International Organization for Standardization: ISO/TR 10017:2003(E)

-- determination of the dominant product wearout characteristics to help improve product design, or to plan the appropriate service maintenance schedule and effort required.

4.9 Sampling

4.9.1 What it is

Sampling is a systematic statistical methodology for obtaining information about some characteristic of a population by studying a representative fraction (i.e. sample) of the population. There are various sampling techniques that may be employed (such as simple random sampling, stratified sampling, systematic sampling, sequential sampling, skip-lot sampling), and the choice of technique is determined by the purpose of the sampling and the conditions under which it is to be conducted.

4.9.2 What it is used for

Sampling can be loosely divided into two broad nonexclusive areas: "acceptance sampling" and "survey sampling".

Acceptance sampling is concerned with making a decision with regard to accepting or not accepting a "lot" (i.e. a grouping of items) based on the result of a sample(s) selected from that lot. A wide range of acceptance sampling plans are available to satisfy specific requirements and applications.

Survey sampling is used in enumerative or analytical studies for estimating the values of one or more characteristics in a population, or for estimating how those characteristics are distributed across the population. Survey sampling is often associated with polls where information is gathered on people's opinions on a subject, as in customer surveys. It can equally be applied to data-gathering for other purposes, such as audits.

A specialized form of survey sampling is exploratory sampling, which is used in enumerative studies to gain information about a characteristic(s) of a population or a subset of the population. So is production sampling, which may be carried out to conduct say, a process capability analysis.

Another application is the bulk sampling of materials (e.g. minerals, liquids and gases) for which sampling plans have been developed.

4.9.3 Benefits

A properly constructed sampling plan offers savings in time, cost and labour when compared with either a census of the total population or 100 % inspection of a lot. Where product inspection involves destructive testing, sampling is the only practical way of obtaining pertinent information.

Sampling offers a cost-effective and timely way of obtaining preliminary information regarding the value or distribution of a characteristic of interest in a population.

4.9.4 Limitations and cautions

When constructing a sampling plan, close attention should be paid to decisions regarding sample size, sampling frequency, sample selection, the basis of sub-grouping and various other aspects of sampling methodology.

Sampling requires that the sample be chosen in an unbiased fashion (i.e. the sample is representative of the population from which it is drawn). If this is not done, it will result in poor estimates of the population characteristics. In the case of acceptance sampling, non-representative samples can result in either the unnecessary rejection of acceptable quality lots or the unwanted acceptance of unacceptable quality lots.

Even with unbiased samples, information derived from samples is subject to a degree of error. The magnitude of this error can be reduced by taking a larger sample size, but it cannot be eliminated. Depending on the specific question and context of sampling, the sample size required to achieve the desired level of confidence and precision may be too large to be of practical value.

4.9.5 Examples of applications

A frequent application of survey sampling is in market research, to estimate (say) the proportion of a population that might buy a particular product. Another application is in audits of inventory to estimate the percentage of items that meet specified criteria.

Sampling is used to conduct process checks of operators, machines or products in order to monitor variation, and to define corrective and preventive actions.

Acceptance sampling is extensively used in industry to provide some level of assurance that incoming material satisfies prespecified requirements.

By means of bulk sampling, the amount or the properties of constituents in bulk material (e.g. minerals, liquids and gases) can be estimated.

4.10 Simulation

4.10.1 What it is

Simulation is a collective term for procedures by which a (theoretical or empirical) system is represented mathematically by a computer program for the solution of a problem. If the representation involves concepts of probability theory, in particular random variables, simulation may be called the "Monte-Carlo method".

4.10.2 What it is used for

In the context of theoretical science, simulation is used if no comprehensive theory for the solution of a problem is known (or, if known, is impossible or difficult to solve), and where the solution can be obtained through brute computer force. In the empirical context, simulation is used if the system can be adequately described by a computer program. Simulation is also a helpful tool in the teaching of statistics.

The evolution of relatively inexpensive computing capability is resulting in the increasing application of simulation to problems that hitherto have not been addressed.

4.10.3 Benefits

Within theoretical sciences, simulation (in particular the Monte-Carlo method) is used if explicit calculations of solutions to problems are impossible or too cumbersome to carry out directly (e.g. w-dimensional integration). Similarly, in the empirical context, simulation is used when empirical investigations are impossible or too costly. The benefit of simulation is that it allows a solution with saving of time and money, or that it allows a solution at all.

The use of simulation in the teaching of statistics is that it can effectively illustrate random variation.

4.10.4 Limitations and cautions

Within theoretical science, proofs based on conceptual reasoning are to be preferred over simulation, since simulation often provides no understanding of the reasons for the result.

Computer simulation of empirical models is subject to the limitation that the model may not be adequate (i.e. it may not represent the problem sufficiently). Therefore, it cannot be considered a substitute for actual empirical investigations and experimentation.

4.10.5 Examples of applications

Large-scale projects (such as the space programme) routinely use the Monte-Carlo method. Applications are not limited to any specific type of industry. Typical areas of applications include statistical tolerancing, process simulation, system optimization, reliability theory and prediction. Some specific applications are

-- modelling variation in mechanical sub-assemblies,

-- modelling vibration profiles in complex assemblies,

-- determining optimal preventive maintenance schedules, and

-- conducting cost and other analyses in design and production processes to optimize allocation of resources.

4.11 Statistical process control (SPC) charts

4.11.1 What it is

An SPC chart, or "control chart", is a graph of data derived from samples that are periodically drawn from a process and plotted in sequence. Also noted on the SPC chart are "control limits" that describe the inherent variability of the process when it is stable. The function of the control chart is to help assess the stability of the process, and this is done by examining the plotted data in relation to the control limits.

Any variable (measurement data) or attribute (count data) representing a characteristic of interest of a product or process may be plotted. In the case of variable data, a control chart is usually used for monitoring changes in the process centre, and a separate control chart is used for monitoring changes in process variability.

For attribute data, control charts are commonly maintained of the number or proportion of nonconforming units or of the number of nonconformities found in samples drawn from the process.

The conventional form of control chart for variable data is termed the "Shewhart" chart. There are other forms of control charts, each with properties that are suited for applications in special circumstances. Examples of these include "cusum charts" that allow for increased sensitivity to small shifts in the process, and "moving average charts" (uniform or weighted) that serve to smooth out short-term variations to reveal persistent trends.

4.11.2 What it is used for

An SPC chart is used to detect changes in a process. The plotted data, which may be an individual reading or some statistic such as the sample average, are compared with the control limits. At the simplest level, a plotted point that falls outside the control limits signals a possible change in the process, possibly due to some "assignable cause". This identifies the need to investigate the cause of the "out-of-control" reading, and make process adjustments where necessary. This helps to maintain process stability and improve processes in the long run.

The use of control charts may be refined to yield a more rapid indication of process changes, or increased sensitivity to small changes, through the use of additional criteria in interpreting the trends and patterns in the plotted data.

4.11.3 Benefits

In addition to presenting the data in a visible form to the user, control charts facilitate the appropriate response to process variation by helping the user to distinguish the random variation that is inherent in a stable process from variation that may be due to "assignable causes" (i.e. to which a specific cause may be assigned) whose timely detection and correction may help improve the process. Examples of the role and value of control charts in process related activities are given below.

-- Process control: variable control charts are used to detect changes in the process centre or process variability and to trigger corrective action, thus maintaining or restoring process stability.

-- Process capability analysis: if the process is in a stable state, the data from the control chart may be used subsequently to estimate process capability.

-- Measurement system analysis: by incorporating control limits that reflect the inherent variability of the measurement system, a control chart can show whether the measurement system is capable of detecting the variability of the process or product of interest. Control charts may also be used to monitor the measurement process itself.

-- Cause and effect analysis: a correlation between process events and control chart patterns can help to infer the underlying assignable causes and to plan effective action.

-- Continuous improvement: control charts are used to monitor process variation, and they help to identify and address the cause(s) of variation. They are found to be especially effective when they are used as part of a systematic programme of continual improvement within an organization.

4.11.4 Limitations and cautions

It is important to draw process samples in a way that best reveals the variation of interest, and such a sample is termed a "rational subgroup". This is central to the effective use and interpretation of SPC charts, and to understanding the sources of process variation.

Short run processes present special difficulties as sufficient data are seldom present to establish the appropriate control limits.

There is a risk of "false alarms" when interpreting control charts (i.e. the risk of concluding that a change has occurred when this is not the case). There is also the risk of failing to detect a change that has occurred. These risks can be mitigated but never eliminated.

4.11.5 Examples of applications

Companies in automotive, electronics, defence and other sectors often utilize control charts (for critical characteristics) to achieve and demonstrate continuing process stability and capability. If nonconforming products are received, the charts are used to help establish the risk and determine the scope of corrective action.

Control charts are used in problem solving in the work place. They have been applied at all levels of organizations to support problem recognition and root cause analysis.

Control charts are used in machining industries to reduce unnecessary process intervention (over-adjustment) by enabling employees to distinguish between variation that is inherent to the process and variation that can be attributed to an "assignable cause".

Control charts of sample characteristics, such as average response time, error rate and complaint frequency, are used to measure, diagnose and improve performance in service industries.

4.12 Statistical tolerancing

4.12.1 What it is

Statistical tolerancing is a procedure based on certain statistical principles, used for establishing tolerances. It makes use of the statistical distributions of relevant dimensions of components to determine the overall tolerance for the assembled unit.

4.12.2 What it is used for

When assembling multiple individual components into one module, the critical factor or requirement in terms of assembly and interchangeability of such modules is often not the dimensions of the individual components but instead the total dimension achieved as a result of assembly.

Extreme values for the total dimension (i.e. very large or small values) only occur if the dimensions of all individual components lie either at the lower or upper end of their relevant individual ranges of tolerances. Within the framework of a chain of tolerances, if the individual tolerances are added up into a total dimension tolerance, then one refers to this as the arithmetical overall tolerance.

For statistical determination of overall tolerances, it is assumed that in assemblies involving a large number of individual components, dimensions from one end of the range of individual tolerances will be balanced by dimensions from the other end of the tolerance ranges. For example, an individual dimension lying at the lower end of the tolerance range may be matched with another dimension (or combination of dimensions) at the high end of the tolerance range. On statistical grounds, the total dimension will have an approximately normal distribution under certain circumstances. This fact is quite independent of the distribution of the individual dimensions, and may therefore be used to estimate the tolerance range of the total dimension of the assembled module. Alternatively, given the overall dimension tolerance, it may be used to determine the permissible tolerance range of the individual components.

4.12.3 Benefits

Given a set of individual tolerances (which need not be the same), the calculation of statistical overall tolerance will yield an overall dimensional tolerance that will usually be significantly smaller than the overall dimensional tolerance calculated arithmetically.

This means that, given an overall dimensional tolerance, statistical tolerancing will permit the use of wider tolerances for individual dimensions than those determined by arithmetical calculation. In practical terms, this can be a significant benefit, since wider tolerances are associated with simpler and more cost-effective methods of production.

4.12.4 Limitations and cautions

Statistical tolerancing requires one to first establish what proportion of assembled modules could acceptably lie outside the tolerance range of the total dimension. The following prerequisites must then be met for statistical tolerancing to be practicable (without necessitating advanced methods):

-- the individual actual dimensions can be considered as uncorrelated random variables;

-- the dimensional chain is linear;

-- the dimensional chain has at least four units;

-- the individual tolerances are of the same order of magnitude;

-- the distributions of the individual dimensions of the dimensional chain are known.

It is obvious that some of these requirements can only be met if the manufacture of the individual components in question can be controlled and continuously monitored. In the case of a product still under development, experience and engineering knowledge should guide the application of statistical tolerancing.

4.12.5 Examples of applications

The theory of statistical tolerancing is routinely applied in the assembly of parts that involve additive relations or cases involving simple subtraction (e.g. shaft and hole). Industrial sectors that use statistical tolerancing include mechanical, electronic and chemical industry. The theory is also applied in computer simulation to determine optimum tolerances.

4.13 Time series analysis

4.13.1 What it is

Time series analysis is a family of methods for studying a collection of observations made sequentially over time. Time series analysis is used here to refer to analytical techniques in applications such as

-- finding "lag" patterns by statistically looking at how each observation is correlated with the observation immediately before it, and repeating this for each successive lagged period,

-- finding patterns that are cyclical or seasonal, to understand how causal factors in the past might have repeated influences in the future, and

-- using statistical tools to predict future observations or to understand which causal factors have contributed most to variations in the time series.

While the techniques employed in time series analysis can include simple "trend charts", in this Technical Report such elementary plots are listed among the simple graphical methods cited in "Descriptive statistics" (4.2.1).

4.13.2 What it is used for

Time series analysis is used to describe patterns in time series data, for identifying "outliers" (i.e. extreme values whose validity should be investigated) either to help understand the patterns or to make adjustments, and for detecting turning points in a trend. Another use is to explain patterns in one time series with those of another time series, with all the objectives inherent in regression analysis.

Time series analysis is used to predict future values of time series, typically with some upper and lower limits known as the forecast interval. It has extensive use in the area of control and is often applied to automated processes. In this case, a probability model is fitted to the historical time series, future values are predicted, and then specific process parameters are adjusted to keep the process on target, with as little variation as possible.

4.13.3 Benefits

Time series analysis methods are useful in planning, in control engineering, in identifying a change in a process, in generating forecasts and in measuring the effect of some outside intervention or action.

Time series analysis is also useful for comparing the projected performance of a process with predicted values in the time series if a specific change were to be made.

Time series methods may provide insights into possible cause-and-effect patterns. Methods exist for separating systematic (or assignable) causes from chance causes, and for breaking down patterns in a time series into cyclical, seasonal and trend components.

Time series analysis is often useful for understanding how a process will behave under specified conditions, and what adjustments (if any) could influence the process in the direction of some target value, or what adjustments could reduce process variability.

4.13.4 Limitations and cautions

The limitations and cautions cited for regression analysis also apply to time series analysis. When modelling a process in order to understand causes and effects, a significant level of skill is required to select the most appropriate model and for using diagnostic tools to improve the model.

Whether included or omitted from the analysis, a single observation or a small set of observations can have a significant influence on the model. Therefore influential observations should be understood and distinguished from "outliers" in the data.

Different time-series estimation techniques can have varying degrees of success, depending on the patterns in the time series and on the number of periods for which predictions are desired, relative to the number of time periods for which time-series data are available. The choice of a model should consider the objective of the analysis, the nature of the data, the relative cost, and the analytical and predictive properties of the various models.

4.13.5 Examples of applications

Time series analysis is applied to study patterns of performance over time, for example, process measurements, customer complaints, nonconformance, productivity and test results.

Forecasting applications include predicting spare parts, absenteeism, customer orders, materials needs, electric power consumption.

Causal time-series analysis is used to develop predictive models of demand. For example, in the context of reliability, it is used to predict the number of events in a given time period and the distribution of time intervals between events such as equipment outages.

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