Process Control Charts
VOCABULARY
• IMPORTANT TERMS:
– Nominal: Data at expected value
– Discrete: Data with only a finite number of values
– Indiscrete: Data of Acceptable vs. Unacceptable
– Variable: Property being measured
– Under Control: Variables fall in a nominal range
– Data Points: Measurements taken
– Datum: A fixed reference point for measurements
Process Control Charts
• Used to test if the process is in control
• Used to see if significant changes have occurred in the
process over time
“Indiscrete” or “Continuous
Data Chart” or “X-R Chart”
• Measurement at time intervals
• Measurements compare the control
over time
Example units of measurement to Use:
 Length (mm)
Volume (cc)
 Weight (gm.)
Power (kwh)
 Time (sec, min, hr.) Pressure (psi)
 Voltage (v)
“Discrete Data Charts”
or “PN-P Charts”
• Inspection on lot or batch of
parts;
• Notation of the number of
good and defective parts
Variable representation:
 The number of parts
inspected in the lot = n
 Fraction of defective in lot = p
 Number of defectives = pn
X-
R CHART CONSTRUCTION
Indiscrete Chart
Classroom Example
In the manufacturing process, car engine valve
stems are being machined with a nominal
diameter of 13 mm. Samples are taken at the
following times of day: 6:00, 10:00, 14:00,
18:00 and 22:00, for 25 consecutive days. The
diameter measurements (data) from these
samples are presented in the table on the
next slide.
5
Steps to formulating
the chart:
Step 1: Collect the data
Step 2: Sort the data into
subgroups, such as
lots, order number,
or days
Step 3: Identify the values for the variables “n” and
“k”, where
n = the size of the sub group (i.e., five times)
k = the number of sub groups (i.e., 25 days)
Steps to formulating
the chart:
Step 4: Calculate the mean
for each group that will be
represented by X
Formula:
𝑿 =
𝑿=
𝑿𝟏 + 𝑿𝟐 +𝑿𝟑 ⋯⋯+𝑿𝒏
𝒏
𝟏𝟒.𝟎+𝟏𝟐.𝟔+𝟏𝟑.𝟐+𝟏𝟑.𝟏+𝟏𝟐.𝟏
𝟓
= 𝟏𝟑. 𝟎𝟎
Steps to formulating
the chart:
Step 5: Calculate the range
for each subgroup
represented by R
Formula:
𝑹 = 𝑿𝒍𝒂𝒓𝒈𝒆𝒔𝒕 𝒗𝒂𝒍𝒖𝒆 − 𝑿𝒔𝒎𝒂𝒍𝒍𝒆𝒔𝒕 𝒗𝒂𝒍𝒖𝒆
𝑹 = 𝟏𝟒 − 𝟏𝟐. 𝟏 = 𝟏. 𝟗
Upper & Lower Control Limits
The Upper Control Limit and the Lower
Control Limit set the tolerance level for the
control of the manufacture of a product.
Learning how to plot two separate charts:
o X Control Chart
o R Control Chart
9
Steps to Calculating the
Upper & Lower Control Limits
Step 6: Find the Overall Mean – noted by the
double bar X or 𝑿
𝑿 is the sum of all the means determined in Step 4
divided by the total number of sub groups denoted by
the variable, k , which in this example is 25.
Formula:
𝑿𝟏 + 𝑿𝟐 + 𝑿𝟑 + ⋯ ⋯ 𝑿𝒌
𝑿=
𝒌
𝟏𝟑. 𝟎𝟎+ ? ? +? ? + 𝑿𝟐𝟓
𝑿=
𝟐𝟓
Continuation of Calculating the
Upper & Lower Control Limits
Step 7: Find the Average Value of the Range –
noted by the bar R or 𝑹
𝑹 is the sum of all the ranges determined in Step 5
divided by the total number of sub groups denoted by
the variable, k , which in this example is 25.
Formula:
𝑹𝟏 + 𝑹𝟐 + 𝑹𝟑 + ⋯ ⋯ + 𝑹𝒌
𝑹=
𝒌
Manufacturing Statistics
Step 8: Complete the control limits for each chart:
• For X Control Chart:
Central Line (CL) = 𝑿
Upper Control Limit (UCL) = 𝑿 + 𝑨𝟐 𝑹
Lower Control Limit (LCL) = 𝑿 − 𝑨𝟐 𝑹
• For R Control Chart:
Central Line (CL) = 𝑹
Upper Control Limit (UCL) = 𝑫𝟒 𝑹
Lower Control Limit (LCL) = 𝑫𝟑 𝑹
A2 is from the table
based on the size of
the subgroup (i.e.,
Five reading times)
D4 & D3 is from the
table based on the
size of the subgroup
(i.e., Five reading
times)
=
∑ x bar
∑R
X double
bar
R bar
CLx
UCLx
LCLx
CLr
UCLr
LCLr
Step 8:
Plot Chart
Plot Chart
Chart
Variations
Future
Prediction
P-Control Chart Construction
Classroom Example
An inspector of car wheel rims, working at the end of a
manufacturing line, near the end of each shift must
inspect the lot of wheel rims made during that shift.
On good days when the welder is running properly,
over 400 wheel rims are made per batch. On poor days,
as low as 50 to 60 wheel rims are made per batch. The
inspector marks on the “check sheet” for each batch
the total number of wheels inspected and the number
of defects returned for rework for each lot.
18
Steps to Calculating P-Control Chart
Step 1. Collect data
Step 2. Divide the data into sub groups (i.e.,
usually days or lot). Each sub group
size should be larger than 50 units,
where
n = the number in each subgroup
pn = the number of defects in each sub
group
Calculating Fraction of Defectives
Step 3. Calculate the fraction of defective parts
using the following formula:
Formula
𝒑𝒏
𝒑=
𝒏
Where
p = fraction (decimal) of the number of defectives
pn = number of defects in each subgroup
n = number in each subgroup
NOTE: To convert result to percentage(%), multiply the result by 100
Calculating
Average Fraction of Defectives
Step 4. Calculate the Average Fraction of
Defectives using the following formula:
Formula
𝒕𝒐𝒕𝒂𝒍 𝒅𝒆𝒇𝒆𝒄𝒕𝒊𝒗𝒆𝒔
𝒑=
=
𝒕𝒐𝒕𝒂𝒍 𝒊𝒏𝒔𝒑𝒆𝒄𝒕𝒆𝒅
Where
𝒑 = average of fraction of defectives
𝒑𝒏 = summation (total) number of defects in each subgroup
𝒏 = summation (total) number in each subgroup
𝒑𝒏
𝒏
Calculating Control Limits
Step 5. Calculating the Control Limit for each
Subgroup:
Formulas:
Central Line (CL) = 𝒑
Upper Control Limit (UCL) = 𝒑 + 𝟑
Lower Control Limit (LCL) = 𝒑 − 𝟑
𝒑(𝟏−𝒑)
𝒏
𝒑(𝟏−𝒑)
𝒏
23
Step 6:
Draw P Control Chart
Plot Chart
PN Control Chart
Classroom Example
On an assembly line of windshield wiper motors, the
inspector selects randomly 100 motors per hour to
examine.
The inspector notes on the “check sheet” the number
of defective motors in each 100 selected.
The inspector samples 100 samples for a total of 30
sampling events.
PN Control Data Chart
Calculating the PN Control Values
Step 1. Collect Data – Lot size is set constant
Step 2. Calculate the Average Fraction of
Defectives, the Center Line, and Upper
and Lower Control Limits
Formulas:
𝑝𝑛
𝑝=
𝑛
𝐶𝐿 = 𝑝 𝑛
𝑈𝐶𝐿 = 𝑝 𝑛 + 3 𝑝𝑛(1 − 𝑝)
𝐿𝐶𝐿 = 𝑝 𝑛 − 3 𝑝𝑛(1 − 𝑝)
Step 3: Plot the Chart
27
Plotting of PN Control Chart
28
Check For Understanding
Please Develop a Control Chart for This Valve Manufacturing line:
Your Company makes gate valves which you guarantee to flow
water at 3 gallons per minute when fully open. Any restriction or
misplaced gaskets in the opening will alter this flow rate. Your
inspector at the end of the line tests one valve each hour by
measuring the flow for one minute in sample valves. The flow
rate is recorded for several days in the table in the next slide. Is
this operation in control ?
Check For Understanding
Flow Rate in Gate Valve Inspection (gal/min)
Day
9 AM
10 AM
11 AM
Noon
1 PM
2 PM
3 PM
1
3.11
3.20
2.99
2.85
3.00
3.08
2.90
2
2.98
3.01
2.86
2.55
3.06
3.18
3.11
3
2.86
2.99
3.01
3.10
3.03
2.99
3.10
4
3.05
3.04
2.95
3.08
3.01
2.96
2.89
5
2.58
3.25
3.16
3.08
3.01
2.99
2.98
6
3.09
3.03
3.06
3.02
2.99
2.95
3.01
7
3.12
3.04
2.65
2.99
3.30
3.03
2.99
8
3.01
2.97
2.98
3.01
3.00
3.01
3.02