August 19, 2015

Six Sigma: Control Phase : 2. Control Chart

Characteristics Types   Variables Attributes Out-of-Control Patterns X-R Charts (when data is readily available and size is small) Run Charts (limited single-point data) XM-MR Charts (limited data -moving average/moving range) X-MR Charts (limited data, I -MR, individual moving range, real data) X-S Charts (when sigma is readily available) Median Charts Short Run Charts p Charts (for defectives -sample size varies) np Charts (for defectives -sample size fixed) c Charts (for defects -sample size fixed) u Charts (for defects -sample size varies) Short Run Varieties of p, np, c and u Charts

What are Control Charts?

Control chart, is a graphical display (chart) of a quality characteristic.

Characteristics of control charts Control Charts.
If a single quality characteristic has been measured or computedfrom a sample, the control chart shows the value of the quality characteristic versus the sample number or versus time. In general, the chart contains a center line that represents the mean value for the in-control process. Two other horizontal lines, called the upper control limit (UCL) and the lower control limit (LCL), arealso shown on the chart. These control limits are chosen so that almost all other data points will fall within these limits as long as the process remains in-control.

Plus or minus “3 sigma” limits are typical

Letting Xdenote the value of a process characteristic, if the system of chance causes generates a variation in Xthat follows the normal distribution, the 0.001 probability limits will be very close tothe 3 limits. From normal tables we glean that the 3 in one direction is 0.00135, or in both directions 0.0027. For normal distributions, therefore, the3 limits are the practical equivalent of 0.001 probability limits.

Types of Charts

There are many variations of possible control charts. The two primary types are:

1. Control Charts for Variables

Plots specific measurements of a process characteristic (temperature, size, weight, sales volume, shipments, etc.).


X-R Charts (when data is readily available) Run Charts (limited single-point data)

XM-MR Charts (limited data -moving average/moving range) X-MR Charts (limited data, I -MR, individual moving range)

X-S Charts (when sigma is readily available) Median Charts Short Run Charts

2. Control Charts for Attributes

Plots general measurement of the total process (the number of complaints per order, number of orders on time, absenteeism frequency, number of errors per letter, etc.). Types:

p Charts (for defectives -sample size varies)

np Charts (for defectives -sample size fixed)

c Charts (for defects -sample size fixed)

u Charts (for defects -sample size varies)

Short Run Varieties of p, np, c and u Charts

Charts for variables are generally most costly since each separate variable (thought to be important) must have data gathered and analyzed. In some cases, the relatively larger sample sizes associated with attribute charts can prove to be more expensive. Often, variable charts are the most valuable and useful because the specific measurement values are known. We will review variable charts first.

X Bar And R chart

If the sample size is relatively small (say equal to or less than 10), we can use the range instead of the standard deviation of a sample to construct control charts on and the range, R. The range of a sample is simply the difference between the largest and smallest observation. There is a statistical relationship (Patnaik, 1946) between the mean range for data from a normal distribution and , the standard deviation of that distribution. This relationship depends only on the sample size,n. The mean of Ris d2, where the value of d2is also a function of n. An estimator ofis therefore R/d2.

The R chart

This chart controls the process variability since the sample range is related to the process standard deviation. The center line of the R chart is the average range.

Typical Out-of-Control Patterns
•Point outside control limits
•Sudden shift in process average
•Hugging the center line
•Hugging the control limits

Out of Control or Nonrandom Trend

1. One or more points more than 3 sigma from the center line

2. Nine points in a row on the same side of center line.

3. Six points in a row, all increasing or decreasing

4. Fourteen points in a row, alternating up and down.

5. Two out of three points more than 2 standard deviations from the centerline (Same side).

6. Four out of five points more than 1 standard deviation from center line (Same side)

7. Fifteen points in a row within 1 standard deviation of centerline (either sides)

8. Eight points in a row more than 1 standard deviation from center line (either side)


If a process is “out-of-control,” then special causes of variation are present in either the average chart or range chart, or both. These special causes must be found and eliminated in order to achieve an in-control process. A process out-of-control is detected on a control chart either by having any points outside the control limits or by unnatural patterns of variability.

Control limits are the boundaries set by the process which alert us to process stability and variability. Remember, our control limits are 3 standard deviations above and below the grand average. If the process is in control, 99.7% of the averages will fall inside these limits.The same is true for the range control limits. Because there are two components to every control chart –the average chart and the range chart –four possible conditions could occur in the process.

Control Chart for Individuals: I and MR (Moving Range)

Charts In today’s world, measurement technology is progressing rapidly, we can measure many process data in real time and 100% monitoring is common. In this case, data are not in subgroups, but in individual pieces. To control Such kind of process, we use individual run chart (I chart) and moving range MR chart to monitor the average and variation in the process.

The data are collected as individual points.

One control chart is I-chart, which is established for ‘individual runs’.The moving ranges, MR (difference between 2 consecutive points) are also calculated and a chart is kept for MR.I-chart is a measure of ‘process mean’, MR is a measure for process variation.Both charts are based on 3-Sigma limits.

Attribute Control Charts

What are Attributes Control Charts?

The Shewhart control chart plots quality characteristics that can be measured and expressed numerically. We measure weight, height, position, thickness, etc. If we cannot represent a particular quality characteristic numerically, or if it is impractical to do so, we then often resort to using a quality characteristic to sort or classify an item that is inspected into one of two “buckets”.

An example of a common quality characteristic classification would be designating units as “conforming units” or “nonconforming units”. Another quality characteristic criteria would be sorting units into “nondefective” and “defective” categories. Quality characteristics of that type arecalled attributes.

Note that there is a difference between “nonconforming to an engineering specification” and “defective” –a nonconforming unit may function just fine and be, in fact, not defective at all, while a part can be “in spec” and not function as desired (i.e., be defective).

Examples of quality characteristics that are attributes are the number of failures in a production run, the proportion of malfunctioning wafers in a lot, the number of people eating in the cafeteria on a given day, etc.

P charts:

The p chart reflected in % The best use of an attribute chart is to:

. Follow trends and cycles

. Evaluate any change in the process Key points to consider when using attribute charts:

. Normally the subgroup size is greater than 50 (for p charts).

. The average number of defects/defectives is equal to 4 or more.

. If the actual p chart subgroup size varies by more than 10 -20% from the average subgroup size, the data point must either be discarded or the control limits calculated for the individual point. With computer generated charts any variation in n should have recalculated limits.

. The most sensitive attribute chart is the p chart. The most sensitive and expensive chart is the X-R

. . The defects and defectives plotted in attribute charts are often categorized in Pareto fashion to determine the vital few. To actually reduce the defect or defective level, a fundamental change in the system is often necessary.


Use of p-Charts
The data are collected in samples, each sample may have unequal number of ‘Inspection unites’.
Each inspection unit can be either classified as ‘pass’or ‘failure’.
For each sample, the ‘rate of pass’, or ‘rate of failure’, p, is calculated.
A control chart will be calculated and kept for , p.
P-chart is based on 3-Sigma limits, it is based on Binomial distribution.

Proportions Control Charts

p is the fraction defective in a lot or population

The proportion or fraction nonconforming (defective) in a population is defined as the ratio of the number of nonconforming items in the population to the total number of items in that population. The item under consideration may have one or more quality characteristics that are inspected simultaneously. If at least one of the characteristics does not conform to standard, the item is classified as nonconforming.

The fraction or proportion can be expressed as a decimal, or, when multiplied by 100, as a percent. The underlying statistical principles for a control chart for proportion nonconforming are based on the binomial distribution.Let us suppose that the production process operates in a stable manner, such that the probability that a given unit will not conform to specifications is p. Furthermore, we assume that successive units produced are independent. Under these conditions, each unit that is produced is a realization ofa Bernoulli random variable with parameter p. If a random sample of n units of product is selected and if Dis the number of units that are nonconforming, the D follows a binomial distribution with parametersn and p

p control charts for lot proportion defective If the true fraction conforming p is known (or a standard value is given), then the center line and control limits of the fraction non conforming control chart is. When the process fraction (proportion) pis not known, it must be estimated from the available data. This is accomplished by selecting mpreliminary samples, each of size n. If there are Di defectives in sample i, the fraction nonconforming in sample.

NP chart deals with same issue as that of p chart, except that the sample size for each run is a constant

Use of c-Charts
Use only when the number of defects per inspection unit can be counted.c stands for the number of defects per inspection unit
–Scratches, chips, dents, or errors per item
–Cracks or faults per unit of distance
–Breaks or Tears per unit of area
–Bacteria or pollutants per unit of volume
–Calls, complaints, failures per unit of time

The data are collected in consecutive inspection units.
For each inspection unit, the number of ‘defects’,c, are counted and recorded
The number of defects, c, are plotted in c-chart.
c-chart is based on 3-Sigma limits, it is based on Poisson distribution.

Counts Control Charts

The literature differentiates between defectand defective, which is the same as differentiating between nonconformity and nonconforming units. This may sound like splitting hairs, but in the interest of clarity let’s try to unravel this man-made mystery.

Consider a wafer with a number of chips on it. The wafer is referred to as an “item of a product”. The chip may be referred to as “a specific point”. There exist certain specifications for the wafers. When a particular wafer (e.g., the item of the product) does not meet at least one of the specifications, it is classified as a nonconforming item.Furthermore, each chip, (e.g., the specific point) at which a specification is not met becomes a defector nonconformity.So, a nonconforming or defective item contains at least one defect or nonconformity. It should be pointed out that a wafer can contain several defects but still be classified as conforming. For example, the defects may be located at noncritical positions on the wafer. If, on theother hand, the number of the so-called “unimportant” defects becomes alarmingly large, an investigation of the production of these wafers is warranted.

Control charts involving counts can be either for the total number of non conformities (defects) for the sample of inspected units, or for the average number of defects per inspection unit.

Poisson approximation for numbers or counts of defects

Let us consider an assembled product such as a microcomputer. The opportunity for the occurrence of any given defect may be quite large. However, the probability of occurrence of a defect in any one arbitrarily chosen spot is likely to be very small. In such a case, the incidence of defects might be modeled by a Poisson distribution. Actually, the Poisson distribution is an approximation of the binomial distribution and applies well in this capacity according to the following rule of thumb: The sample size n should be equal to or larger than 20 and the probability of a single success, p, should be smaller than or equal to .05. If n 100, the approximation is excellent if np is also 10.

The inspection unit

Before the control chart parameters are defined there is one more definition: the inspection unit.We shall count the number of defects that occur in a so-called inspection unit. More often than not, an inspection unit is a single unit or item of product; for example, a wafer. However, sometimes the inspection unit could consist of five wafers, or ten wafers and so on. The size of the inspection units may depend on the recording facility, measuring equipment, operators, etc. Suppose that defects occur in a given inspection unit according to the Poisson distribution, with parameter c(often denoted by np

U-chart is very similar to c-chart, except that several inspection units are inspected and the total number of defects will be divided by the number of inspection units, n.