Statistical Quality Control
Statistical quality control (S. Q. C.) is a branch of Statistics it deals with the application of statistical methods to control and improve that quality of product. In this use statistical methods of sampling and test of significance to monitoring and controlling than quality of product during the production process.
The most important word in statistical Quality control is quality
The quality of product is the most important property while purchasing that product the product fulfill or meets the requirements and required specification we say it have good quality or quality product other wise not quality.
Quality Control is the powerful technique to diagnosis the lack of quality in material, process of production.
Causes of variation:
When the product are produced in large scale there are variation in the size or composition the variation is inherent and inevitable in the quality of product these variation are classified into two causes.
1) chance causes and 2)assignable causes.
1) Chance causes : some stable pattern of variation is inherent in manufacturing process and anybody can not be control this type of variation ,this variation has no reason can be assigned, it has random nature these cause of variation or type of variation are known as chance causes or chance variation. The variation due to chance causes is beyond the control of human and it can not be eliminated in any circumstances. For that to allow variation within this suitable pattern. This pattern is termed as allowable variation. This type of variation is tolerable and it does not affects the quality of process. And the range of tolerable variation is known as tolerance limit of process.
2) Assignable causes: sometime the product shows the variation in process due to causes that are other than chance causes and they are non-random is known as assignable causes. This causes are responsible for the variation in the process such as low quality raw material, faulty equipment and improper handling of machine etc. Also this type of causes of variation are identified immediately so it is also called variation due to identifiable causes. This assignable causes can be eliminated taking necessary action.
Methods of inspection.:-
The quality of product can be checked or measured by inspection of product. There are two methods of inspection. 1) 100% inspection and 2) sample inspection
100% inspection is not applicable method when the product are produced in large scale. Sometime the cost of inspection is may be exceed than cost of production. The items under inspection are destroyed that case 100% inspection is useless. For that situation we inspecting sample and it selected from population with sufficiently large and randomly selected. They give reasonable reliable conclusion about the population. Also sample inspection method is appropriate when production is in large scale. Or production process is continuous.
Statistical quality control :
The main purpose of SQC is to use some statistical methods to identify and separate the chance and assignable causes of variation so taking appropriate actions to eliminating assignable causes and maintaining the quality of product. A production process is said to be in statistical control. Of the process contain only chance causes and assignable causes are absent.
The SQC method are used in two different situations.
1) Controlling the quality during manufacturing process
2) Controlling quality of product
1) Product control : SQC method are useful while purchasing the raw material for the production they are used to inspecting raw material using sampling technique. If the inspection of all lot or whole lot of product is neither possible. That situation the sampling inspection is practicable way to testing the quality of product and sample is randomly selected it sufficiently large size. If the sample shows that the variation are within tolerance limit then the lot is accepted. And sample show the variation beyond the limit of the. We reject the lot. This limits are setting using the statistical level of significance.
2) Process control : in the production process a number of factor’s such as quality of raw material design of product, specification of product, etc. are responsible to quality of product. If any factor is wrong the quality of product is decreases. Using the sample regularly in time interval. The lies in limit the process is in control. And if the beyond the limit the process is out of control.
Control Charts:
The control chart is a graph in which the result of sample inspected are plotted time to time, using the control chart we monitoring the production process and say whether the process is in control or out of control. The control chart was firstly developed by Walter A. Shewhart. these chart gives the warning if the process is out of control.
A typical control chart consist of the following three horizontal lines:
i) Central line : (C.L) Indicates the expected value of product i.e. length or it is indicates the desired standard level of process.
ii)Upper Central Line: (U.C.L.) this line lying above the central line & it indicate upper tolerance limit.
iii) Lower Central Line: (L.C.L) this line lying the below the central line & it indicate the lower tolerance limit.
In the control chart the U.C.L and L.C.L are draw as dotted line.
the sample are selected randomly & inspecting time to time and this result are plotted on graph using this control limit plot control chart. if the all the points lies in the control limits we say the process is in control other wise it is out of control. but the point goes out of control limit it is indicates that the product falls their standard & warning to identify the assignable causes they are enter in process.
Types of control charts :
there are two types of control chart: i. control chart for variable and ii. control chart for attributes.
firstly we see the control chart for variable: The characteristics which can be measured is called variable. for example the length, etc, are called variate. for this we are interested in controlling the average or mean and range of the variable. this type of control chart for mean and range are constructed in control chart for variable.
Control Chart for Mean x̄ ( i.e. x- bar) and R.
This type of control charts are used when the characteristics is measurable and controlling the mean and range of the variable. the mean chart or x̄ (i.e. X-bar) chart used to control the values of average of variable, and R chart used to control the range of variable. for inspection sample are drawn regular interval and measurement take place. such sample is known as sub-group sample or rational sub-group. the sub-group of size is generally taken as 5 -4.
for constructing control chart first we find out the control limits for R- chart and Mean x̄ (i.e. X-bar ) chart.
1. control limits for range charts (i.e. R- Charts)
i. firstly we calculate the R (range ) for each sub-group.
ii. then we find the average range OR R̄ (i.e. R- bar) for all sub- group.
iii. control limits are
U.C.L = D4 (average of Range) = D4 (R-bar) = D4R̄
Central line =R-bar OR R̄
L.C.L = D3 R - bar = D3 R̄
2.Control limits for the Mean chart or x̄ (i.e.X-bar ) chart is
i. firstly we calculate the x̄ (i.e X-bar) for each sub-group.
ii. then final mean X̿ (i.e. X-double Bar) of all Sub-group.
iii. control limits are
U.C.L = X̿ + A2 R̄
C.L.= X̿
L.C.L = X̿ - A2 R̄
Where the A2 ,D3 , D4 are the constant values depending on the size of the sub- group.
Example 1 . The following data gives the 10 sample of size 5 as
Sample No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Mean x̄ | 350 | 348 | 459 | 687 | 458 | 785 | 145 | 254 | 265 | 297 |
Range R | 78 | 124 | 158 | 268 | 254 | 245 | 154 | 189 | 87 | 99 |
Draw the control chart for x̄ and R for the above data. given that the values A2 = 0.58 ,D3 = 0 and
D4 =2.11
Solution : for Drawing the control chart firstly we calculate the control limit for both charts as
Mean ( x̄) of sample is given then we calculate X̿ as average of x̄ and average of range R̄
Sample No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total |
Mean x̄ | 350 | 348 | 459 | 687 | 458 | 785 | 145 | 254 | 265 | 297 | 4048 |
Range R | 78 | 124 | 158 | 268 | 254 | 245 | 154 | 189 | 87 | 99 | 1656 |
X̿ = sum of x̄ / sample size
X̿ = 4048 /10
X̿ = 404.8
the average of range is R̄ calculated as
R̄ = sum of range / sample size
R̄ = 1656 /10
R̄ = 165.6
Control limits for R chart are
U.C.L = D4 (R-bar) = D4R̄ = 2.11 X 165.6 = 349.42
Central line =R-bar = R̄ = 165.6
L.C.L = D3 R - bar = D3 R̄ = 0 X 165.6 = 0
the R chart is
Control Chart for Attributes:
When the characteristics can not be measurable is called attribute, but it can be classified into different classes defective or non-defective items etc. in SQC the attribution refers to the process characteristics can be classified into defective or non- defective, good or bad, etc. the control chart of attribute is used to monitoring the quality of process that produce the defective or non-defective items. they are also known as attribute control chart or control chart for discrete data. there are several types of control charts for attribute, P-chart, NP- chart and C -chart.
we see one by one chart types.
P-chart:
control chart for Fraction Defective ( P-chart).
A P chart is also called the a proportion chart. P-chart is used to monitoring the proportion of defective and non-defective items in a sample. we are interested in controlling the proportion of defective items in lot or sample. we use chart is known as P-chart. P-chart Plot the proportion of defective item (fraction defectives) against the sample number. it is based on the binomial distribution, and if the sample size is large it has approximately normal distribution.
for constructing the P-chart the follow following procedure:
i) Firstly we finding the proportion of defective items in a each sample, and it is calculated as
p = (Number of defective items ) / (number of items are inspected) for each sample we finding the proportion of defectives and p is also known as fraction defective it is proportion of defective items, e.g. the sample of 100 items are selected and out of them 5 are defectives then the fraction defectives is 5/100= 0.05=5%
ii) then we finding the average of proportion as
p̄ = (total number of defective items )/ total number of items inspected for all sample we finding the proportion of defectives
iii) Control Limits :Lot No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
No. Of defectives | 7 | 8 | 6 | 10 | 12 | 5 | 4 | 3 | 15 | 1 |
Fraction Defectives | 0.07 | 0.08 | 0.06 | 0.1 | 0.12 | 0.05 | 0.04 | 0.03 | 0.15 | 0.01 |
Solution: here the fraction defective are given so we direct calculate the p̄
average of proportion as
p̄ = (total number of defective items )/ total number of items inspected
here total number of defectives are = 7+8+6+10+12+5+4+3+15+1 = 71
and total number of inspected items are = 10 x 100 = 1000 {10 lost and each lot have 100 items}
p̄ = 71 / 1000 = 0.071
and we known q̄= 1- p̄ = 1-0.071 = 0.929
Control Limit = p̄ = 0.071
Upper central Line = p̄ + 3 {square root of [(p̄q̄)/n]} = 0.071 + 3 x [√((0.071*0.929)/100)] = 0.1480
Lower central Line = p̄ - 3 {square root of [(p̄q̄)/n]} = 0.071 - 3 x [√((0.071*0.929)/100(] = -0.0061
here the negative value of lower central line is meaning less. is consider as zero i.e. 0
since all the point are lies inside the control limits then the process is in control.
Control chart for Number of defectives (NP-chart)
If we are interested in controlling the number of defectives in production process the we use chart is known as NP- chart
The np-chart it is also known as the number of nonconforming or defectives chart is statistical control chart used to monitor the proportion of nonconforming ( defectives) items or event in a process, this np-chart is particularly useful when the data is in the form of counts or discrete data.
There is small change in procedure of P-chart in P-chart we interested in proportion of defective and in NP-chart we are interested in number of defective on process. The NP-chart is more useful chart than P-chart.
The NP-chart obtained as plotting the number of defective against the sample number.
Procedure for NP- chart is :-
Step I :- firstly we finding the p̄ as
p̄ = ( total number of defective)/ (total number of item inspected)
Step II:- For NP-chart the control limits are:-
Central line (C.L) = np̄
Upper Central Line ( U.C L) = np̄ + 3 {square root of [p̄q̄n]}
Lower Central Line (L.C.L) = np̄ - 3 {square root of [p̄q̄n]}
And where q̄ = 1- p̄
This is very simple type of chart to draw they containing only average number of defectives and for graph we plot number of defective against the sample number. And if the one of the plotted point is go above or below to the control limits then the process is out of control other wise in control.
Example of np- Chart:
Example 1. Construct the np- chart for the following data. in this data each sample containing 100 items
Sample No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
No. of Defectives | 14 | 12 | 7 | 9 | 4 | 2 | 7 | 12 | 11 | 9 |
Solution: we constructing the np- chart Firstly calculating the p̄
total number of defective = 14+7+12+9+4+2+7+12+11+9 = 87
total number of item inspected = 10*100 = 1000 because 10 samples and each sample have 100 items
p̄ = ( total number of defective)/ (total number of item inspected)
p̄ = 87 / 1000
p̄ = 0.087
where q̄ = 1- p̄ = 1-0.087 = 0.913
Now we obtaining the Control Limits
Central line (C.L) = np̄ = 100 x 0.087 = 8.7
Upper Central Line ( U.C L) = np̄ + 3 {square root of [p̄q̄n]} = np̄ + 3 √ (p̄q̄n)
U.C L = np̄ + 3 √ (p̄q̄n) = 8.7 + 3 x √(0.087 x 0.913 ) = 17.1504
Lower Central Line (L.C.L) = np̄ - 3 {square root of [p̄q̄n]} = np̄ - 3 √ (p̄q̄n)
L.C.L = np̄ - 3 √ (p̄q̄n) = 8.8 - 3 x √(0.088 x 0.912 ) = 0.2449
the np -chart is
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Quality control: The np-chart plays a important role in quality assurance by giving the systematic method to monitor the process and controlling the proportion of nonconforming or defective items. also it helps to provided information about the process maintain it quality standard or not.
The np-chart gives the visual representation of the defective items over time, this visual representation is effective to see the performance of process.
Control chart for Number of defectives (np- Chart) :
The np-chart it is also known as the number of nonconforming or defectives chart is statistical control chart used to monitor the proportion of nonconforming ( defectives) items or event in a process, this np-chart is particularly useful when the data is in the form of counts or discrete data.
and the np-chart consist of plotting the number of nonconforming (defectives) items on the y- axis against the sample number on the x- axis. the subgroup size remains the same for each data points.
The np-chart is commonly used in quality control and process improvement and monitoring the proportion on nonconforming items in a process, following are the some uses of np-chart:
1. Monitoring the process performance: The np-chart is helps to monitoring the stability and performance of the process over time. by checking the proportion of nonconforming items, it given visual representation of variation in process, and it is helps to identify the process is in control or not.
2. Early Detection of Process issues: The np-chart detect the variation in the proportion of nonconforming or defective items, which may be identify the issues in process or changes in process, if the issues are identifies then taking appropriate actions maintaining the process quality.
3. Process Improvement: The np-chart is a one of the tool is used to improving the process, using the np-chart we monitoring the process of defective items to measure the effective ness of process change.
4. Quality control: The np-chart plays a important role in quality assurance by giving the systematic method to monitor the process and controlling the proportion of nonconforming or defective items. also it helps to provided information about the process maintain it quality standard or not.
The np-chart gives the visual representation of the defective items over time, this visual representation is effective to see the performance of process.
Example of np- Chart:
Example 1. Construct the np- chart for the following data. in this data each sample containing 100 items
Sample No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
No. of Defectives | 12 | 10 | 6 | 8 | 9 | 9 | 7 | 11 | 7 | 9 |
Solution: we constructing the np- chart Firstly calculating the p̄
total number of defective = 12+10+6+8+9+9+7+11+7+9 = 88
total number of item inspected = 10*100 = 1000 because 10 samples and each sample have 100 items
p̄ = ( total number of defective)/ (total number of item inspected)
p̄ = 88 / 1000
p̄ = 0.088
where q̄ = 1- p̄ = 1-0.088 = 0.912
Now we obtaining the Control Limits
Central line (C.L) = np̄ = 100 x 0.088 = 8.8
Upper Central Line ( U.C L) = np̄ + 3 {square root of [p̄q̄n]} = np̄ + 3 √ (p̄q̄n)
U.C L = np̄ + 3 √ (p̄q̄n) = 8.8 + 3 x √(0.088 x 0.912 ) = 17.2988
Lower Central Line (L.C.L) = np̄ - 3 {square root of [p̄q̄n]} = np̄ - 3 √ (p̄q̄n)
L.C.L = np̄ - 3 √ (p̄q̄n) = 8.8 - 3 x √(0.088 x 0.912 ) = 0.3011
the np -chart is
Statistical quality control.:
Control chart for Number of Defects (C-chart)
A C chart is a one type of chart in Statistical quality control (SQC) to monitor the count or frequency of nonconforming items. it is particularly used when dealing with discrete data or attribute data, where the outcomes are classified in to defective or non defectives or (conforming or nonconforming).
The Primary use of c chart is to monitor the number of defects or nonconforming items in a production or manufacturing process. it helps to identify the the trend of defectives in a process to take decision and action regarding the production process. this allows the continuous improvement in the production process.
A C chart is also useful for tracking and monitoring the occurrence of defects over a time. it allows to identify periods or specific factors that contributes to increasing number of defects.
The C chart helps to evaluate the effectiveness of process optimize and guide to decision making in achieving the better quality outcome.
Control chart for Number of Defects. (C- chart)
If we are interested in a number of defects in unit that makes the unit useless. The number of defects in items are in certain range, then item is acceptable, otherwise we rejecting the items. for this we used c-chart. In this chart the number of defect denotes as 'C'.
For constructing the C-chart follow steps as
Step I:- firstly we finding the arithmetic mean of number of defects. And is calculated as
c̄ = ( Total number of defects) /(number of items are inspected)
Where the number of items are inspected is N.
Step II :- The control limits are
Central line = c̄
Upper Central Line = c̄ +3 √ c̄
Lower Central Limit = c̄ -3√ c̄
Using this control limit we draw the control chart for number of defect. I.e. C-chart
The C-chart is plotted as number of defect against sample number.
we see the example on c chat to better understanding to construction of C- chart. it is simple to construct a c chart for that first we find out the control limit that are central line, upper central line, lower central line. then plotting on graph paper. the number of defectives on x- axis and sample number on y- axis. then drawing the three line of control limit and then plotting point of number of defectives.
Example 1. Construct control chart for following data.
| Sample no. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Number of defects | 9 | 8 | 5 | 8 | 7 | 6 | 5 | 7 | 10 |
Solution :- in the table they gives the number of defects then for this type of data we used c-chart, for construction of C- chart firstly calculating the control limits for the
c̄ = ( Total number of defects) /(number of items are inspected)
c̄ = 65 / 9 = 7.22
Central line = c̄ = 7.22
Upper Central Line = c̄ +3 √ c̄ = 7.22 + 3 x √ 7.22 = 15.28
Lower Central Limit = c̄ -3√ c̄ = 7.22 - 3 x √ 7.22 = -0.84 = 0
( the lower limit is consider as zero because the number of defect are always positive )
the control chart is C- chart
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here all the point lies under the control limits the process is in control.
The c chart is a valuable tool in a Quality control and process improvement efforts. it enables organizations to monitor the occurrence of defectives items, to take appropriate action to maintain the quality and the performance of process, using this chart it helps to organization to improve the quality and reducing the defective items, to product meets to customer expectations.
Advantages of Statistical Quality Control:
Statistical Quality Control is one of the powerful tool to managing and improving quality of the production process in manufacturing industries. Following are the advantages of statistical quality control.
1. Identification of Quality Problems: Statistical Quality control can helps to identify the causes in production process by analyzing the samples. and correcting causes by taking necessary steps.
2. Avoiding loss: In Statistical Quality Control the items are inspected when they are in process of production, therefore the the causes are identified early and taking action to correcting the causes before the rejecting the whole lot.
3. Improved customer satisfaction: using the Statistical Quality Control improving the quality and standard of product to customer satisfy with product.
4.Increased Efficiency : monitoring and analyzing the production process using Statistical Quality Control, inefficiencies can be identified and they are corrected, then the efficiency is increased.
5. Cost Reduction: Statistical Quality Control help to reducing the cost associated with quality of product by identifying and eliminating the defectives product to reduce the rework on that product and scrap to improving the production process.
6. Continuous Improvement: Statistical Quality Control is monitoring the process continuously identifying and correcting the cause are responsible to lose quality of product to improving the process continuously.
Use of SQC :
SQC mean Statistical Quality Control, it is a set of statistical method that are used to monitoring and controlling the Quality of the process or product. the statistical quality controlling techniques are widely used or applicable in various industries to controlling the product or process. following are the some fields there SQC is applicable or used:
Manufacturing : Statistical Quality Control Is play an a important role in a manufacturing industries. this techniques of SQC are used to monitoring the Quality of Raw material, Intermediate products. The techniques such as control chart, process capability analysis are used to controlling the production process and the Quality of the product. any manufacturing industries has used SQC for maintaining the quality of product to increasing the profit.
Health care: The statistical Quality Control techniques find the application in healthcare. In Healthcare setting to monitoring and improving the quality of patients care. the Process control charts are used to monitoring the patients outcome ( response to medicine), infection rate, medication error, and other quality indicators. by analyzing data, it help to health care providers to identify the area for improvement and improving the quality of health care.
Process Improvement: The Statistical Quality Control used for continuous improvement of the process or inviting to improvement. by collecting and analyzing data on process performance to identify the sources of variation, reducing the defectives, improving the process.
these are few example but the SQC has wide range of applications. the goal is to use the Statistical techniques to monitor, analyze and controlling the quality of process and product. to improve the customers satisfaction, reduced the cost and enhanced the efficiency.
Following are the causes of variations in any process.
Causes of variation:
When the product are produced in large scale there are variation in the size or composition the variation is inherent and inevitable in the quality of product these variation are classified into two causes.
1) Chance causes : some stable pattern of variation is inherent in manufacturing process and anybody can not be control this type of variation ,this variation has no reason can be assigned, it has random nature these cause of variation or type of variation are known as chance causes or chance variation. The variation due to chance causes is beyond the control of human and it can not be eliminated in any circumstances. For that to allow variation within this suitable pattern. This pattern is termed as allowable variation. This type of variation is tolerable and it does not affects the quality of process. And the range of tolerable variation is known as tolerance limit of process.
2) Assignable causes: sometime the product shows the variation in process due to causes that are other than chance causes and they are non-random is known as assignable causes. This causes are responsible for the variation in the process such as low quality raw material, faulty equipment and improper handling of machine etc. Also this type of causes of variation are identified immediately so it is also called variation due to identifiable causes. This assignable causes can be eliminated taking necessary action.
using the SQC techniques we also identify and understand the various causes of variation are present in a process. above two are the main causes of variations.
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