Control statistical processes

The statistical control of the processes ( MSP ) ( Statistical Process Control or SPC in English), is control Statistique S of process. Through Curve S analyzing the variation (in + or in -) with a value of reference, to pre-empt the measures to be taken to improve a process of manufacture (Automobile, Metallurgy,…).

It is especially with the Japan after the Second world war that this discipline was established thanks to William Edwards Deming, disciple of Walter A. Shewhart. The improvement of the quality of the Japanese products with the systematic use of the control charts was such, that the Western countries developed in their turn of the tools for the follow-up of quality. This discipline uses certain numbers of techniques such control of reception, the experimental designs, the techniques of regression, the diagrams of Pareto (Loi of Pareto), the Capabilité…. and of course, the control charts.

The purpose of the control of production is to obtain a stable production with a minimum of products nonin conformity with the specifications. The quality control is “dynamic”: it is not interested in the isolated and instantaneous result, but in the follow-up in time: it is not enough that a part is within the limits of the specifications, it is also necessary to supervise the chronological distribution of the parts inside the intervals of tolerances. The SPC or MSP has as an aim a quality increased by the use of statistical tools aiming at a centered production and the least dispersed possible.

Analyzes graphic data

The first phase, after the data-gathering is the visualization of their distribution. Different graphic tools proposed by the software from statistical control can be used for the Charts of statistical data

Group of dots (or chronological cloud)

The group of dots makes it possible to visualize the data in time by a chronological number of sample, a date, etc

Example 1: An industrial company manufactures metal keys and notes the lengths of the parts obtained on a sample of 32 articles. The table below summarizes the lengths in mm per chronological number of sample. The corresponding group of dots is the following:

Histogram

A histogram is a diagram with rectangles (or bars) whose surfaces are proportional to the frequencies. Its correct layout makes it possible to carry out the test of normality Loi of chi-2. Histogram corresponding to example 1: This histogram gave place to the following statistical results:

Moustache box

The principle of the Boîte to moustache is of sorting the data and dividing them into four equal parts. On the left first Quartile are a quarter of the data, half of the data are on the left of the second quartile (or Médiane), while the three quarters of the data are third quartile.
on the left In our example, the first quartile, on the left side of the box is close to 128 Misters the right-sided of the box shows the third quartile (137 mm). This means that half of the keys lie between 128 and 137 Misters the median (vertical line in the box) is approximately 133 Misters One can observe the good symmetry of the data confirmed by the histogram.

Graphics stems and sheets

The graph “stem-and-sheet” (stem and leaf) is based on the ordered numerical values of the data of example 1 and resembles its histogram. It is very close to the moustache box.
Values 124, 125 will be represented by stem 12 and sheets 4 and 5. The found value 127 3 times by stem 12 and sheets 777. Idem for the found value 128 4 times 12 8888. The other stems are 13 and 14. The stems and the sheets form a graph of which the length of each line is identical to the corresponding bar of a horizontal histogram. In this graph “stem and sheet”, the 2:00 of the graph are the first (128) and third (136) quartiles and the M the median (133). One finds here the symmetrical distribution of the data.

Concepts of statistics applied to the quality control

The methods and tools SPC call upon the Statistiques and more precisely with the Statistique mathematics.

Distribution of discrete random variables

General information

A population is a whole of individuals (machine element, sand sample, boxes.) on which one will follow one or more characters (color, temperature, dimension, concentration, pH,… This character can be qualitative (aspect, good or defective) or quantitative (size, weight, many defects.). In this last case, the values taken by the character will constitute a discontinuous or continuous distribution. In practice, the observations will be carried out on part of the population (statistical sample) taken randomly.

Binomial distribution

The Binomial distribution is used during a pulling " not exhaustif" (with handing-over). It can be used when the report/ratio of the sample size (N) on that of the batch (NR) is such as N/NR $\backslash \; Leq$ 0,1. It is the case where the parts taken for the sample are not given in the lot.
The probability so that there is K defective pieces in N nonexhaustive pullings (independent) is:

$\backslash \; mathrm\; \{C\}\; \_\; \{N\}\; ^\; \{K\}\; =\; \{N\; \backslash \; choose\; K\}\; =\; \backslash \; frac\; \{N!\}\{K!\; (n-k)!\}$
Example: It is a question of evaluating the probabilities (P) of observing X = 0,1,2,3,4,5 defective pieces in a sample of size N =5 with p =0,10 and Q =0,90. The part is classified defective if the diameter is too small or too large. The machine tool provides 10% of defective pieces in a sample of size 5.

This law is used very often in the control of reception by attributes or during the control of reception of a batch. The binomial variable follows a continuous distribution being able to take only the whole values 0,1,2,…, N.

Hypergeometric distribution

When a control implies the complete destruction of the parts or elements to be controlled, it is impossible to carry out a nonexhaustive pulling. The hypergeometric Loi is used in the place of the binomial distribution and its chart is close to the latter. The hypergeometric law has as an expression: $p\; (K)\; =\; \backslash \; frac\; \{C\_\; \{Np\}\; ^kC\_\; \{Nq\}\; ^\; \{n-k\}\}\; \{C\_N^n\}$
(Q = 1 - p)
NR: cut lot.
N: cut sample (N < NR).
p: % the defective one in the batch initial.
C: full number of defective in the lot.k: many possible defective products in the sample of size n.

Distribution of Poisson

When N is large and p weak, the Loi of Poisson constitutes an approximation of the binomial distribution. The probability of having K defective pieces in a given sample N is:

$p\; (K)\; =\; P\; (X\; =\; K)\; =\; \backslash \; mathrm\; \{E\}\; ^\; \{-\; \backslash \; lambda\}\; \backslash \; frac\; \{\backslash \; lambda\; ^k\}\; \{K!\}\; \{\backslash \; quad\; with\; \backslash \; quad\}\; (\backslash \; lambda\; =\; Np)$

A current application of the law of Poisson is the prediction of the number of events likely to occur over one determined period of time, for example, the number of cars which are presented to a tollbooth in the one minute space. The law of Poisson is an approximation of the binomial distribution if N > 40 and p < 0,1.

Distributions of continuous random variables

Normal distribution

Analysis of the variables intervening in the production of a machine (diameter of the produced parts, weight…) show that if the adjustment is constant, the distribution follows a bell-shaped curve “” whose two important characteristics are the average and dispersion. The normal Loi is very much used in statistical control, especially in its reduced centered form. wikisource: Function table of distribution of the reduced centered normal law. In practice SPC, a concept like confidence the Interval, is based on the probabilities of the normal law. For example, 68,3% of surface under the curve of a normal distribution mean that 2/3 of the observations will be in the interval: - \ sigma; - \ driven + \ sigma. In the same way one estimates at 95,4% the probability so that an observation (dispersion = 6 times the standard deviation) is in the interval: -2 \ sigma; - \ driven + 2 \ sigma. The interval -3 \ sigma; - \ driven + 3 \ sigma, contains 99.7% of the data. The normal law is defined by its first two moments (m and $\backslash \; sigma$). Its coefficient of assymetry is null and its coefficient of flatness equal to 3. What it is necessary to retain for the quality control is that the normal distribution is symmetrical around the average, which is also the median and is all the more spread out around the standard deviation is larger. The adjustment of the machine is positioned on the average of the parts $\backslash \; scriptscriptstyle\; \backslash \; overline\; \{X\}$. If the production is stabilized, the distribution of the characteristics of a part on a machine, follows a normal law because of the central limit theorem: any system, subjected to many independent factors follows the law of Gauss and in this case, average, median and mode are equal. If N is the number of values, the standard deviation of the population is:

$s=\; \backslash \; sqrt\; \{\backslash \; frac\; \{1\}\; \{n-1\}\; \backslash \; sum\_\; \{i=1\}\; ^n\; (x\_i-\; \backslash \; overline\; \{X\})\; ^2\}$.

Distribution of Student

This distribution, still called Law of Student or law T is based on the difference between the standard deviation of the population (σ) and that of the sample (S). This difference is all the more large as the sample is small so that the distribution is not normal any more when one uses the standard deviation of the sample (S) instead of the standard deviation of the population (σ). The report/ratio:
$T\; =\; \backslash \; frac\; \{\backslash \; overline\; \{X\}\; -\; \backslash \; driven\}\; \{S\; \backslash \; sqrt\; \{N\}\}$
follows a distribution of Student. The parameter (degree of freedom) represents the precision of the standard deviation and determines the shape of the curve which is symmetrical, like the normal distribution. When the number of degrees of freedom increases, S becomes a more precise estimate of σ.

Distribution of the Khi-deux

One uses this distribution to compare two proportions. In control of reception, this test, based on the probablilities, is used to highlight if the percentage of defective between two suppliers has or not a statistical significance or if it is due randomly. Like the distribution of Student, the parameter “degree of freedom” gives a distribution of different form.

Distribution of Fisher

The distribution of Fisher (F) or Test of Fisher is used in regression and analyzes variance. F uses 2 degrees of freedom, one with the numerator (m), the other with denominator (N). It is noted F (m, N)

Law of Weibull

Descriptive statistics

The quality control uses a certain number of technical terms: sample, population, character….clarified in the descriptive Statistical . In the same way, the Criteria of position (average, mode, median) and of dispersion (extent, standard deviation) are necessary to the comprehension of the Histogramme.
The arithmetic mean is the location parameter most representative, because most sensitive to the fluctuations. On the other hand, the median and the mode are simpler to determine, not requiring any calcul.
The estimator of the most significant parameter of the dispersion is the standard deviation based on the variation of the whole of the values compared to the average. The extent, when the sample size is low (N < 7), is a good Estimateur of dispersion. It is frequently used in the control charts of mesures.

Interval estimating of confidence

In practice, one cannot trust with the estimate of parameters such as the average or the standard deviation in a specific way (see Critères of dispersion and Erreur (metrology) to the measures). One seeks to find an interval in which the unknown and estimated parameters have a chance to be. It is the principle of the estimate by Interval confidence.

Confidence interval on the average

Confidence interval on the variance

Normality of a distribution

General information

The tests which follow make it possible to make sure of the normality of the results obtained.

Right-hand side of Henry

On a simple examination of the histogram, it is not obvious to come to a conclusion about the “normality” of the variables. It is practical to compare the graph obtained with the corresponding theoretical data based on a standardized theoretical law and to represent these data on a line using a scaling. The Droite of Henry makes it possible to carry out this transformation. It is noted that the distribution of example 1 is quasi-normale.
The Droite of Henry cut the x axis at the point of X-coordinate m and its slope is equal to 1/$\backslash \; sigma$, which allows an estimate of the standard deviation of the distribution.

Average of average of data

If it is supposed that the population from which one extracted the sample is normal and has as theoretical parameters X = 133 mm (average) and $\backslash \; sigma$ = 5 mm, then one can trace the histogram of reduced normal law centered corresponding on a simulation of 1000 randomly drawn parts. The theoretical average is of 133 mm, the standard deviation of 5 mm and the extent of 34 Misters. Since our sample of 32 data follows a normal law roughly, one can put the question to know if the theoretical average of the population would be more precise if one multiplied the number of the series of observations. Let us simulate for example 9 successive taking away of 1000 parts of theoretical average 133 mm and standard deviation 5 Misters. The theoretical average obtained is of 120mm, the standard deviation of 4.8 mm and the extent of 32 Misters. The distribution of the “average of the averages” seems normal. The histogram is centered on the theoretical average (120), but the distribution of the average is ressérée than in the histogram above. The average of the sample thus gives a more precise average of the theoretical average than only one series of observations X. Either $M\_m$, the average of the moyennes.
It is shown that for N series of observations:

1. Average $M\_m$ = average (X) of a series of observations.
2. Standard deviation of $M\_m$ =$\backslash \; sigma\_X/\backslash \; sqrt\; \{N\}$.
3. Variance of $M\_m$ = variance (X)/N.

Theorem of the central limit

In quality control, the majority of the distributions of data of samples follow a normal law. If however the data do not seem “normal”, the Théorème of the central limit makes it possible to affirm that the average of a independent Variable distributed in an unspecified way becomes a normal variable when the number of observations is rather large.

The arithmetic mean $\backslash \; overline\; \{x\_n\}$ of N values according to the same law of probability tightens towards its expectation E (X). The larger N is, the more the law approaches a normal distribution. Its standard deviation is: $\backslash \; tfrac\; \{\backslash \; sigma\_x\}\; \{\backslash \; sqrt\; \{N\}\}$
Therefore, if N is > 5, for example, the distribution of the averages $\backslash \; overline\; \{x\_n\}$ of the various successive samples drawn from this same population will tighten towards a normal law in the same way average m and standard deviation $\backslash \; tfrac\; \{\backslash \; sigma\_x\}\; \{\backslash \; sqrt\; \{N\}\}$, even if the random variable X does not follow a normal law. This property is used in the control charts to measurements.

Test of the khi-deux one

Control charts

The control chart

The Control chart is the principal tool of the SPC. One proposes to trace the control charts to the average and with extended from the example.

Part : metal key.

Characteristic : Length.

Measuring unit : Misters.

Frequency of control : every 2 hours.

By applying the coefficients of calculation of the limits for 5 taking away per sample, one calculates for each line, average X and the extent R by sample, then the average of the averages and the average extent on the whole of the samples. (20 lines)

A2 = 0,577

D4 = 2,115

D3 = 0

For the chart with the average (in top), the limits of control are:

$LCL\; =\; \backslash \; overline\; \{X\}\; -\; A\_2\; \backslash \; overline\; \{R\}\; =132,15$

$UCL\; =\; \backslash \; overline\; \{X\}\; +A\_2\; \backslash \; overline\; \{R\}\; =\; 137,88$

For the chart with wide (in bottom):

$LCL\; =\; D3.\; \backslash \; overline\; \{R\}\; =\; 0$

$UCL\; =\; D4.\; \backslash \; overline\; \{R\}\; =\; 10,57$

The manufactoring process is under control.

Use of the various charts

Optimization of quality

Variance analyzes

Adjustments

Analyzes causes of nonconformity

References

• Maurice Pintail, To apply the statistical control of the processes. MSP/SPC , Eyrolles, 6th edition, 2005
• Gerald Baillargeon (1980). “Introduction to the statistical methods in quality control”. Editions SMG. ISBN 2-89094-008-X
• Elbekkaye Ziane (1993). Control of total quality. Hermes. ISBN 2-86601-362-X
• Kenneth NR. Berk and Jeffrey W.Steagall. Analysis statistics of the data with Student Systat . International Thomson Publishing France. ISBN 2-84180-003-2

See too

• Control chart
• Capability machine
• Capability proceeded

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