Control charts are statistical tools used to monitor processes and distinguish between common and special cause variations. They graphically display process stability over time and can provide early warnings if a process becomes out of control. The X-bar and R chart is used for variables data with subgroup sizes of 2-15. It involves calculating the mean and range for each subgroup, then determining control limits based on the grand mean and average range. Patterns outside the control limits or showing trends over time indicate the process may need investigation.

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UNIT 3 :UNIT 3 :
CONTROL CHART FOR VARIABLES
© Mechanical Engineering Department

2. LOGOOUTLINEOUTLINE
 Introduction
 Variation
 Purpose of Control Charts
 Patterns in Control Charts
 X – R Charts
 X – S Charts
 Process Capability

3. LOGOINTRODUCTIONINTRODUCTION
CONTROL CHART
“It is a statistical tool used to distinguish between
variation in a process resulting from common cause
& special cause.
Its presents graphical display of process stability or
instability over a time.”
OR
Control charts are particularly useful for monitoring
quality and giving early warnings that a process
may be going “Out of Control” and on its way to
producing defective parts.

4. LOGOVARIATION
“The variation concept is a law of
nature in that no two natural items
in any category are the same”

5. LOGOVARIATION
The variation may be quite large and easily
noticeable
The variation may be very small. It may
appear that items are identical; however,
precision instruments will show difference
The ability to measure variation is necessary
before it can be controlled

6. LOGOVARIATION
There are three categories of variation in piece
part production:
1. Within-piece variation: Surface
2. Piece-to-piece variation: Among pieces
produced at the same time
3. Time-to-time variation: Difference in product
produced at different times of the day

7. LOGOVARIATION
Materials
Tools
Operators
Methods Measurement
Instruments
Human
Inspection
Performance
EnvironmentMachines
Sources of Variation in production processes:
INPUTSINPUTS PROCESSPROCESS OUTPUTSOUTPUTS

8. LOGOPURPOSE OF CONTROL CHART
Monitor process variation over a time
Differentiate between special cause and common cause
variations
Assess the effectiveness of changes to improve a
process
Communicate how a process performed during specific
period
Improve productivity of process
Prevent defects in process
Prevent unnecessary process adjustments
Collect diagnostic information
Collect information about process capability

9. LOGOTYPES OF CONTROL CHARTS
There are two main categories of Control Charts, those that
display attribute data,and those that display variables
data.
Attribute Data: This category of Control Chart displays
data that result from counting the number of occurrences or
items in a single category of similar items or occurrences.
These “count” data may be expressed as pass/fail, yes/no,
or presence/absence of a defect.
Variables Data: This category of Control Chart displays
values resulting from the measurement of a continuous
variable. Examples of variables data are elapsed time,
temperature, and radiation dose.

10. LOGOTYPES OF CONTROL CHARTS
Variable data
x-bar and R-charts
x-bar and s-charts
Charts for individuals (x-charts)
Attribute data
For “defectives” (p-chart, np-chart)
For “defects” (c-chart, u-chart)

11. LOGOTYPES OF CONTROL CHARTS
Continuous
Numerical Data
Categorical or Discrete
Numerical Data

12. LOGOCONTROL CHARTS ELEMENTS
1. Title. The title
briefly describes
the information
which is displayed.
2. Legend. This is
information on how
and when the data
were collected.
3. Data Collection
Section. The
counts or
measurements are
recorded in the
data collection
section.

13. LOGOCONTROL CHARTS ELEMENTS
4a. The Upper Graph
4b. The Lower Graph
5. Y-Axis. This axis
reflects the magnitude
of the data collected.
6. X-Axis. This axis
displays the
chronological order in
which the data were
collected.
7. Control Limits (CL).
Control limits are set
at a distance of 3
sigma above and 3
sigma below the
centerline.
8. Centerline. This line
is drawn at the
average or mean
value of all the plotted
data.

14. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
The X-Bar and R Control Chart is used with variables
data when subgroup or sample size is between 2 and
15. The steps for constructing this type of Control
Chart are:
Step 1 - Determine the data to be collected. Decide
what questions about the process you plan to answer.
Refer to the Data Collection module for information on
how this is done.

15. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
Step 2
•Collect and enter the
data by subgroup.
•A subgroup is made
up of variables data
that represent a
characteristic of a
product produced by a
process.
•The sample size
relates to how large
the subgroups are.

16. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
Step 3
•Calculate and enter
the average for each
subgroup.
•Use the formula
below to calculate the
average (mean) for
each subgroup
n
xxxx n.....21 ++=

17. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
Step 4
•Calculate and enter
the range for each
subgroup.
•Use the following
formula to calculate
the range (R) for
each subgroup.
valuevalueR minmax −=

18. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
Step 5 - Calculate the grand mean of the subgroup’s
average. The grand mean of the subgroup’s
average (X-Bar) becomes the centerline for
the upper plot.
Where:
subgroupsofnumberthek
averagessubgroupeachforaveragethex
averagessubgroupindividualallofmeangrandthex
=
=
=
k
xxxxx n.......321 +++=

19. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
Step 6 - Calculate the average of the subgroup
ranges. The average of all subgroups
becomes the centerline for the lower plotting
area.
subgroupsofnumberthek
subgroupeachforrangeindividualtheR
subgroupsallforrangestheofaveragetheR
i
=
=
=
k
RRRR n.........21
++=

20. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
Step 7 - Calculate the upper control limit (UCL) and lower
control limit (LCL) for the averages of the subgroups.
To find the X-Bar control limits, use the following
formula:
NOTE: Constants, based on the subgroup size (n), are used in determining
control limits for variables charts.
RAXLCL
RAXUCL
x
x
2
2
−=
+=
n A2 n A2 n A2
2 1.880 7 0.419 12 0.266
3 1.023 8 0.373 13 0.249
4 0.729 9 0.337 14 0.235
5 0.577 10 0.308 15 0.223
6 0.483 11 0.285

21. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
Step 8 - Calculate the upper control limit for the ranges. When
the subgroup or sample size (n) is less than 7, there
is no lower control limit. To find the upper control limit
for the ranges, use the formula:
NOTE: Constants, based on the subgroup size (n), are used in determining
control limits for variables charts.
n D4 n D4 n D4
2 3.267 7 1.924 12 1.717
3 2.574 8 1.864 13 1.693
4 2.282 9 1.816 14 1.672
5 2.114 10 1.777 15 1.653
6 2.004 11 1.744
RDLCL
RDUCL
R
R
3
4
=
=

22. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
Step 9
•Select the scales and plot
the control limits,
centerline, and data points,
in each plotting area.
•The scales must be
determined before the data
points and centerline can
be plotted.
•The scales for both the
upper and lower plotting
areas should allow for
future high or low out-
ofcontrol data points.

23. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
Step 10 - Provide the appropriate documentation.
Each Control Chart should be labeled with
who, what, when, where, why, and how
information to describe where the data
originated, when it was collected, who
collected it, any identifiable equipment or
work groups, sample size, and all the
other things necessary for understanding
and interpreting it. It is important that the
legend include all of the information that
clarifies what the data describe.

24. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
Sample Size Mean Factor Upper Range Lower Range
n A2 D4 D3
2 1.880 3.268 0
3 1.023 2.574 0
4 0.729 2.282 0
5 0.577 2.115 0
6 0.483 2.004 0
7 0.419 1.924 0.076
8 0.373 1.864 0.136
9 0.337 1.816 0.184
10 0.308 1.777 0.223
12 0.266 1.716 0.284
CONTROL CHART FACTORS

25. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
EXERCISE 1: A team collected the variables data recorded in the table below.
Use these data to answer the following questions and plot a Control Chart:
1. What type of Control Chart would you use with these data?
2. Why?
3. What are the values of X-Bar for each subgroup?
4. What are the values of the ranges for each subgroup?
5. What is the grand mean for the X-Bar data?
6. What is the average of the range values?
7. Compute the values for the upper and lower control limits for both the upper and
lower plotting areas.
8. Plot the Control Chart.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
X1 6 2 5 3 2 5 4 7 2 5 3 6 4 5 3 6
X2 5 7 6 6 8 4 6 4 3 5 1 4 3 4 4 2
X3 2 9 4 6 3 8 3 4 7 2 6 2 6 6 7 4
X4 7 3 2 7 5 4 6 5 1 6 5 2 6 2 3 4

26. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
EXERCISE 1 ANSWER
KEY:
1. X-Bar and R.
2. There is more than one
measurement within each
subgroup.
5. Grand Mean of X = 4.52
6. Average of R = 4.38
7. UCLx = 7.71
LCLx = 1.33
UCLR = 10.0
LCLR = 0

27. LOGO
CONSTRUCTING AN X-BAR AND
R CHART

28. LOGOX-BAR AND R CHART FORMULA
1 1
g g
i i
i i
i
i
X R
X and R
g g
where
X average of subgroup averages
X average of the ith subgroup
g number of subgroups
R average of subgroup ranges
R range of the ith subgroup
= =
= =
=
=
=
=
=
∑ ∑
Central Lines are obtained using:

29. LOGOX-BAR AND R CHART FORMULA
3 3
3 3
R RX X
R RX X
X
R
UCL X UCL R
LCL X LCL R
where
UCL=upper control limit
LCL=lower control limit
population standard deviation of the subgroup averages
population standard deviation of the range
σ σ
σ σ
σ
σ
= + = +
= − = −
=
=
If σ are known, control limits are obtained using :

30. LOGOX-BAR AND R CHART FORMULA
If σ are unknown, control limits are obtained using :
2 4
2 3
RX
RX
UCL X A R UCL D R
LCL X A R LCL D R
= + =
= − =
NOTE :
A2,D3 and D4 are factors that vary with the subgroup
size and are found in Table.

31. LOGOX-BAR AND R CHART FORMULA
Revised Central LinesRevised Central Lines
d d
new new
d d
d
d
d
X X R R
X and R
g g g g
where
X discarded subgroup averages
g number of discarded subgroups
R discarded subgroup ranges
− −
= =
− −
=
=
=
∑ ∑

32. LOGOX-BAR AND R CHART FORMULA
Standard ValuesStandard Values
0
0 0 0
2
new new
R
X X R R and
d
σ= = =
0 0 2 0
0 0 1 0
RX
RX
UCL X A UCL D
LCL X A LCL D
σ σ
σ σ
= + =
= − =

33. LOGO
CONSTRUCTING AN X-BAR AND
R CHART
Sample Std. Deviation Chart
(x - s control chart )
 Both R and s measure dispersion of
data
 R chart - simple, only use XH
(highest) and XL(lowest)
 s chart - more calculation - use ALL
xi’s ∴ more accurate, need calculate
sub-group sample standard
deviation
 When n < 10 R chart ≅ s chart
 n ≥ 10 - s chart better , R not
accurate any more
sAXUCL 3x +=
sAXLCL 3x +=
sBUCL 4s =
sBLCL 3s =
44
o
o
C
s
C
s
==σ

34. LOGOX-BAR AND S CHART FORMULA
For subgroup sizes >=10, an s chart is more accurate
than an R Chart. Control limits are given by:
1 1
3 4
3 3
g g
i ii i
sX
sX
s X
s X
g g
UCL X A s UCL B s
LCL X A s LCL B s
= =
= =
= + =
= − =
∑ ∑

35. LOGOX-BAR AND S CHART FORMULA
0
0
0 0
4
0 0 6 0
0 0 5 0
4 5 6, , ,
d
new
d
d
new
d
sX
sX
d
X X
X X
g g
s s s
s s
g g c
UCL X A UCL B
LCL X A LCL B
where
s discarded subgroup averages
c A B B factors found in Table B
σ
σ σ
σ σ
−
= =
−
−
= = =
−
= + =
= − =
=
=
∑
∑
Revised Limits for s chartRevised Limits for s chart

36. LOGOPATTERNS IN CONTROL CHARTS
Upper control limit
Target
Lower control limit
Normal behavior. Process is “in
control.”

37. LOGOPATTERNS IN CONTROL CHARTS
One plot out above (or below).
Investigate for cause. Process is
“out of control.”
Upper control limit
Target
Lower control limit

38. LOGOPATTERNS IN CONTROL CHARTS
Upper control limit
Target
Lower control limit
Trends in either direction, 5 plots.
Investigate for cause of progressive
change.

39. LOGOPATTERNS IN CONTROL CHARTS
Upper control limit
Target
Lower control limit
Two plots very near lower (or upper)
control. Investigate for cause.

40. LOGOPATTERNS IN CONTROL CHARTS
Upper control limit
Target
Lower control limit
Run of 5 above (or below) central line.
Investigate for cause.

41. LOGOPATTERNS IN CONTROL CHARTS
Upper control limit
Target
Lower control limit
Erratic behavior. Investigate.

42. LOGOPATTERNS IN CONTROL CHARTS

43. LOGOPATTERNS IN CONTROL CHARTS

44. LOGOPATTERNS IN CONTROL CHARTS

45. LOGOProcess is OUT of control if :
One or multiple points outside the
control limits
Eight points in a row above the average
value
Multiple points in a row near the control
limits

46. LOGOProcess is IN of control if :
The sample points fall between the
control limits
There are no major trends forming, i.e..
The points vary, both above and below
the average value.

47. LOGOSTATE OF CONTROL
When the process is in control:
1. Individual units of the product or service will
be more uniform
2. Since the product is more uniform, fewer
samples are needed to judge the quality
3. The process capability or spread of the
process is easily attained from 6ơ
4. Trouble can be anticipated before it occurs

48. LOGOSTATE OF CONTROL
When the process is in control:
5. The % of product that falls within any pair of
values is more predictable
6. It allows the consumer to use the producer’s
data
7. It is an indication that the operator is
performing satisfactorily

49. LOGOPROCESS CAPABILITY (CP)
 The range over which the natural variation of
a process occurs as determined by the
system of common causes
 Measured by the proportion of output that
can be produced within design specifications

50. LOGOPROCESS CAPABILITY (CP)

51. LOGOPROCESS CAPABILITY (CP)
Cp =
Upper Specification - Lower Specification
6σ
 A capable process must have a Cp of at
least 1.0
 Does not look at how well the process
is centered in the specification range
 Often a target value of Cp = 1.33 is used
to allow for off-center processes
 Six Sigma quality requires a Cp = 2.0

52. LOGOPROCESS CAPABILITY (CP)
Example
Cp = Upper Specification - Lower Specification
6σ
Insurance claims process
Process mean x = 210.0 minutes
Process standard deviation σ = .516 minutes
Design specification = 210 ± 3 minutes

53. LOGOPROCESS CAPABILITY (CP)
Calculation
Cp = Upper Specification - Lower Specification
6σ
Insurance claims process
Process mean x = 210.0 minutes
Process standard deviation σ = .516 minutes
Design specification = 210 ± 3 minutes
= = 1.938
213 - 207
6(.516)
Process is
capable

54. LOGOPROCESS CAPABILITY INDEX (Cpk)
 A capable process must have a Cpk of at
least 1.0
 A capable process is not necessarily in the
center of the specification, but it falls within
the specification limit at both extremes
Cpk = minimum of ,
Upper
Specification - x
Limit
3σ
Lower
x - Specification
Limit
3σ

55. LOGO
Example
New Cutting Machine
New process mean x = .250 inches
Process standard deviation σ = .0005 inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches
PROCESS CAPABILITY INDEX (Cpk)

56. LOGO
Calculation
New Cutting Machine
New process mean x = .250 inches
Process standard deviation σ = .0005 inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches
Cpk = = 0.67
.001
.0015
New machine is
NOT capable
Cpk = minimum of ,
(.251) - .250
(3).0005
.250 - (.249)
(3).0005
Both calculations result in
PROCESS CAPABILITY INDEX (Cpk)

57. LOGO
Interpreting Cpk
Cpk = negative number
Cpk = zero
Cpk = between 0 and 1
Cpk = 1
Cpk > 1
PROCESS CAPABILITY INDEX (Cpk)

58. LOGOEXERCISE
Dalam suatu proses pengeluaran sebuah
mentol lampu, didapati purata jangka hayat
µ=900jam dan σ=77jam. Nilai norminal bagi
julat kelegaan adalah 1000 jam dengan
spesifikasi atas = 1200jam dan spesifikasi
bawah = 800jam. Jika anda sebagai seorang
pengurus operasi, tentukan adakah proses
berupaya untuk menghasilkan spesifikasi
mentol lampu. Huraikan dapatan anda.

59. LOGO
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