SQC Project 01

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STATISTICAL QUALITY CONTROL
(QUAL 53273)
Project Report
Prepared By: Nisarg Shah
Submitted to: Dr. Mozammel Khan

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Table of Contents:
Sr. No. Topic Page No.
0.0 Abstract 3
1.0 Raw Data 4
2.0 Decoded data 5
2.1 Control Limit calculations and control chart 6
2.2 Revised control limit calculations and control chart 7
3.0 Process Capability 9
3.1 Frequency Distribution 10
3.2 Skewness and Kurtosis 10
3.3 Histogram and out of spec parts 11
4.0 Confidence Interval Calculation 12
5.0 Discussion 13
6.0 Conclusion 14

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0.0: Abstract:
The use of statistical methods of production monitoring and the parts inspection is known as
Statistical Quality Control, wherein statistical data are collected, analyzed and interpreted to
solve quality problems. Statistics in general is a collection of techniques used for decision making
for a process or population based on analysis of the information contained in a sample.
In Statistical Quality Control, various probability distributions are used to describe or model
critical characteristics of a process. Generally those follows normal distribution. Control Charts
are used for monitoring of the process and identifying special cause variation in the system.
Here, measurement from samples of an aircraft engine component were used to perform
analysis on the variation of outer diameter. 25 samples were measured, with sample size of 5.
The measurements were analyzed on 𝑋̅ and R Control Charts, there were no any outliers in 𝑋̅
chart, however there were 3 outliers in R chart. Those outliers were eliminated assuming
assignable causes. Process capabilities and capability index were calculated and identified that
the process is capable of producing the conforming parts. Histogram of the process suggested
that the process follows normal distribution. Skewness and Kurtosis analysis showed that the
process is almost symmetric and the population is normally distributed.

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1.0: Raw Data (Coded Value)
Table:1 – Raw Data
1 2 3 4 5
1 0.751 0.747 0.752 0.750 0.751
2 0.750 0.748 0.749 0.750 0.752
3 0.749 0.749 0.752 0.750 0.748
4 0.748 0.749 0.749 0.751 0.748
5 0.753 0.749 0.752 0.751 0.751
6 0.755 0.752 0.753 0.744 0.749
7 0.754 0.751 0.752 0.750 0.750
8 0.748 0.753 0.749 0.748 0.748
9 0.751 0.750 0.751 0.752 0.751
10 0.751 0.753 0.751 0.752 0.751
11 0.752 0.752 0.751 0.751 0.751
12 0.750 0.749 0.749 0.748 0.751
13 0.748 0.749 0.751 0.747 0.750
14 0.749 0.750 0.750 0.751 0.751
15 0.752 0.751 0.751 0.752 0.751
16 0.752 0.750 0.750 0.748 0.750
17 0.756 0.754 0.752 0.744 0.747
18 0.749 0.749 0.750 0.751 0.749
19 0.752 0.750 0.753 0.750 0.754
20 0.754 0.753 0.750 0.750 0.751
21 0.750 0.750 0.750 0.750 0.748
22 0.750 0.752 0.751 0.753 0.750
23 0.750 0.751 0.749 0.748 0.748
24 0.754 0.745 0.747 0.746 0.755
25 0.749 0.749 0.749 0.749 0.750
Sample
No
Sample Measurement

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2.0 Decoded Data
Table 2- Actual data after adding shift constant
𝑋̿ =
∑ 𝑋𝑖
̅25
𝑖=1
𝑛
=
1955.72
25
= 78.2288
𝑅̅ =
∑ 𝑅𝑖
25
𝑖=1
𝑛
=
10
25
= 0.40
1 2 3 4 5
1 78.30 77.90 78.40 78.20 78.30 78.22 0.50
2 78.20 78.00 78.10 78.20 78.40 78.18 0.40
3 78.10 78.10 78.40 78.20 78.00 78.16 0.40
4 78.00 78.10 78.10 78.30 78.00 78.10 0.30
5 78.50 78.10 78.40 78.30 78.30 78.32 0.40
6 78.70 78.40 78.50 77.60 78.10 78.26 1.10
7 78.60 78.30 78.40 78.20 78.20 78.34 0.40
8 78.00 78.50 78.10 78.00 78.00 78.12 0.50
9 78.30 78.20 78.30 78.40 78.30 78.30 0.20
10 78.30 78.50 78.30 78.40 78.30 78.36 0.20
11 78.40 78.40 78.30 78.30 78.30 78.34 0.10
12 78.20 78.10 78.10 78.00 78.30 78.14 0.30
13 78.00 78.10 78.30 77.90 78.20 78.10 0.40
14 78.10 78.20 78.20 78.30 78.30 78.22 0.20
15 78.40 78.30 78.30 78.40 78.30 78.34 0.10
16 78.40 78.20 78.20 78.00 78.20 78.20 0.40
17 78.80 78.60 78.40 77.60 77.90 78.26 1.20
18 78.10 78.10 78.20 78.30 78.10 78.16 0.20
19 78.40 78.20 78.50 78.20 78.60 78.38 0.40
20 78.60 78.50 78.20 78.20 78.30 78.36 0.40
21 78.20 78.20 78.20 78.20 78.00 78.16 0.20
22 78.20 78.40 78.30 78.50 78.20 78.32 0.30
23 78.20 78.30 78.10 78.00 78.00 78.12 0.30
24 78.60 77.70 77.90 77.80 78.70 78.14 1.00
25 78.10 78.10 78.10 78.10 78.20 78.12 0.10
Average: 78.2288 0.4000
Sample
No
Sample Measurement Average Range
(R )𝑋

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2.1 Control Limit Calculations and Control Chart:
𝑈𝐶𝐿 𝑋̅ = 𝑋̿ + 𝐴2 𝑅̅ = 78.2288 + 0577 × 0.4 = 78.4595
𝐿𝐶𝐿 𝑋̅ = 𝑋̿ − 𝐴2 𝑅̅ = 78.2288 − 0577 × 0.4 = 77.9981
𝑈𝐶𝐿 𝑅 = 𝐷4 𝑅̅ = 2.114 × 0.4 = 0.8456
𝐿𝐶𝐿 𝑅 = 𝐷3 𝑅̅ = 0 × 0.4 = 0
Chart 1: 𝑋̅ and R Control Chart
252321191715131197531
78.4
78.3
78.2
78.1
78.0
Sample
SampleMean
__
X=78.2288
UC L=78.4595
LC L=77.9981
252321191715131197531
1.2
0.9
0.6
0.3
0.0
Sample
SampleRange
_
R=0.4
UC L=0.846
LC L=0
1
1
1
Xbar-R Chart of 1, ..., 5
From the above control chart, it can be seen that, there are 3 outliers in R chart, and there are no any
outliers in 𝑋̅ chart. Values of sample 6, 17 and 24 are the outliers. As given in instruction that if any values
are outliers, assuming assignable causes and eliminating them from the data, creating a revised limits and
control charts.

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2.2 Revised Control Limits and Control Chart:
Eliminating the values of sample 6, 17 and 24 the following table shows the revised data.
Table 3: Revised data after eliminating the outlier:
𝑋̿ 𝑛𝑒𝑤 =
∑ 𝑋𝑖
̅25
𝑖=1
𝑛
=
1955.72 − 78.26 + 78.26 + 78.14
25 − 3
= 78.23
𝑅̅ 𝑛𝑒𝑤 =
∑ 𝑅𝑖
25
𝑖=1
𝑛
=
10 − 1.1 + 1.2 + 1.0
25 − 3
= 0.3045
1 2 3 4 5
1 77.80 77.40 77.90 77.70 77.80 77.72 0.50
2 77.70 77.50 77.60 77.70 77.90 77.68 0.40
3 77.60 77.60 77.90 77.70 77.50 77.66 0.40
4 77.50 77.60 77.60 77.80 77.50 77.60 0.30
5 78.00 77.60 77.90 77.80 77.80 77.82 0.40
7 78.10 77.80 77.90 77.70 77.70 77.84 0.40
8 77.50 78.00 77.60 77.50 77.50 77.62 0.50
9 77.80 77.70 77.80 77.90 77.80 77.80 0.20
10 77.80 78.00 77.80 77.90 77.80 77.86 0.20
11 77.90 77.90 77.80 77.80 77.80 77.84 0.10
12 77.70 77.60 77.60 77.50 77.80 77.64 0.30
13 77.50 77.60 77.80 77.40 77.70 77.60 0.40
14 77.60 77.70 77.70 77.80 77.80 77.72 0.20
15 77.90 77.80 77.80 77.90 77.80 77.84 0.10
16 77.90 77.70 77.70 77.50 77.70 77.70 0.40
18 77.60 77.60 77.70 77.80 77.60 77.66 0.20
19 77.90 77.70 78.00 77.70 78.10 77.88 0.40
20 78.10 78.00 77.70 77.70 77.80 77.86 0.40
21 77.70 77.70 77.70 77.70 77.50 77.66 0.20
22 77.70 77.90 77.80 78.00 77.70 77.82 0.30
23 77.70 77.80 77.60 77.50 77.50 77.62 0.30
25 77.60 77.60 77.60 77.60 77.70 77.62 0.10
Average 77.7300 0.3045
Range
(R )
Sample
No
Sample Measurement Average
𝑋

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Revised Control Limits:
UCL 𝑋̅ 𝑛𝑒𝑤 = 𝑋̿ 𝑛𝑒𝑤 + 𝐴2 𝑅̅ 𝑛𝑒𝑤 = 77.73 + 0.577 x 0.3045 = 78.4057
LCL 𝑋̅ 𝑛𝑒𝑤 = 𝑋̿ 𝑛𝑒𝑤 − 𝐴2 𝑅̅ 𝑛𝑒𝑤 = 77.73 – 0.577 x 0.3045 = 78.0543
UCL R = 𝐷4 𝑅̅ 𝑛𝑒𝑤 = 2.114 x 0.3045 = 0.6437
LCL R = 𝐷3 𝑅̅ 𝑛𝑒𝑤 =0 x 0.3045 = 0
Chart2: Revised Control Chart
21191715131197531
78.4
78.3
78.2
78.1
Sample
SampleMean
__
X=78.23
UC L=78.4057
LC L=78.0543
21191715131197531
0.60
0.45
0.30
0.15
0.00
Sample
SampleRange
_
R=0.3045
UC L=0.6440
LC L=0
Xbar-R Chart of 1, ..., 5

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3.0 Process Capability:
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑎𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 𝜎0 =
𝑅0
𝑑2
=
0.3045
2.326
= 0.1309
USL = 75.5 + 32 = 78.7
LSL = 74.5 + 3.2 =77.7
𝐶 𝑝 =
𝑈𝑆𝐿−𝐿𝑆𝐿
6𝜎
=
78.7−77.7
6 𝑥 0.1309
=
1
0.7854
= 1.273
𝐶 𝑃𝐾 = Minimum of [
𝑈𝑆𝐿−𝑋̅
3𝜎
or
𝑋̅−𝐿𝑆𝐿
3𝜎
]
𝐶 𝑃𝐾 = Minimum of [
78.7−78.23
3 𝑥 0.1309
or
78.23−77.7
3 𝑥 0.1309
]
𝐶 𝑃𝐾 = Minimum of [ 1. 196 or 1.349 ] = 1.196
Chart 03- Capability analysis for 22 subgroups
78.678.478.278.077.8
LSL USL
LSL 77.7
Target *
USL 78.7
Sample Mean 78.23
Sample N 110
StDev (Within) 0.134376
StDev (O v erall) 0.155363
Process Data
C p 1.24
C PL 1.31
C PU 1.17
C pk 1.17
Pp 1.07
PPL 1.14
PPU 1.01
Ppk 1.01
C pm *
O v erall C apability
Potential (Within) C apability
PPM < LSL 0.00
PPM > USL 0.00
PPM Total 0.00
O bserv ed Performance
PPM < LSL 40.04
PPM > USL 234.68
PPM Total 274.72
Exp. Within Performance
PPM < LSL 323.19
PPM > USL 1242.44
PPM Total 1565.63
Exp. O v erall Performance
Within
Overall
Process Capability of 1, ..., 5

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From the above chart and calculations, we can say that the process is not centered as Cp ≠ Cpk. Also the
Capability Ratio (Cp) compares Process Width with Specification Width (Voice of Customer). From the
values we can say that the process is capable of producing the parts with required specification, still its
not a six sigma process. There is chance of improvement of the process.
Also the Process Capability Index(Cpk) indicates that the process is not fairly centered about the mean and
is closer to the upper specification limit.
3.1 Frequency Distribution:
Frequency distribution of the data is shown in below table, from the data it can be clearly seen
that the central tendency is clearly towards 78.2 and 78.3. The table represents the frequency
distribution of 110 samples after eliminating the outliers.
Table 4 – Frequency distribution of the data
3.2 Skewness and Kurtosis
Variable Mean Median Mode Skewness Kurtosis
Measurement 78.23 78.2 78.2 0.19 -0.32
The above value of Skewness 0.19 shows that the distribution is approximately symmetric and
slightly skewed on right tail, and vakue of kurtosis -0.32 shows that distribution is slightly flatter
than the normal distribution and have a litter wider peak.
Measurement Count
77.9 2
78.0 13
78.1 19
78.2 27
78.3 26
78.4 14
78.5 6
78.6 3
Grand Total 110

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3.3 Histogram and Out of Specification Parts:
Chart 4- Histogram of the samples
78.678.578.478.378.278.178.077.9
30
25
20
15
10
5
0
Measurement
Frequency
Mean 78.23
StDev 0.1554
N 110
Histogram of Measurement
Normal
From the above histogram it can be seen that the data are approximately normally distributed. Now,
determining the percentage non-conforming.
For Non-conforming below LSL
𝑍1 =
𝐿𝑆𝐿 − 𝑋̅
𝜎
=
77.7 − 78.23
0.1309
= −7.10
𝐴1 ≈ 0
For Non-Conforming above USL
𝑍2 =
𝑈𝑆𝐿 − 𝑋̅
𝜎
=
78.7 − 78.23
0.1309
= 3.59
𝐴2 = 1 − 0.99983
𝐴2 = 0.00017 = 0.017%
From the above calculations. It can be said that there will be no any non-conforming part below
specification, but there will be 0.017% non-conforming part above upper specification.

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4.0 Confidence Interval calculation:
𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = 𝑋̅ ± 𝑧
𝑠
√ 𝑛
𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = 78.23 ± 1.96 ×
0.1554
√110
𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = 78.23 ± 1.96 ×
0.1554
√110
𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = 78.2009 𝑡𝑜 78.2590
From the above calculation, with 95% confidence we can say that the process mean and output
will lie between 78.2009 and 78.2590.

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5.0 Discussion:
Control Charts are used to differentiate between common cause and special cause of variation.
Here Table 1 basically shows coded values that were multiplied by a 100 and a shift constant of
3.2 was added to decode the values and produce table 2. In table 2, there are 25 subgroups, each
with a sample size of 5. The values are given in mm. For each subgroup, the average (𝑋̅) and the
range (R) are also stated. This was easily commuted using Microsoft Excel.
Next, the control limits were determined. Mean of the data was calculated, which worked out to
be 78.2288. This was the central line for the (𝑋̅) chart. For the UCL and LCL of the (𝑋̅) chart, the
formula taught in class was used and the values were determined to be 78.4595 mm and 77.998
mm, respectively. Similarly, the mean of individual ranges (R) was commuted. This was the centre
line or the average of the ranges, which was equal to 0.4 mm. For the UCL and LCL of the R chart,
the formula taught in class was employed and the values were determined to be 0.8456 mm and
0 mm, respectively. This referred to the 𝑋̅ and R chart in Chart 1. No points are outside the UCL
and LCL in (𝑋̅) chart; however, subgroups 6, 17, and 24 fall above the range UCL in R chart. This
meant that although the process seemed to be in control based on the (𝑋̅) chart, there was a
statistical variation in the range such that the subgroups 6, 17, and 24 were above the UCL based
on the R chart. Assignable / unnatural causes of variation were assumed and these subgroups
were eliminated for the rest of the report.
Table 3 shows that the subgroups 6, 17, and 24 are removed, even though the subgroup numbers
were not changed to show that they have been eliminated. It should be noted that subgroup size
at this point was reduced to 22, and there is no subgroup 6, 17, and 24 in table 3. Again, new
control limits are established. The 𝑋̿ 𝑛𝑒𝑤and corresponding UCL and LCL were 78.23 mm, 78.4057
mm, and 78.0543 mm, respectively. The 𝑅̅ 𝑛𝑒𝑤 and corresponding UCL and LCL were computed
0.3045, 0.6435 and 0 respectively. The control chart prepared from data can be seen in Chart-2
Next, the process capability was estimated. First the standard deviation for the process was
determined. This was determined to be 0.1309. Process capability was estimated as percent non-
conforming at 0.017% using a normal probability table. The process capability ratio (Cp) and
process capability index (Cpk) was determined to be 1.273 and 1.196 respectively. Process
Capability analysis can be seen in Chart-3.
The frequency distribution is shown in Table-4 and Histogram can be seen in Chart-4 for the in-
control points of the data. Skenewss of 0.19 and Kurtosis of -0.32 suggests that the distribution
is slightly skewed to the right (positively skewed) and it is somewhat flatter compared to a normal
distribution and the data is approximately normally distributed.
The confidence interval is finally calculated which predicts the possibility that the true process
center is within a specified interval based on a certain confidence level. In this case, the
confidence interval is 78.2009 mm to 78.2590 mm based on 95% confidence level.

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6.0 Conclusion:
Statistics allows us to make interference based on the information contained in samples. Control
charts has many benefits in decision making on the basis of the sample information.
Initially the given set of values were decoded that represented the outer diameter of a certain
component of aircraft engines. These values were plotted on 𝑋̅ and R control charts in order to
determine the ones that were out of control. Further, out of control points were removed from
the data set.
After removal of the out of control subgroups, new values were plotted on new control charts.
The capability index and ratio was then determined based on specified specification limits. The
process seemed to be a fairly good process. The frequency distribution of in-control points was
determined to see where the central tendency is. Skewness and kurtosis was determined for the
distribution to see how closely it resembles a normal distribution. It was concluded that data was
approximately normally distributed.

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