X‾ AND R CHARTSSEMINAR ON

1.AKHIL KRISHNAN G2.MADHUSOODHANAN 3.MOHAMMED SHAFEEQ P K4.VARUN RAJ M5.VISHNU S

INTRODUCTIONX‾ -R Chart maximum utilization of information

available from data & provide detailed information in process average & variation for control of individual dimensions.

Samples(subgroup size) are drawn at intervals and measures are taken.

Control charts for X‾ & R are constructed for a) A process is behaving normally.

b) No assignable cause are present.

c) Ensure product quality level.

The chart is advantageous in the following situationsThe sample size is relatively small (say, n

≤ 10)— x‾and s charts are typically used for larger sample sizes)

The sample size is constantHumans must perform the calculations for

the chart

Construction of X‾ & R Chart.

1. Begin taking samples and place the numbers on the chart in the order they are taken.

2. Calculate the average of each sample.

3. Divide sum by the total number of samples taken for any particular time.

4. Calculate the overall average by adding on the figure in the average X‾ row and dividing that total by the number of readings in the row.

5. Find the range by subtracting the smaller number from the larger number.

6. Calculate the average range R‾ by the summing all range entries and dividing by the number of entries.

Construction of X‾ & R Chart.

7. To calculate the graph scales begin by first finding the larger and smallest average X‾ and the largest and smallest range.

8. Plot the data using the average data for the top graph and the range data for the lower graph and connect the dots forming a line for the averages and another for ranges.

9. Draw heavy line at those points from one end of each graph to the other and label them.

CASE STUDYHere we are considering 100

finished work pieces from fitting workshop for our analysis

Width of each work piece was measured and as taken as desired dimension

Study leads to the following results

NO. Subgroup Subgroup Avg.

Range X̿� R‾

1 2 3 4 5

1 39 39 38 37 37 38 2            

39.7

           

2.9

2 37 39 40 39 38 38.6 3

3 39 37 36 38 37 37.4 3

4 38 38 36 37 38 37.4 2

5 38 38 37 36 39 37.6 3

6 39 39 40 38 38 38.8 2

7 35 41 39 38 38 38.2 6

8 38 39 37 36 38 37.6 3

9 36 38 39 35 37 37 4

10 38 39 37 39 38 38.2 2

11 39 39 36 37 38 37.8 3

12 38 40 36 38 38 38 4

13 37 37 38 38 38 37.6 1

14 38 36 37 39 39 37.8 3

15 37 36 37 36 38 36.8 2

16 37 38 38 39 40 38.4 3

17 35 38 38 39 38 37.6 4

18 37 37 36 37 38 37 2

19 38 39 39 36 38 38 3

20 39 37 37 36 38 37.4 3

COMPUTATION OF MEAN AND RANGE

CalculationsX<=(X₁‾+X₂‾+----+X₂₀‾)/20 = 39.7R‾=(R₁+R₂+------+R₂₀)/20 = 2.9For subgroup size, n=5 A₂=0.58 D₃=0 D₄=2.11 D₂=2.326

Control limitsFor X‾ chartUCL= X4+A₂R‾ =41.382LCL= X4-A₂R‾ =38.018

For R chart• UCL=D₄R‾ =6.119• LCL=D₃R‾ =0

R CHART

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

1

2

3

4

5

6

7

Range

Range

sub grp no

UCL (6.2)

LCL(0)

X‾ CHART

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2034

35

36

37

38

39

40

41

42

Subgroup Avg.

Subgroup ...

UCL(41.382)

LCL(38)

All the values of ranges are lying between UCL and LCL of R chart

In X‾ chart some values are out of control

So we have to eliminate those groups and calculate revised control limits

REVISED CONTROL LIMITSR Chart : UCL = 6.752 LCL =0

X‾ Chart: UCL = 40.296 LCL =36.584

SOME COMMENTS Now all the points in both charts

are under control limits the process is seems to be under control

Means only chance causes of variations are present in the process

Now we can calculate process average, upper natural limit, lower natural limit, etc to comment about the process control

X‾’= Process average= X<(revised)=38.44σ‾=R‾(revised)/D2= 1.375Process Capability=6σ‾=8.25UNL=X‾’+3σ‾=42.565LNL=X‾’-3σ‾=34.315Here 6σ‾=UNL-LNL. The process is under

strict controlNow these limits can use for future

references

THANKYOU