Componentes de la Inteligencia Cultural

sample taken, and the other the range of the sample, that is the maximum minus the minimum value. Both are necessary. For example, the average of a sample of five may be fine, but the range could be unacceptably wide. And the range could be small, but located in the wrong place; that is it has an undesirable average. So typically, from time to time throughout the day, the operator will take, say, the five most recent products produced and set them aside. This is a sample. The particular product dimension is measured and the average and range values of the sample calculated. These two results are plotted on the chart, usually by the process operator. The chart indicates if the process is acceptable. If it is acceptable work continues. If not, work stops to investigate. This is SPC in practice.

Refer to Figure 13. Note the rotated normal distribution shown next to the average chart indicating the expectation that samples will be normally distributed and that the upper and lower control limits correspond to plus and minus three standard deviations. Notice that both the average and the range charts have an upper and a lower ‘control limit’. These limits are the bounds beyond which unacceptable performance is indicated.

Figure 13: Statistical process control charts

N

This brings us to the concept of natural variation; you met this earlier in our discussions on Deming. Every process has natural variation. In other words, it is impossible to make any product with absolute consistency. The inconsistency will

Types of chart SPC Variables quantitative data x and R chart Attributes qualitative (yes/no) data

p chart, c chart

Examples of x and R chart Corresponding to

Average chart

(x) x

Upper control limit (UCL_x)

Lower control limit (LCL_x) x

x xxx x x x x x

R

Upper control limit (UCL_r)

x x x x x x x x x Range chart (R) x _(LCL r)

be caused by chance variations, however small, in, perhaps, the material, tool wear, positioning of the piece, speed of the machine, actions by the operator, and so on. These are called common causes. This variation can be measured and, using statistics, its spread can be predicted. It turns out that the spread follows the normal distribution, irrespective of the type of process. This is a consequence of the central limit theorem. Therefore if points are plotted which fall outside the distribution, then special events are occurring. The special events are assignable to unusual or unexpected changes or events, which may cause defects to be produced. This knowledge is very convenient for two reasons.

First, if the variation does not follow a normal distribution then we know that some special event is taking place. The special event may be an untrained operator, a change in the type of material, tool or bearing wear-out and so on. These special events can, with perseverance, be tracked down and the cause eliminated. Second, the spread of a normal distribution can be measured by calculating the control limits. The formula is given in Figure 14. It turns out that within these limits which equal plus or minus three ‘standard deviations’ on either side of the process average value, lies virtually all of the natural variation. So if an operator takes a measurement and finds that it lies outside these control limits, then it is virtually certain that something has happened to the process. The process is then referred to as being out of control. The process should be stopped and the situation investigated.

Be careful to distinguish between: the control limits (which are a characteristic of the process and are set at plus and minus three standard deviations from the average), and the tolerance limits (also known as the specification limits) which are set by the designers or engineers. Ideally, of course, the control limits should lie within the tolerance limits, and a designer should take into account the normal variation of the process when designing a product.

S

A chart should ideally be set up for each process, that is for each machine, making a particular type of product. (Pre-printed SPC charts are available from some quality societies or in books, and these make data entry and chart plotting very easy.) When setting up a chart it is important that there is consistency, so you should take samples over a representative period of time. You will need to decide on a sample size and the number of samples. Typical numbers are a sample size of 5 and at least 25 samples. For each sample calculate the average (‘mean’) and the range. Refer to Figure 14. Then calculate the average of the averages, and the average of the ranges. Now you will need to look up the control limit factors for the sample size you have used. If you have used a sample size of 5, the factors are given in the Figure 14. Now use the formulas in Figure 14 to calculate the control limits. When these are drawn in you can begin to use the charts for control purposes. You will have to decide what is a reasonable interval for samples to be taken. Generally, the higher the ‘Cpk’ value, the less frequent does the sampling have to be. We discuss Cpk, or capability ratios, later.

Figure 14: Calculations for mean and range charts

C

There are other criteria, apart from falling outside of the control limits, that indicate an out of control condition. These other criteria can be identified by operators, or automatically where SPC data is entered by computer, so that early action can be taken. Let us consider the logic. With natural variation occurring you would expect measurements to be spread more or less evenly on either side of the average value. To be more precise, with the standard deviation known, you would expect a certain proportion of measurements to fall within plus and minus one, two, and three standard deviations of the average value. If this does not occur, again there is an indication of trouble. As an example, the probability of a measurement falling above the average is, of course, 50%. The probability of two successive measures above the mean is 25% (.5 ×.5). And the probability of three successive measures above the mean is 12.5% (.5 ×.5 ×.5). Four successive measures above is 6.25%, and so

x and R charts

Assume samples of the size 3 are taken the readings are x1_{, x}2_{, x}3

x =

These two values are plotted on the charts To set up the chart, take 20 (minimum) random samples, each of sample size (say) 3

Calculate x and R as above, then x1_{+ x}2_{+ x}3

3 R = largest x – smallest x

x1_{+ x}2_{+ x}3_{+...+ x}20 _R1_{+ R}2_{+ ...+ R}20

20 20

x = _{R =}

These give the average (x and R) lines on the chart Then calculate the control limits using the formulas UCLx = X + A2_{x R UCLr = D}4_{x R}

LCLx = X – A2_{x R LCLr = D}3_{x R}

where the values A2_{, D}3_{, D}4_{depend on the sample size}

Sample size A2 _D3 _D4 3 1.023 0 2.575 4 0.729 0 2.282 5 0.577 0 2.115 6 0.483 0 2.004 7 0.419 0.076 1.924 ,

on. If we get to seven successive measures above the average the probability is less than 1%, and we could reasonably conclude that something strange (a special event) has taken place. The other criteria are linked with the probabilities of successive measurements falling beyond a particular number of standard deviations. The interpretation of process control charts is a skill that can be developed. Particular chart patterns are indicative of particular problems that may be developing. Some indications of the possibilities are given in Figure 15.

Figure 15: SPC chart interpretation

It is not always possible to measure variables. Some defects, such as scratches, tears, and holes are either there or they are not. The products either pass inspection or they do not. There are two basic types of attribute chart: p charts and c charts. p charts are used where there are batches of product and the percentage that

are defective can be determined. c charts are used where there are a number of possible types of defect associated with a particular product, for instance the number of scratches or stains or dents on a table.

5.2 Attribute charts