Control chart

A control chart is a tool making it possible to determine the moment when appears the assignable cause involving the drift. Thus, the process will be stopped at the good moment, i.e. before it produces parts nonin conformity (out of the interval of Tolerance).

The most used control charts are the control charts by measurement of the average and the extent. These charts are established together and interpreted together. Indeed, the distribution of dimensions manufactured modelled by a law Normale is characterized by the average and dispersion (standard deviation).

SPC and quality control

The creator of the control chart is Walter A. Shewhart which worked on the Beautiful Telephone Laboratory of the Electric Western. Shewart published into 1931 the principles of the variability of a process by distinguishing natural random variability and accidental variability. natural variability is resulting from “common causes of dispersion” or “normal disturbances” integrated in the manufacturing process “under control”. Accidental variability must with occasional and uncontrolled “special causes” (raw materials with the fluctuating characteristics, machines badly regulated, different work hours, qualification of labor, changes of temperature or pressure, bad lubrication…). The SPC has the role of determining if the process is under control or not. A more detailed analysis of the causes of the variations will make it possible to improve its performances and its regularity. The control charts are a graphic tool for visualization of the manufacturing process in time and of description of its stability (monitoring of the special causes).

The control chart

They make it possible to carry out a convenient adjustment of the manufactoring process and to know its Capabilité machine. This tool is presented as a graph whose points represent the follow-up in the time of a characteristic of the process whose central value (often the average) is represented by an horizontal line as well as the limits lower (LCL) and higher (UCL) (UCL: Upper Limit Control, LCL: Lower Limit Control).

These two values are the limits inside of which the process is under control. The values of the controlled characteristic must be inside these limits, if not these values are “except control” and must be examined.

Various control charts

Two control charts are necessary to supervise the position and dispersion.

Charts for quantitative variables

The quantitative variables are continuous measurements (weight, length, thickness, temperature, diameter.). One checks on the control chart of the average (mean chart) or on the chart of wide (chart arranges) that the studied character will be stable in time. The sample size is from 4 to 6.

Charts for qualitative variables

To measure qualitative variables (% the defective one, ¨% of breakdowns…), one makes use of charts to the attributes p, Np or C to control the attributes in time. The size of sample is about 50 to 100.

Extent and control chart to the average

These two parameters are independent and complementary. The median value can vary without dispersion not varying and conversely.

These control charts make it possible to visualize the evolution of the dispersion of dimensions manufactured. The control chart of the standard deviation is " more juste" and less dispersed than the control chart of the extent. On the other hand, calculations has to realize to trace the control chart of the extent are less complicated and thus more reliable. The control chart of the standard deviation will be implemented if the layout of the chart is computerized or assisted by an automatic computational tool. The control chart of the extent will be in work if the layout of the control chart is manual on a paper medium.

Goal

This control chart makes it possible to visualize the evolution and the variation of the median value of dimensions manufactured. This control chart is traced by successive points representing the median value of sample taken with regular intervals.

The goal is to compare the average performances of production in time using a chart which characterizes the tendency of the central value. One carries out several individual observations on several sub-groups numbered at a frequency of time given (every hour, three times per day…). On each sub-group K chronological one carries out N observations. One defers on the chart of average the average of the sub-group according to his chronological number which will be deferred on the horizontal axis of the control charts. Because of the Théorème limits central, the average of the values on the control chart follows a normal Loi that the observations are normally distributed or not.
This law is valid even for samples of small size, which is frequent of quality control. A production will be known as “stable”, if the tendency and dispersion are statistically constant in time. The control chart to the average supervises the adjustment of the process, the chart of wide dispersions.

Limits of control

The standard deviation is known

The calculation of the limits of the chart of average differs according to whether the standard deviation is known or not.
When the standard deviation of the process is known, the characteristics of the normal distribution make it possible to calculate the limits of control. The interval -3 \ sigma; - \ driven + 3 \ sigma, contains 99.7% of the data and represents limits LCL and UCL of the control charts. The probability so that a point is outside the limits is thus 0,3%. The value of the average differs appreciably according to whether the standard deviation of the population is known or not. If the true value of the average µ of the observations is unknown, one can replace it by the average by sample $\ overline {X}$

$LCL = \ overline {X} - \ frac {3 \ sigma} {\ sqrt {N}}$

$UCL = \ overline {X} + \ frac {3 \ sigma} {\ sqrt {N}}$

94,74% of the points must be between the limits if the process is under control. The interpretation of the chart of the average is the same one if the true average of the process is known or unknown.

The standard deviation is unknown

In much of case, the standard deviation $\ sigma$ is unknown and must be estimated per T (Loi of Student) instead of the standard normal law. One makes use then of a A2 coefficient which depends on the number of observations in each sample and on the type of chart used. The limits of control are estimated by using the average extent of the observations inside a sub-group like measures variability.

$LCL = \ overline {X} - A_2 \ overline {R}$

$UCL = \ overline {X} +A_2 \ overline {R}$

with:
$\ overline {R}$ : extended average in the échantillon.
Elle is equal to the average of wide (maximum value of the sub-group - minimal value) of each sub-group. The limits are tightened when the size of the sub-group of the samples increases.

an example

The Emballex company manufactures metal boxes which one controls the weight at the exit of manufacture. This characteristic depends primarily on the composition of the raw materials and the quality of alloy carried out. The company decides to control the manufactoring process using control charts for the average weight of the parts and the extent of the weight for each sampling carried out. Control consists in taking 5 boxes on the outlet side of the machine and to weigh each box. the weight expressed in grams. The weights were recorded on a series of 20 taken samples.

To trace the control charts for charts X and R.

The average weight is 84,5 g.
The higher limit is 87,81 g.
The lower limit is 81,18 g.
The manufactoring process is under control.

Chart with the extent

Control charts of the wide one: This control chart makes it possible to visualize the evolution and of the variation of wide (image of dispersion) of dimensions manufactured. This control chart is traced by successive points representing the extent of taken samples with regular intervals.

The limits of control are the following ones:

$LCL = D3. \ overline {R}$

$UCL = D4. \ overline {R}$

with:
$\ overline {R}$ : extended average in the sample = R1 +R2 +… +Rk/k.
Wide IH of sub-group I. K: many sub-groups. The average of the extents indicates the importance of the natural variability of the process. For the example above, the average extent is 6,750 g.
The lower limit is: 0 g.
The higher limit is: 14,28 g.
The manufactoring process is under contrôle.

Coefficients

These coefficients are used to calculate the limits of control according to sample and the type size of chart used.

Control charts to the attributes

If one does not wish to carry out a control of variables by measurements, or if that is not possible, one will prefer to it quality control by attributes which consists in noting the presence or the absence of a qualitative criterion on the controlled parts. Examples: visual monitoring (absence of defect or not), too small or too large dimension (work clearance in a gauge)….
The principal control charts to the attributes are:
1- The chart p for the proportion of défectueux.
2- The chart Np for the follow-up of the number of défectueux.
3- The chart C for the follow-up of the number of defects.

Chart p

Principle

One uses this chart to follow the proportion p of defective contents in a sample coming from a batch or of a machine. On a taking away randomly, with regular interval, of a sample of N parts, one notes the number of defective found that one divides by the manpower of sample N (N > 50) to obtain p.
p = many defective pieces/many parts in the sample = D/n

One periodically defers p on a control chart where one will reveal the average of proportion of defective and lower and higher limits correspondantes.
The average proportion the defective one on the whole of the taking away is: $\ overline {p} = \ Sigma {D}/\ Sigma {N}$
The limits of control are at 3 standard deviations on each side of the average proportion.

$LIC = \ overline {p} - 3 \ sqrt {\ frac {\ overline {p} (1 - \ overline {p})}{N}}$

$LSC = \ overline {p} + 3 \ sqrt {\ frac {\ overline {p} (1 - \ overline {p})}{N}}$

The law which governs the control chart p is the Binomial distribution: observation of D defective pieces (two methods) on a taking away randomly of N parts. When N is large, an approximation by the normal law is legitimate. The interval -3 \ sigma; - \ driven + 3 \ sigma, then contains 99.7% of the data contained within limits LCL and UCL of the control charts.

Example

Company GLASSEX manufactures plastic handles used in the furnishing of the kitchens and bathrooms. One carries out a visual monitoring every hour on a sample of 65 handles to detect the principal defects: scratches, doubtful color, microscopic cracks, etc… in order to follow the rate of defective in the manufactoring process. The results on 28 series of samples are mentioned in table ci-contre.
One obtains an average for p of 0,079 with an average standard deviation of 0,031 of défectueux.
The higher limit of the chart p is 0.179.

Chart Np

The use of the chart Np is recommended if manpower N of the sample remains the same one for each series of samples. One defers on the chart Np the number of defective observed chronologically in the successive taking away. The number of defective in a sample of size N is D = Np. For K taking away of manpower N, the median number of defective is: $n \ overline {p} = \ Sigma {D}/K$
with
$\ overline {p} = \ Sigma {D}/n.k$
One can calculate the limits of control based on intervals of three standard deviations around the median number of défectueux.
$LIC = N \ overline {p} - 3 \ sqrt {N \ overline {p} (1 - \ overline {p})}$
$LSC = N \ overline {p} + 3 \ sqrt {N \ overline {p} (1 - \ overline {p})}$

Chart C

The chart C is used chronologically to follow the number of defects per controlled unit (100 meters of cable, 20 meters of wallpaper roller,…). It is different from the charts p and Np, because the followed criterion is the number of defects and not the number of defective (refused), a part presenting of the defects being able to be or not accepted. According to the criterion of gravity of the defect (critical, major or minor), the part will be or not regarded as defective. The median number of defects observed on K controlled units is: $\ overline {C} = \ Sigma {c_i}/K$
$c_i$ is the number of defects observed for I ième controlled unit. The limits of control are thus:
$LIC = \ overline {C} - 3 \ sqrt {\ overline {C}}$
$LSC = \ overline {C} + 3 \ sqrt {\ overline {C}}$

See too

Management of statistical quality
Control of the processes

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