Control charts monitor process variation over time to distinguish between common and special causes. Information from control charts helps improve processes by addressing special causes. Control charts aim to prevent errors like treating common variation as special causes or vice versa. When a process is in control, its control chart will not indicate any out-of-control conditions and it will contain only common causes. If common variation is too large, the process must be altered. Different control charts like p, R, x-bar, C, and capability indices like Cp, CPL, CPU, Cpk are used to monitor different quality characteristics.

1. Quality Management
- A control chart monitors variation in a characteristic of a product or service
overtime.
- Information gained from analyzing a control chart forms the basis for process
improvement.
- A principal benefit of a control chart is the attempt to separate special causes
of variation from common causes of variation.
Causes of variation
Special Common
- Represent large fluctuations or
patterns in the data that are not
part of a process.
- They represent problems that can
be corrected without the change
of the system.
- Also, they are called assignable
causes of variations.
- Inherent variability that exits in a
process.
- They represent small causes of
variability that operate randomly,
and can be reduced by changing
the system.
- Also, they are called chance
causes of variations.
- Note: The systematic changes are the responsibility of the management (i.e.
not the workers involved in the process).
- Control charts help to prevent 2 types of errors
Type of error Details
First - Treating common cause
variation as a special cause.
- Results in over adjusting the
process (i.e. tampering).
- Tampering increases the
variation of the process.
Second - Treating special causes
variation as common cause
variation.
- Results in not taking immediate
corrective action as necessary
- To construct a control chart, you collect samples from the output of a process
overtime (i.e. subgroups).
- For each subgroup, you calculate the value of a statistic associated with a CTQ
variable (i.e. critical_ to_ quality variable).

2. - Common used statistics include sample proportion for categorical variables,
and the mean & range of a numerical variable.
- Plot the values over time and add control limits to the chart where
Upper control limit(UCL)=Process mean+3 * standard deviation
Lower control limit(LCL) =Process mean –3*standard deviation
- You evaluate the control chart by trying to find
i- Any pattern that might exist in the values over time.
ii- Whether any points fall outside the control limits.
PanelA
- Process
stable.
- Contains
common
causes only.
PanelB
- Special
causes exist
PanelC
- Consecutive
points above
the mean
line
- Consecutive
points below
the mean
line
- Two other simple rules allow you to detect a shift in the mean (i.e. trend
detection)
i- Eight or more consecutive points that lie above/below the
center line.
ii- Eight or more consecutive points that move upward/downward
in value.
- A process whose control chart indicates an out_of_control condition (i.e. a
point outside the limits and/or exhibit a trend) is said to be out of control.
- An out of control process includes common and special causes of variation.
- A process whose control chart does not indicate any out_of_control condition
is said to be in control.
- An in control process contains only common causes of variation.

3. - When the process is an in_ control process, you must determine whether the
amount of common cause variation in the process is small enough to satisfy
the customers of the product or service.
- If common variations are
i- Small: use chart continuously to monitor the process and to
make sure it remains in control.
ii- Large: alter the process. These alterations are the responsibility
of the management.
Control chart for the proportion (p chart)
- Used for categorical variables.
- Used to monitor and analyze the proportion of nonconforming items in
repeated samples selected from a process.
𝑃̅ ± 3√
𝑃̅(1 − 𝑃̅)
𝑛̅
Where
- 𝑛̅ =
∑ 𝑛 𝑖
𝑘
1
𝑘
- 𝑃̅ =
∑ 𝑥𝑖
𝑘
1
∑ 𝑛 𝑖
𝑘
1
- 𝑥𝑖 =⋕ of nonconforming items in subgroup i.
- 𝑛𝑖 = size for subgroup i.
- 𝑘 =⋕ of subgroups selected.
- 𝑛̅ = mean subgroup size.
- 𝑃̅ =average proportion of nonconforming items.
Note: any negative value of LCL means that the LCL does not exist.

4. Control charts for the range and the mean
- For numerical variables such as money, weight, etc.
- Charts are used in pairs (I.e. range chart for dispersion or variability and mean
chart for the process mean).
- Range chart is examined first. If it indicated out of control then the
interpretation of mean chart is misleading.
R chart
- used when sample size ≤10 (i.e we consider the range of the sample/subgroup
data as a measure of dispersion). This is the most common case.
- If sample size>10, we consider the standard deviation of the sample/subgroup
data as a measure of dispersion.
- Control limits for the range
UCL =𝐷4 𝑅̅
LCL=𝐷3 𝑅̅
Where
𝑅̅ =
∑ 𝑅𝑖
𝑘
1
𝑘
k=⋕ of subgroups , and 𝐷3 & 𝐷4 are obtained from the given
table.
𝑥̅ chart
- Control limits for the 𝑥̅ chart
UCL= 𝑥̿ + 𝐴2 𝑅̅
LCL= 𝑥̿ - 𝐴2 𝑅̅
Where we get 𝐴2 from the given table, k=⋕ of subgroups, and 𝑥̿ =
𝛴 𝑖=1
𝑘
𝑥̅
𝑘
.

5. The C Chart
- Nonconformities are defects, faults, errors, or imperfections in a product or a
service.
- An area of an opportunity is an individual unit of a product or service, a unit
of time, unit of space, or unit of area.
- To monitor and analyze number of nonconformities in an area of opportunity,
use C chart.
- Examples of number of nonconformities in an area of opportunity may be as
follows:
a- The number of typographical errors on a printed page.
b- The number of hotel customers filing complaints in a given week.
- C chart process assumes Poission distribution.
Control limits for the C chart
𝐶̅±3√𝐶̅
UCL=𝐶̅+3√𝐶̅
LCL=𝐶̅-3√𝐶̅
𝐶̅=
∑ 𝑐 𝑖
𝑘
𝑖=1
𝑘
k= number of units sampled
ci= number of nonconformities in unit i
-

6. Process capability
- A customer who believes that a product or service has met or exceeded his or
her expectations will be satisfied. Hence, quality is defined by customers.
- Specification limits are technical requirements set by management in response
to customers' needs and expectations.
- The upper/lower specification limit (USL/LSL) is the largest/smallest value a
CTQ can have and still conform to customers expectations.
- Hence, the process capability is the ability of a process to consistently meet
specified customer driven requirements. This can be done by estimating the
percentage of products or services that are within specifications.
P(an outcome will be within specification) = P( LSL<x<USL)
=P(
𝐿𝑆𝐿−𝑥̿
𝑅̅
𝑑2
< Z <
𝑈𝑆𝐿−𝑥̿
𝑅̅
𝑑2
)
Where
𝑑2 is obtained from the given table, and
We assumed normal distribution for the x values.
Capability index
- 𝐶 𝑝=
𝑈𝑆𝐿−𝐿𝑆𝐿
6(
𝑅̅
𝑑2
)
- Larger 𝐶 𝑝 means better capability of the process.
- 𝐶 𝑝is a measure of process potential, and not the actual performance.
- To measure the capability of the process in terms of actual process
performance use CPL, CPU, and 𝐶 𝑝𝑘
Where
CPL=
𝑥̿− 𝐿𝑆𝐿
3(
𝑅̅
𝑑2
)
CPU==
𝑈𝑆𝐿−𝑥̿
3(
𝑅̅
𝑑2
)
𝐶 𝑝𝑘=min(CPL,CPU)
The last measures the actual process performance