is t-distributed with n-1 degrees of freedom. If the population standard deviation is known in advance, then this known value is substituted for the estimate s and the resulting test statistic would be normally distributed. The t distribution is similar in appearance to a standard normal distribution, although the spread or variability in the function decreases as the degrees of freedom parameter increases. As the number of degrees of freedom becomes very large, the t-distribution coincides with the normal distribution.
With an upper limit, the calculations are similar, and the probability that the average value of a population is less than a particular upper limit can be calculated from the test statistic:
With both upper and lower limits, the sum of the probabilities of being above the upper limit or below the lower limit can be calculated.
The calculations to estimate the fraction of items above an upper limit or below a lower limit are very similar to those for the population average. The only difference is that the square root of the number of samples does not appear in the test statistic formulas:
where tAL is the test statistic for all items with a lower limit and tAU is the test statistic for all items with a upper limit. For example, the test statistic for items above an upper limit of 5.5 with = 4.0, s = 3.0, and n = 5 is tAU = (8.5 - 4.0)/3.0 = 1.5 with n - 1 = 4 degrees of freedom.
Instead of using sampling plans that specify an allowable fraction of defective items, it saves computations to simply write specifications in terms of the allowable test statistic values themselves. This procedure is equivalent to requiring that the sample average be at least a pre-specified number of standard deviations away from an upper or lower limit. For example, with = 4.0, U = 8.5, s = 3.0 and n = 41, the sample mean is only about (8.5 - 4.0)/3.0 = 1.5 standard deviations away from the upper limit.
To summarize, the application of sampling by variables requires the specification of a sample size, the relevant upper or limits, and either (1) the allowable fraction of items falling outside the designated limits or (2) the allowable probability that the population average falls outside the designated limit. Random samples are drawn from a pre-defined population and tested to obtained measured values of a variable attribute. From these measurements, the sample mean, standard deviation, and quality control test statistic are calculated. Finally, the test statistic is compared to the allowable trigger level and the lot is either accepted or rejected. It is also possible to apply sequential sampling in this procedure, so that a batch may be subjected to additional sampling and testing to further refine the test statistic values.
With sampling by variables, it is notable that a producer of material or work can adopt two general strategies for meeting the required specifications. First, a producer may insure that the average quality level is quite high, even if the variability among items is high. This strategy is illustrated in Figure 13-3 as a "high quality average" strategy. Second, a producer may meet a desired quality target by reducing the variability within each batch. In Figure 13-3, this is labeled the "low variability" strategy. In either case, a producer should maintain high standards to avoid rejection of a batch.
Figure 13-3 Testing for Defective Component Strengths
Example 13-5: Testing for defective component strengths
Suppose that an inspector takes eight strength measurements with the following results:
4.3, 4.8, 4.6, 4.7, 4.4, 4.6, 4.7, 4.6
In this case, the sample mean and standard deviation can be calculated using Equations (13.8) and (13.9):
= 1/8(4.3 + 4.8 + 4.6 + 4.7 + 4.4 + 4.6 + 4.7 + 4.6) = 4.59
s2=[1/(8-1)][(4.3 - 4.59)2 + (4.8 - 4.59)2 + (4.6 - 4.59)2 + (4.7 - 4.59)2 + (4.4 - 4.59)2 + (4.6 - 4.59)2 + (4.7 - 4.59)2 + (4.6 - 4.59)2] = 0.16
The percentage of items below a lower quality limit of L = 4.3 is estimated from the test statistic tAL in Equation (13.12):
Construction is a relatively hazardous undertaking. As Table 13-1 illustrates, there are significantly more injuries and lost workdays due to injuries or illnesses in construction than in virtually any other industry. These work related injuries and illnesses are exceedingly costly. The Construction Industry Cost Effectiveness Project estimated that accidents cost $8.9 billion or nearly seven percent of the $137 billion (in 1979 dollars) spent annually for industrial, utility and commercial construction in the United States. Included in this total are direct costs (medical costs, premiums for workers' compensation benefits, liability and property losses) as well as indirect costs (reduced worker productivity, delays in projects, administrative time, and damage to equipment and the facility). In contrast to most industrial accidents, innocent bystanders may also be injuried by construction accidents. Several crane collapses from high rise buildings under construction have resulted in fatalities to passerbys. Prudent project managers and owners would like to reduce accidents, injuries and illnesses as much as possible.
TABLE 13-1 Nonfatal Occupational Injury and Illness Incidence Rates
As with all the other costs of construction, it is a mistake for owners to ignore a significant category of costs such as injury and illnesses. While contractors may pay insurance premiums directly, these costs are reflected in bid prices or contract amounts. Delays caused by injuries and illnesses can present significant opportunity costs to owners. In the long run, the owners of constructed facilities must pay all the costs of construction. For the case of injuries and illnesses, this general principle might be slightly qualified since significant costs are borne by workers themselves or society at large. However, court judgements and insurance payments compensate for individual losses and are ultimately borne by the owners.
The causes of injuries in construction are numerous. Table 13-2 lists the reported causes of accidents in the US construction industry in 1997. A similar catalogue of causes would exist for other countries. The largest single category for both injuries and fatalities are individual falls. Handling goods and transportation are also a significant cause of injuries. From a management perspective, however, these reported causes do not really provide a useful prescription for safety policies. An individual fall may be caused by a series of coincidences: a railing might not be secure, a worker might be inattentive, the footing may be slippery, etc. Removing any one of these compound causes might serve to prevent any particular accident. However, it is clear that conditions such as unsecured railings will normally increase the risk of accidents. Table 13-3 provides a more detailed list of causes of fatalities for construction sites alone, but again each fatality may have multiple causes.
TABLE 13-2 Fatal Occupational Injuries in Construction, 1997 and 1999
TABLE 13-3 Fatality Causes in Construction, 1998
Various measures are available to improve jobsite safety in construction. Several of the most important occur before construction is undertaken. These include design, choice of technology and education. By altering facility designs, particular structures can be safer or more hazardous to construct. For example, parapets can be designed to appropriate heights for construction worker safety, rather than the minimum height required by building codes.
Choice of technology can also be critical in determining the safety of a jobsite. Safeguards built into machinery can notify operators of problems or prevent injuries. For example, simple switches can prevent equipment from being operating when protective shields are not in place. With the availability of on-board electronics (including computer chips) and sensors, the possibilities for sophisticated machine controllers and monitors has greatly expanded for construction equipment and tools. Materials and work process choices also influence the safety of construction. For example, substitution of alternative materials for asbestos can reduce or eliminate the prospects of long term illnesses such as asbestiosis.
Educating workers and managers in proper procedures and hazards can have a direct impact on jobsite safety. The realization of the large costs involved in construction injuries and illnesses provides a considerable motivation for awareness and education. Regular safety inspections and safety meetings have become standard practices on most job sites.
Pre-qualification of contractors and sub-contractors with regard to safety is another important avenue for safety improvement. If contractors are only invitied to bid or enter negotiations if they have an acceptable record of safety (as well as quality performance), then a direct incentive is provided to insure adequate safety on the part of contractors.
During the construction process itself, the most important safety related measures are to insure vigilance and cooperation on the part of managers, inspectors and workers. Vigilance involves considering the risks of different working practices. In also involves maintaining temporary physical safeguards such as barricades, braces, guylines, railings, toeboards and the like. Sets of standard practices are also important, such as:
While eliminating accidents and work related illnesses is a worthwhile goal, it will never be attained. Construction has a number of characteristics making it inherently hazardous. Large forces are involved in many operations. The jobsite is continually changing as construction proceeds. Workers do not have fixed worksites and must move around a structure under construction. The tenure of a worker on a site is short, so the worker's familiarity and the employer-employee relationship are less settled than in manufacturing settings. Despite these peculiarities and as a result of exactly these special problems, improving worksite safety is a very important project management concern.
Example 13-6: Trench collapse
To replace 1,200 feet of a sewer line, a trench of between 12.5 and 18 feet deep was required down the center of a four lane street. The contractor chose to begin excavation of the trench from the shallower end, requiring a 12.5 deep trench. Initially, the contractor used a nine foot high, four foot wide steel trench box for soil support. A trench box is a rigid steel frame consisting of two walls supported by welded struts with open sides and ends. This method had the advantage that traffic could be maintained in at least two lanes during the reconstruction work.
In the shallow parts of the trench, the trench box seemed to adequately support the excavation. However, as the trench got deeper, more soil was unsupported below the trench box. Intermittent soil collapses in the trench began to occur. Eventually, an old parallel six inch water main collapsed, thereby saturating the soil and leading to massive soil collapse at the bottom of the trench. Replacement of the water main was added to the initial contract. At this point, the contractor began sloping the sides of the trench, thereby requiring the closure of the entire street.
The initial use of the trench box was convenient, but it was clearly inadequate and unsafe. Workers in the trench were in continuing danger of accidents stemming from soil collapse. Disruption to surrounding facilities such as the parallel water main was highly likely. Adoption of a tongue and groove vertical sheeting system over the full height of the trench or, alternatively, the sloping excavation eventually adopted are clearly preferable.
- Ang, A.H.S. and W.H. Tang, Probability Concepts in Engineering Planning and Design: Volume I - Basic Principles, John Wiley and Sons, Inc., New York, 1975.
- Au, T., R.M. Shane, and L.A. Hoel, Fundamentals of Systems Engineering: Probabilistic Models, Addison-Wesley Publishing Co., Reading MA, 1972
- Bowker, A.H. and Liebermann, G. J., Engineering Statistics, Prentice-Hall, 1972.
- Fox, A.J. and Cornell, H.A., (eds), Quality in the Constructed Project, American Society of Civil Engineers, New York, 1984.
- International Organization for Standardization, "Sampling Procedures and Charts for Inspection by Variables for Percent Defective, ISO 3951-1981 (E)", Statistical Methods, ISO Standard Handbook 3, International Organization for Standardization, Paris, France, 1981.
- Skibniewski, M. and Hendrickson, C., Methods to Improve the Safety Performance of the U.S. Construction Industry, Technical Report, Department of Civil Engineering, Carnegie Mellon University, 1983.
- United States Department of Defense, Sampling Procedures and Tables for Inspection by Variables, (Military Standard 414), Washington D.C.: U.S. Government Printing Office, 1957.
- United States Department of Defense, Sampling Procedures and Tables for Inspection by Attributes, (Military Standard 105D), Washington D.C.: U.S. Government Printing Office, 1963.