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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 �̅� and R Control Charts, there were no any outliers in �̅�

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

Page | 5

2.0 Decoded Data

Table 2- Actual data after adding shift constant

�̿� = ∑ 𝑋�̅�

25𝑖=1

𝑛=

1955.72

25= 78.2288

�̅� = ∑ 𝑅𝑖

25𝑖=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�̅� = �̿� + 𝐴2�̅� = 78.2288 + 0577 × 0.4 = 78.4595

𝐿𝐶𝐿�̅� = �̿� − 𝐴2�̅� = 78.2288 − 0577 × 0.4 = 77.9981

𝑈𝐶𝐿𝑅 = 𝐷4�̅� = 2.114 × 0.4 = 0.8456

𝐿𝐶𝐿𝑅 = 𝐷3�̅� = 0 × 0.4 = 0

Chart 1: �̅� and R Control Chart

252321191715131197531

78.4

78.3

78.2

78.1

78.0

Sample

Sa

mp

le M

ea

n

__X=78.2288

UC L=78.4595

LC L=77.9981

252321191715131197531

1.2

0.9

0.6

0.3

0.0

Sample

Sa

mp

le R

an

ge

_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 �̅� 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:

�̿�𝑛𝑒𝑤 = ∑ 𝑋�̅�

25𝑖=1

𝑛=

1955.72 − 78.26 + 78.26 + 78.14

25 − 3= 78.23

�̅�𝑛𝑒𝑤 = ∑ 𝑅𝑖

25𝑖=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 �̅�𝑛𝑒𝑤 = �̿�𝑛𝑒𝑤 + 𝐴2�̅�𝑛𝑒𝑤 = 77.73 + 0.577 x 0.3045 = 78.4057

LCL �̅�𝑛𝑒𝑤 = �̿�𝑛𝑒𝑤 − 𝐴2�̅�𝑛𝑒𝑤 = 77.73 – 0.577 x 0.3045 = 78.0543

UCL R = 𝐷4 �̅�𝑛𝑒𝑤 = 2.114 x 0.3045 = 0.6437

LCL R = 𝐷3 �̅�𝑛𝑒𝑤 =0 x 0.3045 = 0

Chart2: Revised Control Chart

21191715131197531

78.4

78.3

78.2

78.1

Sample

Sa

mp

le M

ea

n

__X=78.23

UC L=78.4057

LC L=78.0543

21191715131197531

0.60

0.45

0.30

0.15

0.00

Sample

Sa

mp

le R

an

ge

_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=

0.30452.326

= 0.1309

USL = 75.5 + 32 = 78.7

LSL = 74.5 + 3.2 =77.7

𝐶𝑝 = 𝑈𝑆𝐿−𝐿𝑆𝐿

6𝜎 =

78.7−77.7

6 𝑥 0.1309 =

1

0.7854 = 1.273

𝐶𝑃𝐾 = Minimum of [ 𝑈𝑆𝐿−�̅�

3𝜎 or

�̅�−𝐿𝑆𝐿

3𝜎 ]

𝐶𝑃𝐾 = Minimum of [ 78.7−78.23

3 𝑥 0.1309 or

78.23−77.7

3 𝑥 0.1309 ]

𝐶𝑃𝐾 = 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

Fre

qu

en

cy

Mean 78.23

StDev 0.1554

N 110

Histogram of MeasurementNormal

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 − �̅�

𝜎=

77.7 − 78.23

0.1309= −7.10

𝐴1 ≈ 0

For Non-Conforming above USL

𝑍2 =𝑈𝑆𝐿 − �̅�

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 = �̅� ± 𝑧𝑠

√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 (�̅�) 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 (�̅�) chart. For the UCL and LCL of the (�̅�) 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 �̅� and R chart in Chart 1. No points are outside the UCL

and LCL in (�̅�) 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 (�̅�) 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 �̿�𝑛𝑒𝑤and corresponding UCL and LCL were 78.23 mm, 78.4057

mm, and 78.0543 mm, respectively. The �̅�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.

Page | 14

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 �̅� 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.