statistical quality control telephone
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Quality Control 2 Midterm
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STATISTICAL QUALITY CONTROL
Quality Control 2
College of Pharmacy
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STATISTICAL QUALITY CONTROL
Dr. Walter Shewhart (1920’s, Bell
Telephone Laboratories)
Introduced the concept of “controlling the
quality” rather than inspecting it into the
product.
Devised the Shewhart control chart technique
for in-process manufacturing operations
Introduced the concept of statistical sampling
inspection.
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HISTORY
Pioneered by Walter
W. Edward Deming applied it in US during
WWII = Succeefully improved quality
Deming introduce to Japanese industry
(after the war has ended) = high quality
Japanese products
Shewart crated the basis and concept of
SPC 3
The quality of a manufactured product is
defined as its conformity to given
standards of specifications.
When measured, quality is always subject
to a certain amount of variation.
Variation is present in any process, deciding
when the variation is natural or when it
needs correction is the key to quality control
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SOURCES OF VARIATION
The two general causes of variation are:
1. Chance, or common
Inevitable or unavoidable, usually large in number and random in nature
e.g. slight differences in process variables like diameter, weight, service time, temperature
2.Assignable
- nonrandom, can be identified and eliminated e.g. poor employee training, worn tool, machine needing
repair
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To maintain quality at optimum level with a
minimum of non-uniformity, quality control
is faced with the problem of
Evaluating the magnitude of chance variation
Detecting the existence of assignable causes
of variation which can be corrected.
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Statistical methodology, which is now an
integral part of many quality control
systems, has been shown to be applicable
in solving this problem.
These charts indicate to the workers or
group leaders whether the production is in-
control status or unsatisfactory production
when out-of-control.
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Statistical quality control
is the monitoring of quality by the application of statistical methods in all stages of production. – Descriptive statistics
it consists of proper sampling, determining quality variation of the sample, and making inferences to the entire batch under investigation. - Acceptance sampling inspection
It makes use of control charts, a tool which may influence decisions related to the functions of specification, production or inspection. – Statistical process control
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TRADITIONAL STATISTICAL TOOL
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• Descriptive Statistics include
– The Mean- measure of central tendency
– The Range- difference between largest/smallest observations in a set of data
– Standard Deviation measures the amount of data dispersion around mean
n
x
x
n
1i
i
1n
Xx
σ
n
1i
2
i
OTHER STATISTICAL TOOLS
Mode
• the most commonly occurring value
ex: 6 people with ages 21, 22, 21, 23, 19, 21 - mode = 21
Median
• the center value
• the formula is N+1/2
ex: 6 people with ages 21, 22, 24, 23, 19, 21
line them up in order form lowest to highest
19, 21, 21, 22, 23, 24
and take the center value - mode =21.5
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VARIABLES
Categorical (aka Qualitative)
takes on values that are names or labels
Quantitative (aka Numerical)
represent a measurable quantity
Examples:
Color of the ball
Population number in Metro Manila
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VARIABLES
Independent Variable
is the variable that is changed or controlled in a scientific experiment to test the effects on the DV ( treatment and controls)
Dependent Variable
is the variable being tested and measured scientifc experiment (result of the experiment)
Examples:
A scientist wants to see if the brightness of light has any effect on moth being attracted to the light.
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Quality Control 2 Midterm
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DRY MIX
Dependent Variable
Responding Variable
Y-axis (vertical axis)
Manipulated variable
Independent variable
X-axis (Horizontal axis)
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QUALITY CONTROL CHARTS
There are two basic quality control
charts, which are based on the
measurability of the quality characteristics,
namely:
1. Variable Chart – This is a chart using
actual records of numerical measurement
on a full continuous scale such as meter,
grams, and liter. Examples of variable
charts are the X (mean) and R (range)
charts.
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The average (mean) is the most used
in quality control chart, while the range
and standard deviation are used as
measures of dispersion.
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2. Attribute Chart – This is a chart, which makes use of discrete data classifying the number of items conforming and the number of items failing to conform to any specified requirements. An example of an attribute chart is the control chart for fraction defective known as P chart. Another type is c chart which shows the number of defects per unit.
It is also known as a go or no-go chart. Graphically, these charts show the proportion of the production or per unit that is not acceptable.
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A control chart consists of a solid line and
two horizontally parallel lines on either
side of the solid line. The control solid
line is the target value of the historical
process average and/ or range.
The two dotted parallel lines indicate
the limits within which practically all
observations should fall as long as the
process is under normal variation or
known as “statistically controlled”.
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The upper dotted line is the upper control limit (UCL) and the lower line is the lower control limit (LCL), both of these are three standard deviations above and below, respectively, the central line.
The six standard deviations, if the process is in control, spread between the upper and lower control limits encompass 99.7 % of the values in a normal distribution with its mean at the central line.
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In statistics, the 68–95–99.7 rule, also known as the three-sigma rule or empirical rule, states that nearly all values lie within three standard deviations of the mean in a normal distribution
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The control limits on the chart are so
placed as to disclose the presence or
absence of assignable causes. Although
their actual elimination is usually an
engineering job, the control chart tells
when, and in some instances, suggests
where to look.
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• Control Charts show sample data plotted on a graph with CL, UCL, and LCL
• Control chart for variables are used to monitor characteristics that can be measured, e.g. length, weight, diameter, time
• Control charts for attributes are used to monitor characteristics that have discrete values and can be counted, e.g. % defective, number of flaws in a shirt, number of broken eggs in a box
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STATISTICAL CONTROL OF QUALITY CHARACTERISTICS
The principle of the control chart technique is that quality measurements obtained from samples from production will vary due to chance causes or assignable causes. When all observations are found within the limits, the process is in control. If an observation is found outside the limit lines, this variation is due to an assignable cause.
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General Method:
1. Select a sample size, (n) at random from
the production.
2. Compute an average for each set of
sample measurements. MEAN
3. Compute the appropriate standard
deviation of the average used. SD
4. From the computed standard deviation,
compute for the UCL and LCL.
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5. Prepare the control chart by drawing a solid horizontal line extending from the vertical quality scale at the average value. A pair of dotted lines or broken lines (control limits) are drawn on either side of this central line at a distance x times the standard deviation.
6. Plot the averages obtained from the sample average values. If any of the plotted points fall outside of the established control limits, the process is out-of control.
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Figure 3-1. A control chart for variables
VARIABLE CHARTS
Variables refer to characteristics that can be
measured. Measurements of parts may vary in
length, diameter, tensile strength and so on.
Mean chart – the first chart designed for
variable with the purpose that is to portray the
fluctuation in the sample means. The mean of
these sample means is denoted as double bar x
or the mean of means.
Range Chart – shows variation in the ranges of
the samples.
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Use x-bar charts to monitor the changes in the mean of a process (central tendencies)
Use R-bar charts to monitor the dispersion or variability of the process
System can show acceptable central tendencies but unacceptable variability
Use the two charts together
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CONSTRUCTING A X-BAR CHART: A QUALITY CONTROL INSPECTOR AT LIQUID SECTION HAS
TAKEN FOUR SAMPLES WITH THREE OBSERVATIONS (HOURLY) EACH OF THE VOLUME OF
BOTTLES FILLED. IF THE STANDARD DEVIATION OF THE BOTTLING OPERATION IS .2 OUNCES, USE
THE BELOW DATA TO DEVELOP CONTROL CHARTS WITH LIMITS OF 3 STANDARD DEVIATIONS FOR
THE 16 OZ. BOTTLING OPERATION.
xx
xx
n21
zσxLCL
zσxUCL
sample each w/in nsobservatio of# the is
(n) and means sample of # the is )( where
n
σσ ,
...xxxx x
kk
Hour 1 Hour 2 Hour 3
Sample 1 15.8 16.1 16.0
Sample 2 16.0 16.0 15.9
Sample 3 15.8 15.8 15.9
Sample 4 15.9 15.9 15.8
Sample means (X-bar)
Sample ranges (R)
© Wiley 2007
Center line and control limit
formulas
SOLUTION AND CONTROL CHART (X-
BAR)
Center line (x-double bar):
Control limits for±3σ limits:
15.923
15.915.97515.875x
15.624
.2315.92zσxLCL
16.224
.2315.92zσxUCL
xx
xx
© Wiley 2007
Sum of the means of the subgroup (Samples)
X = ----------------------------------------------------- Number of the sample means
Compute for the standard error of the distribution of sample means designated and found by
δ
δx = ----------------------
Square root of n
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= =
UCL = X + 3 δ LCL = X - 3δ
---- --
square root of n square root of n
Or
= _ = _
UCL = X + A2 R LCL = X - A2R
=
Where: X = mean of the sample means
_
R = mean of the ranges
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Observations
in Sample
Size
(n)
Factors for
Mean Chart
(X)
Factors for Range
Chart
(R)
A2 D3 D4
2 1.88 0.00 3.27
3 1.02 0.00 2.57
4 0.73 0.00 2.28
5 0.58 0.00 2.11
6 0.48 0.00 2.00
7 0.42 0.08 1.92
8 0.37 0.14 1.86
9 0.34 0.18 1.82
10 0.31 0.22 1.78 Use Table 3-2 to estimate the 3 standard deviation
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SECOND METHOD FOR THE X-BAR CHART
USING
R-BAR AND THE A2 FACTOR (TABLE 6-1)
Use this method when sigma for the process
distribution is not know
Control limits solution:
______ .2330.7315.91RAxLCL
______.2330.7315.91RAxUCL
.2333
0.20.30.2R
2x
2x
© Wiley 2007
Quality Control 2 Midterm
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CONTROL CHART FOR RANGE (R)
0.00.0(.233)RDLCL
.532.28(.233)RDUCL
.2333
0.20.30.2R
3
4
R
R
© Wiley 2007
Center Line and Control Limit
formulas:
Observations
in Sample
Size
(n)
Factors for
Mean Chart
(X)
Factors for Range
Chart
(R)
A2 D3 D4
2 1.88 0.00 3.27
3 1.02 0.00 2.57
4 0.73 0.00 2.28
5 0.58 0.00 2.11
6 0.48 0.00 2.00
7 0.42 0.08 1.92
8 0.37 0.14 1.86
9 0.34 0.18 1.82
10 0.31 0.22 1.78 Use Table 3-2 to estimate the 3 standard deviation
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Problem Solving:
The volume of 5 vials was determined
during the filling of an injectable.
Determine the UCL and LCL using the
formulas below:
= _ = _
UCL = X + A2 R LCL = X - A2R
Or
_ _
UCL = D4 R LCL = D3 R
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No. Of Volume (ml) of 5 vials every 30 mins
Inspection A B C D E X R
1 10.5 10.3 10.2 10.6 10.7
2 10.4 10 10.3 10.2 10.1
3 10.1 9.8 9.9 9.6 9.7
4 9.8 9.8 10 10.2 10.1
5 10.6 10.5 10.5 10.5 10.4
6 10.7 10.5 10.6 10.7 10.9
7 10.7 10.3 10.5 10.4 10.4
8 10.8 10.2 10.7 10.8 10.9
9 10 10.3 10.4 10.1 10
10 10 10 10 10.4 10.3
11 9.8 9.9 9.8 9.8 9.9
12 10.7 10.6 9.6 10.9 10.5
13 10.1 10.6 10.4 10.7 10.3
14 10.5 10.9 10.5 10.6 10.4
15 10.3 10.3 10.2 10.5 10.3
X= R= 44
ATTRIBUTE CHARTS
Percent Defective Chart – also known
as P-chart or “p bar chart” – “p”. This chart
shows graphically the proportion of the
production that is not acceptable.
C Bar Chart - c chart portrays the
number of defects per unit, this is to show
graphically how many defects appear in a
unit of a production.
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To construct a control chart for fraction
defective the following steps are
performed:
1. Record the number inspected (n) and the
number of defectives found (d).
2. Compute for fraction defective (p) which
is the ratio of the number of defectives
found to the total number of units actually
inspected in the batch.
p = d
n 50
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3. Compute for the average fraction
defective (p) obtained by dividing the total
number of defectives found by the total
number of units inspected in a series of
batch.
p = ∑ d
∑ n
4. Calculate the UCL and LCL through the
following formulas:
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UCL and LCL = p ± 3
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n
)p(1p
5. Plot the p’s on the control chart with p on
the center line.
6. When all the points fall within the control
limits, the product is said to be in
statistical control.
Percent defective (100P), a more
convenient value may be used in
constructing the P chart.
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P-CHART EXAMPLE: A PRODUCTION MANAGER FOR A TIRE COMPANY HAS
INSPECTED THE NUMBER OF DEFECTIVE TIRES IN FIVE RANDOM SAMPLES
WITH 20 TIRES IN EACH SAMPLE. THE TABLE BELOW SHOWS THE NUMBER
OF DEFECTIVE TIRES IN EACH SAMPLE OF 20 TIRES. CALCULATE THE
CONTROL LIMITS.
Sample Number of
Defective Tires
Number of Tires in each
Sample
Proportion Defective
(fraction defective)
1 3 20 .15
2 2 20
3 1 20
4 2 20
5 2 20
Total
3(.067).1σzpLCL
3(.067).1σzpUCL
20
(.1)(.9)
n
)p(1pσ
100
10
Inspected Total
Defectives#pCL
p
p
p
Solution:
© Wiley 2007
Problem Solving:
A batch of ointment was filled into tubes
during ten working days. 500 tubes were
filled each day. The inspector withdraw a
random sample of based on the master
table and noted the number of leaking
tubes below.
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Day No. Of Leaking Tubes
1 4
2 6
3 7
4 5
5 3
6 1
7 6
8 3
9 2
10 0 56
Quality Control 2 Midterm
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Day n d p
1 4
2 6
3 7
4 5
5 3
6 1
7 6
8 3
9 2
10 0
∑n= ∑d= p=
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a. Construct a control chart for the crimping
process.
b. Is the process statistically controlled?
c. If the AQL is 2.5, give the acceptance criteria
C-CHART EXAMPLE: THE NUMBER OF WEEKLY CUSTOMER
COMPLAINTS ARE MONITORED IN A LARGE HOTEL USING A C-
CHART. DEVELOP THREE SIGMA CONTROL LIMITS USING THE
DATA TABLE BELOW. NUMBER OF UNITS USED ARE PERIOD OF
TIME, SURFACE AREA OR VOLUME.
Week Number of Complaints
1 3
2 2
3 3
4 1
5 3
6 3
7 2
8 1
9 3
10 1
Total 22
02.252.232.2ccLCL
6.652.232.2ccUCL
2.210
22
samples of #
complaints#CL
c
c
z
z
Solution:
© Wiley 2007
The arithmetic mean number of defects
per tuner (c) is found by:
sum of the defects
c = --------------------------
total number of units
UCL and LCL = c ± 3 √ c
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Problem Solving:
Ten of the newly-designed containers
were inspected for defects; the following
numbers were found 8, 5, 6, 4, 3, 8, 8, 10,
9, 9. construct the c bar chart and plot the
unit defects.
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SEATWORK
Time 1 2 3 4 5
8 am 6.04 6.01 6.05 6.02 6.06
9 am 6.01 6.02 6.03 6.02 6.02
10 am 6.01 6.05 6.07 6.03 6.04
11 am 6.02 6.04 6.04 6.03 6.02
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QC Inspector checks five pieces of the output of a shearing
machine every hour. She measure and records each piece.
Design x bar chart, plot the important data for four hours.
Design a range chart. Interpret the charts.
PROCESS CAPABILITY
The ability of production process to meet or
exceed preset specifications
Product specification
- Preset ranges of acceptable quality characteristics
- Tolerance
- an allowance above or below the nominal value.
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Quality Control 2 Midterm
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MEASURING PROCESS CAPABILITY RATIO
specification width (USL –LSL)
Cp = ----------------------------------------
Process width (6σ)
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PROCESS CAPABILITY VALUES
Cp = 1 The process variability just meets specification, process is minimally capable
Cp ≤ 1 The process variability is outside the specification, process is not capable
Cp ≥1 The process variability fall within specification limits, process exceeds minimal capability
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20 25 30
Upper
specification
Lower
specification
Nominal
value
PROCESS CAPABILITY
Process is capable
Process distribution
Process is not capable
20 25 30
Upper
specification
Lower
specification
Nominal
value
Process distribution
PROCESS CAPABILITY CONFIRMING PROCESS CAPABILITY
(USL - µ , µ - LSL)
Cpk = ------------------------------
( 3σ , 3σ )
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Process Capability Index, Cpk, is an index that measures the potential for a process to generate defective outputs relative to either upper or lower specifications
Quality Control 2 Midterm
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EXAMPLE 1: INTENSIVE CARE LAB
The intensive care unit lab process has an average
turnaround time of 26.2 minutes and a standard
deviation of 1.35 minutes.
The nominal value for this service is 25 minutes
with an upper specifications limit of 30 minutes
and a lower specifications of 20 minutes.
The administrator of the lab wants to have three-
sigma performance for her lab. Is the lab process
capable of this level of performance?
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EXAMPLE 1: INTENSIVE CARE LAB
ASSESSING PROCESS CAPABILITY
Upper Specifications: 30 minutes
Lower Specifications: 20 minutes
Average service: 26.2 minutes
σ: 1.35 minutes
specification width (USL –LSL)
Cp = ----------------------------------------
Process width (6σ)
Cp = 1.23 70
EXAMPLE 1: INTENSIVE CARE LAB
ASSESSING PROCESS CAPABILITY
Upper Specifications: 30 minutes
Lower Specifications: 20 minutes
Average service: 26.2 minutes
σ: 1.35 minutes
CpK = Minimum of
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3,
3
xficationupperspecificationlowerspecix
EXAMPLE 2
A bottling filling machine for Paracetamol syrup are evaluated for their capability
Specification is 15.8 – 16.2 ml and mean is 15.9
Which of the machines is capable?
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Bottling machine Standard deviation
A .05
B 0.1
C 0.2
SAMPLE 3
Compute for Cpk measure of process capability
for this machine and interpret your results.
What value would you have obtained with Cp
measure?
USL = 100 LSL = 60
Process σ = 4 process mean = 80
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SAMPLE 4 Food served at a restaurant should be between 38°C and 49°C when it is delivered to the customer. The process used to keep the food at the correct temperature has a process standard deviation of 2°C and the mean value for these temperature is 40. What is the process capability of the process and the capability process index?
Given,
USL (Upper Specification Limit) =49°C
LSL (Lower Specification Limit) =38°C
Standard Deviation =2°C Mean = 40
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