all control charts

7/25/2019 All Control Charts

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A Guide to Control Charts

Control charts have two general uses in an improvement project. The most common application is as a tool to monitor process

stability and control. A less common, although some might argue more powerful, use of control charts is as an analysis tool. The

descriptions below provide an overview of the different types of control charts to help practitioners identify the best chart for anymonitoring situation, followed by a description of the method for using control charts for analysis.

Identifying Variation

When a process is stable and in control, it displays common cause variation, variation that is inherent to the process. A process is in

control when based on past experience it can be predicted how the process will vary (within limits) in the future. If the process is

unstable, the process displays special cause variation, non-random variation from external factors.

Control charts are simple, robust tools for understanding process variability.

The Four Process States

Processes fall into one of four states: 1) the ideal, 2) the threshold, 3) the brink of chaos and 4) the state of chaos (Figure 1).3

When a process operates in theideal state, that process is in statistical control and produces 100 percent conformance. This

process has proven stability and target performance over time. This process is predictable and its output meets customer

expectations.

A process that is in thethreshold stateis characterized by being in statistical control but still producing the occasional

nonconformance. This type of process will produce a constant level of nonconformances and exhibits low capability. Although

predictable, this process does not consistently meet customer needs.

Thebrink of chaos statereflects a process that is not in statistical control, but also is not producing defects. In other words, the

process is unpredictable, but the outputs of the process still meet customer requirements. The lack of defects leads to a false sense

of security, however, as such a process can produce nonconformances at any moment. It is only a matter of time.

The fourth process state is thestate of chaos. Here, the process is not in statistical control and produces unpredictable levels of

nonconformance.

Figure 1: Four Process States

Every process falls into one of these states at any given time, but will not remain in that state. All processes will migrate toward the

state of chaos. Companies typically begin some type of improvement effort when a process reaches the state of chaos (although

arguably they would be better served to initiate improvement plans at the brink of chaos or threshold state). Control charts are robust

and effective tools to use as part of the strategy used to detect this natural process degradation (Figure 2).3


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Figure 2: Natural Process Degradation

Elements of a Control Chart

There are three main elements of a control chart as shown in Figure 3.

1. A control chart begins with a time series graph.

2. A central line (X) is added as a visual reference for detecting shifts or trends this is also referred to as the process

location.

3. Upper and lower control limits (UCL and LCL) are computed from available data and placed equidistant from the central

line. This is also referred to as process dispersion.

Figure 3: Elements of a Control Chart

Control limits (CLs) ensure time is not wasted looking for unnecessary trouble the goal of any process improvement practitioner

should be to only take action when warranted. Control limits are calculated by:

1. Estimating thestandard deviation, ?, of the sample data

2. Multiplying that number by three

3. Adding (3 x ? to the average) for the UCL and subtracting (3 x ? from the average) for the LCL

Mathematically, the calculation of control limits looks like:

(Note: The hat over the sigma symbol indicates that this is an estimate of standard deviation, not the true population standard

deviation.)

Because control limits are calculated from process data, they are independent of customer expectations or specification limits.

Control rules take advantage of the normal curve in which 68.26 percent of all data is within plus or minus one standard deviation

from the average, 95.44 percent of all data is within plus or minus two standard deviations from the average, and 99.73 percent of
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data will be within plus or minus three standard deviations from the average. As such, data should be normally distributed (or

transformed) when using control charts, or the chart may signal an unexpectedly high rate of false alarms.

Controlled Variation

Controlled variationis characterized by a stable and consistent pattern of variation over time, and is associated with commoncauses. A process operating with controlled variation has an outcome that is predictable within the bounds of the control limits.

Figure 4: Example of Controlled Variation

Uncontrolled Variation

Uncontrolled variationis characterized by variation that changes over time and is associated with special causes. The outcomes of

this process are unpredictable; a customer may be satisfied or unsatisfied given this unpredictability.

Figure 5: Example of Uncontrolled Variation

Please note: process control andprocess capabilityare two different things. A process should be stable and in control before

process capability is assessed.

Figure 6: Relationship of Control Chart to Normal Curve
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Control Charts for Continuous Data

Individuals and Moving Range Chart

The individuals and moving range (I-MR) chart is one of the most commonly used control charts for continuous data; it is applicable

when one data point is collected at each point in time. The I-MR control chart is actually two charts used in tandem (Figure 7).

Together they monitor the process average as well as process variation. With x-axes that are time based, the chart shows a history

of the process.

TheI chartis used to detect trends and shifts in the data, and thus in the process. The individuals chart must have the data time-

ordered; that is, the data must be entered in the sequence in which it was generated. If data is not correctly tracked, trends or shifts

in the process may not be detected and may be incorrectly attributed to random (common cause) variation. There are advanced

control chart analysis techniques that forego the detection of shifts and trends, but before applying these advanced methods, the

data should be plotted and analyzed in time sequence.

TheMR chartshows short-term variability in a process an assessment of the stability of process variation. The moving range is

the difference between consecutive observations. It is expected that the difference between consecutive points is predictable. Points

outside the control limits indicate instability. If there are any out of control points, the special causes must be eliminated.

Once the effect of any out-of-control points is removed from the MR chart, look at the I chart. Be sure to remove the point by

correcting the process not by simply erasing the data point.

Figure 7: Example of Individuals and Moving Range (I-MR) Chart


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The I-MR chart is best used when:

1. The natural subgroup size is unknown.

2. The integrity of the data prevents a clear picture of a logical subgroup.

3. The data is scarce (therefore subgrouping is not yet practical).

4. The natural subgroup needing to be assessed is not yet defined.

Xbar-Range Charts

Another commonly used control chart for continuous data is the Xbar and range (Xbar-R) chart (Figure 8). Like the I-MR chart, it is

comprised of two charts used in tandem. The Xbar-R chart is used when you can rationally collect measurements in subgroups of

between two and 10 observations. Each subgroup is a snapshot of the process at a given point in time. The charts x-axes are time

based, so that the chart shows a history of the process. For this reason, it is important that the data is in time-order.

TheXbar chartis used to evaluate consistency of process averages by plotting the average of each subgroup. It is efficient at

detecting relatively large shifts (typically plus or minus 1.5 ? or larger) in the process average.

TheR chart, on the other hand, plot the ranges of each subgroup. The R chart is used to evaluate the consistency of process

variation. Look at the R chart first; if the R chart is out of control, then the control limits on the Xbar chart are meaningless.

Figure 8: Example of Xbar and Range (Xbar-R) Chart


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Table 1 shows the formulas for calculating control limits. Many software packages do these calculations without much user effort.

(Note: For an I-MR chart, use a sample size,n, of 2.) Notice that the control limits are a function of the average range (Rbar). This

is the technical reason why the R chart needs to be in control before further analysis. If the range is unstable, the control limits will

be inflated, which could cause an errant analysis and subsequent work in the wrong area of the process.

Table 1: Control Limit Calculations

Table 2: Constants for Calculating Control Limits

n(Sample Size) d2 D3 D4

2 1.128 3.268

3 1.693 2.574


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4 2.059 2.282

5 2.326 2.114

6 2.534 2.004

7 2.704 0.076 1.924

8 2.847 0.136 1.864

9 2.970 0.184 1.816

10 3.078 0.223 1.777

11 3.173 0.256 1.744

12 3.258 0.283 1.717

13 3.336 0.307 1.693

14 3.407 0.328 1.672

15 3.472 0.347 1.653

Can these constants be calculated? Yes, based ond2, whered2is a control chart constant that depends on subgroup size.

The I-MR and Xbar-R charts use the relationship of Rbar/d2as the estimate for standard deviation. For sample sizes less than 10,

that estimate is more accurate than the sum of squares estimate. The constant,d2, is dependent on sample size. For this reason

most software packages automatically change from Xbar-R to Xbar-S charts around sample sizes of 10. The difference between

these two charts is simply the estimate of standard deviation.

Control Charts for Discrete Data

c-Chart

Used when identifying the total count of defects per unit (c) that occurred during the sampling period, thec-chart allows the

practitioner to assign each sample more than one defect. This chart is used when the number of samples of each sampling period is

essentially the same.

Figure 9: Example ofc-Chart


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u-Chart

Similar to ac-chart, theu-chart is used to track the total count of defects per unit (u) that occur during the sampling period and can

track a sample having more than one defect. However, unlike ac-chart, au-chart is used when the number of samples of each

sampling period may vary significantly.

Figure 10: Example ofu-Chart

np-Chart

Use annp-chart when identifying the total count of defective units (the unit may have one or more defects) with a constant sampling

size.

Figure 11: Example ofnp-Chart


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p-Chart

Used when each unit can be considered pass or fail no matter the number of defects ap-chart shows the number of tracked

failures (np) divided by the number of total units (n).

Figure 12: Example ofp-Chart

Notice that no discrete control charts have corresponding range charts as with the variable charts. The standard deviation is

estimated from the parameter itself (p,uorc); therefore, a range is not required.

How to Select a Control Chart

Although this article describes a plethora of control charts, there are simple questions a practitioner can ask to find the appropriate

chart for any given use. Figure 13 walks through these questions and directs the user to the appropriate chart.


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Figure 13: How to Select a Control Chart

A number of points may be taken into consideration when identifying the type of control chart to use, such as:

Variables control charts (those that measure variation on a continuous scale) are more sensitive to change than attribute

control charts (those that measure variation on a discrete scale).

Variables charts are useful for processes such as measuring tool wear.

Use an individuals chart when few measurements are available (e.g., when they are infrequent or are particularly costly).

These charts should be used when the natural subgroup is not yet known.

A measure of defective units is found withu andc-charts.

In au-chart, the defects within the unit must be independent of one another, such as with component failures on a printed

circuit board or the number of defects on a billing statement.

Use au-chart for continuous items, such as fabric (e.g., defects per square meter of cloth).

Ac-chart is a useful alternative to a u-chart when there are a lot of possible defects on a unit, but there is only a small

chance of any one defect occurring (e.g., flaws in a roll of material).

When charting proportions,p andnp-charts are useful (e.g., compliance rates or process yields).

Subgrouping: Control Charts as a Tool for Analysis

Subgrouping is the method for using control charts as an analysis tool. The concept of subgrouping is one of the most important

components of the control chart method. The technique organizes data from the process to show the greatest similarity among the

data in each subgroup and the greatest difference among the data in different subgroups.

The aim of subgrouping is to include only common causes of variation within subgroups and to have all special causes of variation

occur among subgroups. When the within-group and between-group variation is understood, the number of potential variables that

is, the number of potential sources of unacceptable variation is reduced considerably, and where to expend improvement efforts

can more easily be determined.

Within-subgroup Variation

For each subgroup, the within variation is represented by the range.


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Figure 14: Within Subgroup Variation

The R chart displays change in the within subgroup dispersion of the process and answers the question: Is the variation within

subgroups consistent? If the range chart is out of control, the system is not stable. It tells you that you need to look for the source of

the instability, such as poor measurement repeatability. Analytically it is important because the control limits in the X chart are a

function of R-bar. If the range chart is out of control then R-bar is inflated as are the control limit. This could increase the likelihood of

calling between subgroup variation within subgroup variation and send you off working on the wrong area.

Within variation is consistent when the R chart and thus the process it represents is in control. The R chart must be in control to

draw the Xbar chart.

Figure 15: Example of R Chart

Between Subgroup Variation

Between-subgroup variation is represented by the difference in subgroup averages.

Figure 16: Between Subgroup Variation

Xbar Chart, Take Two

The Xbar chart shows any changes in the average value of the process and answers the question: Is the variation between the

averages of the subgroups more than the variation within the subgroup?


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If the Xbar chart is in control, the variation between is lower than the variation within. If the Xbar chart is not in control, the

variation between is greater than the variation within.

Figure 17: Xbar Chart Within Variation

This is close to being a graphical analysis of variance (ANOVA). The between and within analyses provide a helpful graphical

representation while also providing the ability to assess stability that ANOVA lacks. Using this analysis along with ANOVA is a

powerful combination.

Conclusion

Knowing which control chart to use in a given situation will assure accurate monitoring of process stability. It will eliminate erroneous

results and wasted effort, focusing attention on the true opportunities for meaningful improvement.

Examples:

Statistical Process Control (SPC) Analysis: Control Charts

What

One major challenging in maintaining service quality is identifying when quality has

drifted away from that which is acceptable. When a service is heterogeneous, each

incident of service can present dierent standards of quality. Further, these quality

standards can be very subjective, measured by the opinions of the customers.

When service quality standards (a are e!pected to be consistent over time, and (b

are quantitatively measured, then statistical techniques can be used to help identify

service quality drift. "!amples of quality measurements and standards that may #t

this criteria include$

transaction error rates at ban%s

on&time departure for airlines

recurrence rates for repair services

scaled service ratings on customer comment cards


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'f we loo% at the quality measurements over time we may #nd that they may be

mostly consistent, with only occasional random variation. 'n other situations, the

measurements may change with a pattern that results from a fundamental aw in

the service delivery process. )nfortunately, it can be quite di*cult to tell the

dierence between a aw in the service delivery process and a mere random

variation.

For e!ample, imagine a hotel that measures customer satisfaction with a survey

that records a rating from + (worst quality to + (best quality. "very wee% the

hotel calculates average ratings for the wee%. -he manager desires to %now if

quality is consistent or changing over time. 'magine that wee%ly ratings for a series

of wee%s are ., .+, ./, /.0, .0, .1, and 2.3. -he manager is concerned about

the last wee%4s rating of 2.3. 5oes it represent a statistical anomaly, or is it simply

a result of random variation. -he fact is, customers assign dierent meaning to

dierent points on the scale, and quality could have actually improved that wee%

but the customers were simply frugal in giving high ratings. 6o, one thing the

manager may want to %now is if that 2.3 #ts the random pattern of the other data,

or if it does not.

How

One way to investigate this is with Statistical Process Control, or 678. 678 allows us

to ma%e assumptions about data to help separate simple random variation9also

called natural variation9from variation that is caused by changes in the process9

called assignable variation.

-he foundation of 6tatistical 7rocess 8ontrol is the central limit theorem, which

states that the sum (or average of a number of measurements from any single

given probability distribution will appro!imate a normal distribution. -his holds trueregardless of the distribution of the individual measurements. :ow close the sums

(or averages appro!imates the normal distribution depends on how many

measurements are in a sample, which number is called the sample size.

-he two general types of quantitative quality measures

are attributes and variables. ;n attribute is dichotomous, meaning that it ta%es on

the values of


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variation. One way to analy>e the control of a process is with a control chart. -o

test for statistical control of the variable mean we would use an !&bar control chart.

-he term !&bar, denoted , represents the mean of a sample of measurements.

"ach measurement in a sample is anXvalue, i.e.X+,X0,X?, etc. 'f the sample si>e

is n, then nof theXvalue measurements are averaged to calculate an !&bar value.(-he bar denotes e is #ve. -he sample si>e is NOTseven, which is the

number of samples.

-he !&bar values are calculated as the mean of the measurements in each sample.

-he column labeled B contains the sample range values. -he range of a sample is

simply the ma!imum measurement in the sample minus the minimum

measurement in the sample. -hese B values will be used later.

We can calculate !&bar&bar to be C1.3? as the average of the !&bar values. Of

course, we would e!pect some of the !&bar values to be above the !&bar&bar value,

and some to be below it. 'f there is just natural variation, then we would e!pect a

random distribution above and below the mean. 'f we plot the data we see the

following$


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We observe that the #rst three points are above the mean line, but that otherwise,

the data seems to be randomly dispersed above and below the mean. -he

movement around the mean line in this e!ample appears to be natural variation.

;nother thing it would be good to %now about the !&bar values is how dispersed

they are. 'n particular, we would li%e to %now if a particular !&bar value was outside

of a reasonable range. -he !&bar values are going to vary somewhat simply due to

natural variation. )nusual variation may indication that there is a special cause, or

a speci#c reason, for variation. -hat special cause may be a problem that needs to

be addressed.

6tatistically, we can calculate a range of reasonable variation in the !&bar chart.

-hat reasonable range is bounded by control limits. -heupper control limit()8E

indicates the ma!imum value that is statistically reasonable, and the lower control

limit(E8E indicates the minimum reasonable value. y


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where G!is the standard deviation of individual measurements and n is the sample

si>e. -he control limits are calculated by the following equations$

and

where the > value is the number of standard deviations (sigmas from the mean to

put the control limits. ; H?, which considers

measurements within three standard deviations of the mean to be


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cause. (6ince only .? percent of truly random values will be outside of the three&

sigma control limits. -herefore, it would be good to investigate the situation

further.

'f we do not have the standard deviation of measurements or sample means, we

can use the B (range values to estimate the standard deviation by using tables that

contain control limit factors. -he following is a three&sigma factor table.

Three-sigma Factor Table

sample

si>e ;&factor &factor 8&factor

0 +. ?.03

? +.0? 0.12C

C .20/ 0.00

1 .122 0.++C

3 .C? 0.C

2 .C+/ .23 +./0C

.?2? .+?3 +.3C

/ .??2 .+C +.+3

+ .? .0?? +.222

(6ource$ Quality Control Handbook,

K.A. Kuran, editor. Dew Lor%$

AcMraw&:ill, +/2/.

-he ;&factor from the table pertains to !&bar charts. -he other two factors will be

used in a dierent chart. (6ome te!t boo%s call these factors by dierent names,

such as ;0, 5?, and 5C. -he sample si>e is the number of measurement in each

sample. e careful to not confuse the sample si>e with the number of samples. -he

sample si>e is notthe number of samples in the chartN 't is the number of

measurements within each sample.

;lthough you can construct a control chart with a sample si>e of 0, it is probably not

the best. -he bigger the sample si>e, the more statistically representative will be

your control chart values. ; sample si>e of 1 or 3 is probably o%ay. Earger sample

si>es are the best, but they often ta%e more eort to gather.

-he upper control limit ()8E for an !&bar chart is calculated as


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where r&bar is calculated as the average of the sample range values, as discussed

previously. -he lower control limit (E8E for an !&bar chart is similarly calculated as

6ince the factors were from a three&sigma factor table, we e!pect that three

standard deviations of !&bar values will be between the E8E and the )8E. -he

central limit theorem shows that the !&bar values tend to follow a normal

distribution. -hree standard deviations from the mean of normal distribution

includes //.2 percent of the probability density. -his means that //.2 percent of the

time, a random number drawn from a normal distribution will be within three

standard deviations from the mean. 't is quite unli%ely that values would fall

outside of that range on a regular basis.

From the data above, we can calculate control limits for the e!ample !&bar chart.

We have nH1, @&bar&barHC1.3?, B&barH1.C?, and ;H.122. -herefore we have

E8EHC1.3?9.1221.C?HC0.1 and )8EH C1.3?I.1221.C?HC.23 . Dote that

these control limits are almost the same as were calculated above using the

standard deviation of measurements.

R-charts

;n !&bar chart tells us if the central tendency (i.e. mean of the samples appears to

be in control over time. ;nother chart, the R-chart, tells us if the variance within

each sample tends to be in control over time. 't is certainly possible for the sample

means to be in control over time, yet the sample variance is getting worse and

worse.

;n B&chart is created similar to how we create an !&bar chart e!cept for the

following$

P the points which are plotted are the B, or range, values calculated above.

P the central line is the B&bar value, which is the mean of the B values.

P the upper control limit, E8EB, is simply the &factor times B&bar.

P the lower control limit, )8EB, is simply the 8&factor times B&bar.

;gain, since we are using three&sigma factors, we would e!pect the B values to fall

within the control limits //.2 percent of the time.

For the data from the e!ample above we calculate )8EBH0.++C1.C?H++.C, and

E8EBH1.C?H. Dote that lower control limits for B&charts are bounded by >ero,

since it is impossible to have B values less than >ero.


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-he control chart is as follows$

We see that in period 2 the range of values appears to have gone up beyond what is

statistically common. 't is interesting to note that the !&bar value for that sample

was #ne, meaning that on average the measurements in the sample were incontrol. :owever we had more variance in that sample that we might have usually

e!pected, so we should investigate for special causes or changes in the process.

5o we care if an B value is at or below the E8EQ (;ssuming we have a non&>ero

E8E 't seems that less variance would be better. 'n fact it might be good to

investigate improvementsin the process so that they can be assured to continue in

the future.

P-charts and C-charts

De!t we will loo% at control charts for attributes, which are not continuous variables

but are things that can be counted. ;p-chartconsiders the portion of a sample thatis defective, where each item in the sample is either defective or not. For e!ample,

an airline might trac% on&time arrivals of ights on each given day. 'f there are 1

ights being trac%ed each day (the sample si>e, we could determine a p value as

the portion of ights that arrive late. On a given day if + ights arrive late then the

p value is +R1H.0.


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't loo%s li%e the p values are well within the control limits, with no unusual patterns.

We might conclude that the late arrival process, while not ideal, is in statistical

control.

Finally, a c-chartconsiders attributes that can be counted, without a speci#c sample

si>e. 'nstead we have a sample frame which de#nes a range of defects to be

counted. For e!ample, the airline might count the number of complaint letters that

come in day to day. -here is no sample si>e since any number of letters could

arrive (and an individual might even send multiple letters.

-he plot values for a c&chart are the c values, which are the counts for each sample

frame. -he center line is c&bar, the average of a series of c values. We often

assume that the count values follow a 7oisson distribution, which has the standard

deviation as follows$

-he control limits for a c&chart are simply$

and

For e!ample, if the airline might count the number of complaint letters that arrive

over the past seven days as follows$

day c

+ 1

0 +1

? ++


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C +

1 1

3 0

2 +C

c&bar +.

)sing the equation above, GcHsqrt(+H?.+30. For a three&sigma control chart

(>H? we calculate E8EH.1+? and )8EH+/.C2. ;s with p&charts, the c values will

never be negative, so if the E8E computes negative then use >ero. :ere is a c&chart

for this data$

-his data is in statistical control. Dote, however, that if we used two&sigma control

limits (>H0 then points would be outside of the control limits, suggesting the counts

of complaints is not in statistical control.

Summary of Things to oo! For

What are we loo%ing for in a control chart that might cause us to suspect the

process has changedQ -he following are some e!amples$

;n unusual tendency for the sample values to be above or below the !&bar&

bar or B&bar line.


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Salues that appear outside of the control limits. 'f three&sigma control limits

are used, then it is quite unusual for even one value to appear outside of the

control limits.

Other peculiar patterns in the data, such as erratic uctuations above and

below the mean, or patterns that repeat over a #!ed number of samples.

;gain, with each of these occurrences we investigatefor special causes. 6uch

patterns of behavior can happen with mere natural variation, but they are not li%ely.

all control charts

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