Control Charts For Variable Normal Curve

Control Charts For Variable Variations Variations due to Assignable causes - Greater magnitude - because of difference among M/cs, workers, material etc. Variations due to chance causes - Random - example – little play between nut and screw of table

Control Charts For Variable If many random samples of any given size are taken from universe, the averages of the samples will themselves form a F.D. having its own central tendency and dispersion or spread. If the universe is normal, statistical theory tells that expected F.D. of the average values will also be normal. In long run 1. Average of samples = Average of the universe 2. S. D. of the expected F.D. of the average = S. D. Of the universe / √ sample size

Control chart : Procedure Making and recording measurement Calculate the average characteristic value and Range for each subgroup Calculate the grand average and average range Calculation of control limits on control chart for X-chart Calculation of control limits on R-chart Plot the X and R chart Draw conclusions from control chart

Some control chart patterns Chance pattern

Some control chart patterns Chance pattern characteristics 1. Most of the points will lie near the central line 2. Very few points will be near the control limits 3. None of the points (except 3 in 1000) fall outside the control limits

Some control chart patterns Extreme variations

Some control chart patterns Extreme variations Causes Error in measurement and calculations. Sample chosen at peak position of pressure, temp. etc. Wrong setting of machine, tool etc. Sample chosen at the commencement or end of an operation

Some control chart patterns Trend

Some control chart patterns Causes of Trend Tool wear Wear of threads on clamping device Effects of temperature and humidity Accumulation of dirt and clogging of fixtures and holes On R chart Increasing trend – gradual wearing of operating machine part Decreasing trend – Improvement in operations, better maintenance, improved control on back process

Some control chart patterns Shift

Some control chart patterns Causes of Shift Change in material Change in operator, inspector, inspection equipment Change in machine setting New operator carelessness of operator Loose fixture etc.

Some control chart patterns Erratic fluctuations

Some control chart patterns Causes of Erratic fluctuations Frequent adjustment of machine Different types of material being processed Change in operator. Machine, test equipment etc.

Process capability Minimum spread of a specific measurement variation which will include 99.7% of the measurements from the given process. Measure of a process spread (6σ) – Natural tolerance Process capability analysis consists of Finding whether the process is inherently capable of meeting the specified tolerance limit. Discovering why a process capable is failing to meet specification

Specifications and process capability Case I: 6σ < ( Xmax - Xmin ) or (USL – LSL)

Conclusion In a, b, c practically all the products manufactured will meet specification as long as the process stays in control X¯ may be permitted to go out of control or the distribution may be allowed to move between b and c. cost of frequent machine setup and cost and time of hunting causes will reduce. If ( Xmax - Xmin )/6σ is considerably large, frequency of control chart may be reduced. For economical advantages the specification limits may be tightened.

Case II: 6σ > ( Xmax - Xmin ) or (USL – LSL)

Conclusion Defective parts will always be there. The remedy will be Increase the tolerance Reduce the dispersion Suffer and sort out the defectives Important to maintain the centering of the process.

Case III: 6σ = ( Xmax - Xmin ) or (USL – LSL)

Conclusion A little change in centering will cause defective parts. Calls for continuous use of control charts. To increase tolerance if they are tighter than necessary. Reduce dispersion if economical

Control Charts for Attributes Practical limitations of the control charts for Variables: Can be used for quality characteristics that can be measured and expressed in numbers. However many quality characteristics can be observed only as attributes. Can be used only for one measurable quality characteristic at a time. For the reason of economy even in some cases , where the direct measurement of variable quality characteristics is possible, it is common practice to classify them as good or bad on the basis of inspection by GO-No-GO gauges.

P-Chart (control chart for fraction defectives) The day’s production or other lot of any manufactured article can be thought of as a sample from a larger quantity with some unknown fraction defective. This unknown universe fraction defective depends upon a complex set of causes influencing the production and inspection operations. Fraction defective in the sample may vary considerably. But as long as the universe fraction defective remains unchanged, relative frequencies of various sample fraction defectives may be expected to follow the binomial law.

Inspection results of magnets for nineteen observations Week No. No. of magnets inspected No. of defective magnets Fraction defectives 1 724 48 0.066 2 763 83 0.109 3 748 70 0.094 4 85 0.114 5 45 0.062 6 727 56 0.077 7 726 8 719 67 0.093 9 759 37 0.049 10 745 52 0.07 11 736 47 0.064 12 739 50 0.068 13 723 0.065 14 57 0.076 15 770 51 16 756 71 17 53 0.074 18 757 34 0.045 19 760 29 0.038 Total 14091 1030  

Calculate the average fraction defective and 3 sigma control limits, construct the control chart and state whether the process is in statistical control. The avg. sample size = 14091/19 = 741.63 n = 742 (say) Total defectives in all samples The avg. Fraction defectives = Total inspected in all samples 1030 = 14091 = 0.0731

Calculation of 3 sigma limits UCLp = 0.1018 LCLp = 0.0444

P-Chart

Conclusion From the resulting control chart Sample numbers 2nd and 4th go above the upper control limit and the sample number 19th goes below the lower control limit. Therefore the process does not exhibit statistical control.

Problem : A certain product is given 100% inspection as it is manufactured and resultant data are summarized by the hour. In the following table, 16 hours of data are recorded. Calculate the 3 sigma limits, construct the control chart. Hour No. of units inspected No. of defective units Fraction defective 1 48 5 0.104 2 36 0.139 3 50 4 47 0.106 6 54 0.056 7 8 42 0.024 9 32 0.156 10 40 0.05 11 0.043 12 0.085 13 46 0.022 14 15 0.063 16 39 Total 720  

C-Chart (Control chart for Number of defects) Control limits on C chart based on Poisson distribution. Two conditions: The area of opportunity for occurrence of defects should be fairly constant from period to period. Opportunities for defects should be large, while the chances of a defect occurring in any one spot should be small.

C-Chart (Control chart for Number of defects) Area of application: Number of surface defects in a roll of coated paper Number of defective rivets in an aircraft wing Number of small air holes in glass bottle Number of imperfections observed in a cloth of unit area Number of defects such as blow holes, cracks, undercuts etc in a casting or welded piece.

Problem : The following table gives the numbers of missing rivets noted at aircraft final inspection. Find C¯ compute trial control limits, and plot control chart for C’ would you suggest for the subsequent period? Air plane no. no. of missing rivets 1 8 10 12 19 11 2 16 23 20 9 3 14 21 4 13 22 5 25 7 6 15 24 28 17   18

C-Chart

Comparison between Attribute charts and variable charts Attirbute charts Example X¯, R, σ charts P, np, C, u charts 2. Data required Variable data ( Measured values of characteristics) Attribute data ( Using Go-No-Go gauges) 3. Field of applications Control of individual characteristics Control of proportion of defectives or number of defects 4. Advantage provides detailed information on process average and variation Provides overall picture of quality history 5. Disadvantage Not easily understood Do not recognize different degree of defectiveness