X‾ and r charts

X‾ -R Chart maximum utilization of information available from data & provide detailed information in process average & variation for control of individual dimensions.

1. X ‾ AND R CHARTS SEMINAR ON 1.AKHIL KRISHNAN G 2.MADHUSOODHANAN 3.MOHAMMED SHAFEEQ P K 4.VARUN RAJ M 5.VISHNU S

2. INTRODUCTION X ‾ -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.

3. Control charts for X ‾ & R are constructed for A process is behaving normally. No assignable cause are present. Ensure product quality level.

4. The chart is advantageous in the following situations The sample size is relatively small (say, n ≤ 10)—   x‾and s charts  are typically used for larger sample sizes) The sample size is constant Humans must perform the calculations for the chart

5. Construction of X‾ & R Chart. Begin taking samples and place the numbers on the chart in the order they are taken. Calculate the average of each sample. Divide sum by the total number of samples taken for any particular time. 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. Find the range by subtracting the smaller number from the larger number. Calculate the average range R ‾ by the summing all range entries and dividing by the number of entries.

6. Construction of X‾ & R Chart. To calculate the graph scales begin by first finding the larger and smallest average X‾ and the largest and smallest range. 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. Draw heavy line at those points from one end of each graph to the other and label them.

7. CASE STUDY Here 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

8. 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

9. Calculations X ̿ =(X ₁‾+X₂‾+----+X₂₀‾)/20 = 39.7 R‾=(R₁+R₂+------+R₂₀)/20 = 2.9 For subgroup size, n=5 A₂ =0.58 D₃ =0 D₄ =2.11 D₂=2.326

10. Control limits For X ‾ chart UCL= X̿+A₂R‾ =41.382 LCL= X̿-A ₂R ‾ =38.018 For R chart UCL=D₄R ‾ =6.119 LCL=D₃R ‾ =0

11. R CHART

12. X ‾ CHART

13. 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

14. REVISED CONTROL LIMITS R Chart : UCL = 6.752 LCL =0 X ‾ Chart: UCL = 40.296 LCL =36.584

15. 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

16. X ‾’= Process average= X ̿(revised)=38.44 σ ‾= R ‾(revised)/D2= 1.375 Process Capability=6 σ ‾=8.25 UNL= X ‾’+3σ‾=42.565 LNL= X ‾’-3 σ ‾=34.315 Here 6 σ ‾=UNL-LNL. The process is under strict control Now these limits can use for future references

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