Statistical Quality Control

Anjali Roll no. 10101008

Sameer Pahwa Roll no. 10101046

Introduction

In this era of ever-growing competition, it has become absolutely necessary for a manufacturer/producer to keep continuous watch over the quality of the goods produced, but due to large scale production it is not possible for a producer to check the quality of each and every item produced. Therefore, to control quality of the manufacturer goods the study of statistical quality control , abbreviated as S Q C is very and useful.

Meaning Statistical quality control S Q C refers to the use of stat.

techinques in controlling the quality of manufactured goods, it is the means of estabilising and achieving quality specifications

which requiers use of tools and techinques of stats. It is an important applications of theory of prob and theory of sampling

for the maintance of uniform quality in a continuous flow of manufactured products . One of the major tools of SQC is the control chart first introduced by W.A.SHEWART through the

application of noraml ditribution.

Definations

“ Statistical quality control can be simply defined as an econoic and effective system of maintaing and improving the quality of outputs throughout the whole operating process of specification. Production and inspection based on continuous testing with random samples.”

Causes of variation in quality characterstics

1. Assignable causes: these causes as the name suggests , refers to those changes in the quality of the products which can be assigned to or attributed to any particular cause like defecive materials, defective labour , defective machines ,etc.however,the effect of such variations can be eliminated with a better system of control like SQC.

2. Chance causes: these cause,as the name suggests , takes place as per chance or in a random fashion as a result of the cumulative effect of a multiplicity of several minor causes which cannot be identified. Such types of causes is inhererent in every type of production and hence It is acceepted as an allowable variation in any scheme of production.

Out of two types of causes , nothing can be done about the chance causes,hoever, assignable variations can be detected and corrected .

Control charts The control charts are the graphics devices developed by walter a.shewart for detecting unnatural pattern of variation in the production process ant determing the permissible limits of variation.these are based the theory of prob and sampling . Contolcharts are simple to construct and easy to intrepret n they tell the production manger at a glance whether or not the proocess in in control i.e., within the tolerance limits. A control chart consists of three horizontal lines::1.Central line (CL) :the central line is themiddle line of the chart . It indicates the grand average of the measurements of the samples.it shows the standard or level of the process.2.UPPER CONTROL LIMIT (UCL) : The upper control limit is usually obtained by adding 3 sigma(3 sd) to the process avg.it is denoted by Mean +3sd.3.Lower control limits (LCL): the lower cntrl limit is usually obtned by subtracting 3 sigma to the process avg.it s denoted by mean -3.

Dr. shewart has proposed te 3sd limits fr the contol charts. From the prob,If a variable X’ IS NORAMLY DISTRIBUTED ,THE PROB tht a randm variable ll be between x’ + -3sd,whre x’ is mean and sd is stndrd deviation is 0.9973 which is extremely high.thus.,the prob of a random variable fall outside these limits is 0.0027, which is very low.(Which is like 3 out of 1000.)

Purpose and uses of cntrl charts

1.Determind the qquality stndrd of the products while in process.2.Detecting the chance and assignable variations in qulity stndrds.3.Reveals variations4.Indicates whether the production porocess is in contrl or nt so as to tak eneccessary steps fr its correction.5.Simple to contruct n east to intrepret 6.Less inspection cost n time 7.Tells at a glance whether or nt process is in control.

TYPES OF CONTROL :-

X’ chart : constructed fr controlling the variations in the avg quality stndrd of products ina prduction process.Procedude:steps:-1.Compte the mean of each sample i.e 2.C ompute the mean of samples means by dividing the sum of the sample means by the number of samples i.eThe grand mean (X’’) represents the central line.(CL)3.Determine THE CONTROL LIMITS BY USING THE FOLLOWING FORMULA:a) On the basis of stndrrd devitaion of the popultion (sd)b) On the basis of quality control factors A2 and R2.c) (WHRE A2, is quality control factor whosevalue is obtained from the control chrt table

with refrence to the size of th sample.)d) Construct te mean chart(x’-chart) by plotting the sample number on x-axix n sample

mean .ucl.lcl, n central linein y-axise) Intrpret the x’ chart .if all the sample means x’ fall withing the control , the production

process is in a state of cntrol othe wise it is beyond control.

R-CHART :The range chart (r-chart) is constructed fr controlling the variation in the dispersion or variabilty of the quality stndrd of the prodcuts in a prdction process.Procedure: steps::a) Compute the range of eaach sample using the formula : r = L-S L=largest value ,

S=smallest value.b) Comp;pute th mean ofranges by divinding the sun of the samples ranges {# R} by the

number of samples.i.e

c) Determine the control limits b using following formula: 1.on the basis of quality control factors D3 and D4 and R (whe d3 n d4 ae the quality control factors n their values are obtained frm the control

charts table with refrence to the seize of the sample. 2. on the basis of qulity control factors d1 n d2 andpopoulation sd.

d) Construct the r chart by pltting the smple no. o the x axis n sample range ucl lcl n central line on the y axis.

e) Intrort the r chart . If all th sample ranges fall withing the control lmits the production process isin a state of contol otherwise it is beyond cntrol.

Example 1.

Example 2.

3. Sd chartTo get a better picture of the vaiations in the quality stndrd in a process than is obtained frm the arange a chart prvided the stndrd deviation of the varius samples are readily available.Procedure:steps:a) Find sd of each sample,if nt given.b) Compute the mean of sd by using the formla : the mean of stdrd sd reprents the central line clc) Find the upper n lower control limits by using 1. on the basis of quality control fators b1, b2 and population stndrd dviation…sd ucl = b2 sd lcl = b1 sd2. On the basis of quality control factors b3 n b4 n estimated population sd,sdWhre b1 b2 b3 b4 are equally quality contrl factors….

d) Cnstruct sd-chart by pltting the sample no. the x axis.. Nsample sd, ucl , lcl ncl on the y axis…

e) Intrperet the chart drawn.

Example pp – 287

Control charts for attributesThese charts are used when the quality or charaterstics of a procdt cannot be measured in quantitative form and thedata Is studied on the basis ogf totality of attributes like defective n nono –defective. Such charts re of three types ::1. p-chart2. C-chart3. np-chart

P-chart:: fr contrlling the quality stndrd in the avg fraction defective of the roducs in a process when the observed sample items are classified into defective n non defctive.Procedure:steps::1. Find the frction defetives in each sample2. Find the mean of th fraction 3. Detemine the contorl limits by unsing yhe formula: i..e4. Cnstruct p-chart by pltting the sample no. the x axis.. Nsample sd, ucl , lcl ncl on the y

axis…5. intrpret the p chart. If all th sample fraction defective (p) fall withing the control lmits

the production process isin a state of contol otherwise it is beyond cntrol.

Example pp- 290

Np-chart: fr controlling the qaulity stndrd af attending in a proces o whre the sample size is equal n it is required to plot the nno. Of defectives ((np) in samples intead of fraction distibution.Procedure:steps::1. Find the avg. no. of defects2. Find the value of p’ by using the formula..3. Determine the control limits by using he formula. i.e4. Cnstruct np-chart by pltting the sample no. the x axis.. N sample no. of defective sd,

ucl , lcl ncl on the y axis…5. intrpret the np chart. If all the sample no. of defectives fall withing the control lmits the

production process isin a state of contol otherwise it is beyond cntrol.

Example 11.

C-chart:::fr contrlling no. of defects of no. of defects per unit say a piece of cloth/glass/paper/bottle which may cotain more then one defect.the prob of occurnce of each defect ends to a remain very small.procedure::steps::1. Determine the no. of defecs per unit in the samples of equalt size2. Find themena of the no. of defcts counted in sevral units by using the formula: i.e

3. Determine te control limits i.e

4. Cnstruct c-chart by pltting the sample no. the x axis.. N sample no. of defects observed per unit , ucl , lcl ncl on the y axis

5. intrpret the c-chart. If observed values of no. of defects per unit fall withing the control lmits the production process isin a state of contol otherwise it is beyond cntrol.

Example 14 pp- 297.

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