This part of BS 5702 gives guidance on the application of control charts to individual observations. It demonstrates the benefits, versatility and usefulness of a simple but powerful pictorial method for monitoring, audit and surveillance with the objectives to control and improve many types of processes. These processes can, for example, be in industry, commerce, public service, health care, environment, food, information technology or finance. Case studies are included to illustrate this adaptability.<\/p>\n
It describes how and when to apply, select, construct, interpret and use, control charts:<\/p>\n
This standard distinguishes between the use of control charts to achieve product control as opposed to steering a process (reducing variation and making estimates of, and securing progressive improvements in, process performance).<\/p>\n
This standard is aimed at all those concerned with statistical related initiatives based on individual observations and the continuous improvement of business processes.<\/p>\n
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| 3<\/td>\n | Contents 1 Scope 1 2 Normative references 1 3 Terms, definitions and symbols 2 4 Role of control charts in process control and improvement 8 5 Classes of data and selection of appropriate chart 11 6 Control charts for individual measurements 12 7 Construction and use of standard individual and moving range control charts 14 8 Construction and use of standard moving mean and moving range charts 25 9 Construction and use of median control charts 30 10 Construction and use of basic cusum charts 33 11 Construction and use of individual measurement cusum charts for process control 36 12 Construction and use of individual and moving range charts for attribute data 37 Annexes Annex A (informative) Case studies for typical applications 45 Annex B (informative) Control chart templates 55 Annex C (normative) Relationship between Cpk and percentages and parts-per-million (ppm) out-of-specification 57 Bibliography 58 List of figures Figure 1 – Application sequence for process control, performance assessment and improvement 9 Figure 2 – Focus on influential process characteristics rather than product yield 10 Figure 3 – Road map for control charts 12 Figure 4 – Provisional individual and moving range chart for silicon content in steel 19 Figure 5 – Provisional control chart for silicon content 21 Figure 6 – Provisional histogram of silicon content 22 Figure 7 – Performance capability analysis using provisional data 22 Figure 8 – Comparison of before and after results 24 Figure 9 – Histogram and capability analysis of data after adjustment of mean 24 Figure 10 – Spreadsheet-based moving mean and moving range control chart for silicon content 29 Figure 11 – Median control chart for spend versus budget 32 Figure 12 – Conventional individuals chart for spend versus budget 32 Figure 13 – Basic cusum chart for % silicon content 34 Figure 14 – Refined individuals control chart for % silicon content 35 Figure 15 – Cusum chart for % silicon content in control mode 36 Figure 16 – Cusum chart for nonconforming deliveries 39 Figure 17 – np control chart for nonconforming deliveries 40 Figure 19 – Cusum chart for fabric nonconformities 41 Figure 20 – Cusum chart for fabric nonconformities per roll 42 Figure 21 – c chart for fabric nonconformities per roll 42 <\/td>\n<\/tr>\n | ||||||
| 4<\/td>\n | Figure 22 – c chart of outages per week 22 Figure 23 – Individuals chart of system outage rate per month 44 Figure A.1 – Individuals chart for strontium-90 in cows\u2019 milk in UK location 47 Figure A.2 – Individuals chart for caesium-137 in cows\u2019 milk 48 Figure A.3 – Individuals chart of daily fasting blood sugar levels 49 Figure A.4 – Individuals chart for peak expiratory flow rate 51 Figure A.5 – Cusum chart of the number of trades executed at the LSE per month 52 Figure A.6 – Individuals chart of the number of trades executed at the LSE per month 53 Figure A.7 – FTSE all share index by month 54 Figure B.1 – Individual and moving range control chart template 55 Figure B.2 – Moving mean and moving range control chart template 56 List of tables Table 1 – Key data for constructing a conventional individual and moving range chart 16 Table 2 – Key constants for control limits 16 Table 3 – Provisional data for plotting of silicon content in steel 21 Table 4 – Silicon content in steel data after adjustment 23 Table 5 – Key data for constructing a moving average mean and moving range chart 27 Table 6 – Key constants for control limits 27 Table 7 – % Silicon content provisional data 28 Table 8 – Key data for constructing a median control chart 31 Table 9 – Derivation of the medians for spend against budget 31 Table 10 – Example of construction of cusum chart for % silicon content 34 Table 11 – Count of system outages per week over a period of one year 43 Table 12 – Conversion of outage count per month to monthly outage rate 43 Table A.1 – Strontium-90 in cows\u2019 milk 46 Table A.2 – Concentrations of caesium-137 in cows\u2019 milk in a UK location 48 Table A.3 – Daily fasting blood sugar levels (mg\/dl) 49 Table A.4 – Project desirables and associated means to improve asthma care 50 Table A.5 – Daily readings of peak expiratory flow rate (PEFR): Days 1-10 50 Table A.6 – Daily readings of peak expiratory flow rate (PEFR): Days 11-20 51 Table A.7 – Daily readings of peak expiratory flow rate (PEFR): Days 21-30 52 Table C.1 – Proportion of values expected outside of an upper and lower limit in terms of values of capability indices for a normally distributed characteristic 57 <\/td>\n<\/tr>\n | ||||||
| 5<\/td>\n | Foreword <\/td>\n<\/tr>\n | ||||||
| 7<\/td>\n | 1 Scope 2 Normative references <\/td>\n<\/tr>\n | ||||||
| 8<\/td>\n | 3 Terms, definitions and symbols 3.1 Terms and definitions <\/td>\n<\/tr>\n | ||||||
| 13<\/td>\n | 3.2 Symbols <\/td>\n<\/tr>\n | ||||||
| 14<\/td>\n | 4 Role of control charts in process control and improvement 4.1 Principle 4.2 Improvement and control a) Is the process in control? b) What is the capability of the process? c) Is there evidence of significant improvement? <\/td>\n<\/tr>\n | ||||||
| 15<\/td>\n | Figure 1 Application sequence for process control, performance assessment and improvement <\/td>\n<\/tr>\n | ||||||
| 16<\/td>\n | 4.3 Monitoring of influential process characteristic in preference to after-the-event output Figure 2 Focus on influential process characteristics rather than product yield 4.4 Process improvement not process judgement <\/td>\n<\/tr>\n | ||||||
| 17<\/td>\n | 4.5 SPC is not confined to control charts a) reduce variation; b) increase knowledge about the process; c) steer the process in the desired way. 5 Classes of data and selection of appropriate chart 5.1 Data classification <\/td>\n<\/tr>\n | ||||||
| 18<\/td>\n | 5.2 Control chart selection Figure 3 Road map for control charts 6 Control charts for individual measurements 6.1 Scope of application <\/td>\n<\/tr>\n | ||||||
| 19<\/td>\n | 6.2 Statistical and attitudinal aspects <\/td>\n<\/tr>\n | ||||||
| 20<\/td>\n | 7 Construction and use of standard individual and moving range control charts 7.1 Purpose a) the characteristic is measurable; b) the sample size is 1; c) the measurements are independent of each other; d) the expectation is that the characteristic will exhibit a constant mean and variation; e) a timely response is required to any significant change in the value of the characteristic; f) the outcome of the individuals is normally distributed; g) the individual measurements are taken periodically. 7.2 Construction 7.2.1 Appropriate characteristics 7.2.2 Sampling time and frequency <\/td>\n<\/tr>\n | ||||||
| 21<\/td>\n | 7.2.3 Preferred value for the chosen characteristic 7.2.4 Method of construction (manual) a) enter the administration details for process, characteristic, aim, specification, and sampling frequency taking into account 7.2.1, 7.2.2 and 7.2.3. b) take individual measurements according to the predetermined sampling frequency and enter the results in row X. Enter also the number, date and time as appropriate. c) calculate the moving range between individual measurements and enter the results in row Rmoving. The short term variation in … d) bearing in mind the highest and lowest values for X and R, select appropriate scales for the individuals X chart and the R chart and enter in the template. <\/td>\n<\/tr>\n | ||||||
| 22<\/td>\n | e) plot the individual X and the moving range R values and construct a tally chart in the grid marked \u201cDistribution of X\u201d. f) mark in on the chart any significant process events. g) calculate and enter the mean of all the individuals and the mean of all the moving ranges . h) check visually that the tally chart for the individuals is approximately normal. i) calculate provisional centre lines and estimated plus\/minus three standard deviation control limits using the data inTable 1 and Table 2. j) enter the provisional centre lines and control limits. k) if any points are out of control reflect on whether these points should be discarded and revised control limits established. Table 1 Key data for constructing a conventional individual and moving range chart Table 2 Key constants for control limits 7.2.5 Interpret and act for control 7.2.5.1 General a) Rule 1: Any plotted point is beyond the control limit. b) Rule 2: Seven points in a row are on one side of the centre line. c) Rule 3: Seven intervals in a row are consistently increasing or consistently decreasing. d) Rule 4: Any obvious non-random patterns. <\/td>\n<\/tr>\n | ||||||
| 23<\/td>\n | 7.2.5.2 Moving range chart 7.2.5.3 Individuals chart 7.2.6 Interpret and act for performance capability <\/td>\n<\/tr>\n | ||||||
| 24<\/td>\n | 7.2.7 Interpret and act for improvement a) Steering the process to produce a shift in the mean towards the preferred value or aim. b) Securing an improvement in process variability shown by a sustained decrease in the value of the range. 7.3 Case study (manual method) 7.3.1 Scenario <\/td>\n<\/tr>\n | ||||||
| 25<\/td>\n | Figure 4 Provisional individual and moving range chart for silicon content in steel <\/td>\n<\/tr>\n | ||||||
| 26<\/td>\n | 7.3.2 Provisional control chart 7.3.3 Improvement action 7.4 Case study (computer generated approach) 7.4.1 General 7.4.2 Construction of provisional control chart and histogram <\/td>\n<\/tr>\n | ||||||
| 27<\/td>\n | Table 3 Provisional data for plotting of silicon content in steel Figure 5 Provisional control chart for silicon content <\/td>\n<\/tr>\n | ||||||
| 28<\/td>\n | Figure 6 Provisional histogram of silicon content 7.4.3 Initial control and performance analysis Figure 7 Performance capability analysis using provisional data <\/td>\n<\/tr>\n | ||||||
| 29<\/td>\n | 7.4.4 Improvement action Table 4 Silicon content in steel data after adjustment <\/td>\n<\/tr>\n | ||||||
| 30<\/td>\n | Figure 8 Comparison of before and after results Figure 9 Histogram and capability analysis of data after adjustment of mean <\/td>\n<\/tr>\n | ||||||
| 31<\/td>\n | 8 Construction and use of standard moving mean and moving range charts 8.1 Purpose a) the characteristic is measurable; b) the sample size is 1; c) the measurements are independent of each other; d) the expectation is that the characteristic will exhibit a constant mean and variation; e) a smoothing effect is required to any significant change in the value of the characteristic; f) the outcome of the individuals might not be normally distributed. <\/td>\n<\/tr>\n | ||||||
| 32<\/td>\n | 8.2 Construction 8.2.1 Appropriate characteristics 8.2.2 Sampling time and frequency 8.2.3 Preferred value for the chosen characteristic 8.2.4 Method of construction (manual) a) enter the administration details for process, characteristic, aim, specification, and sampling frequency, taking into account clauses 8.2.1 to 8.2.3. b) take individual measurements according to the predetermined sampling frequency and enter the results in row X. Enter also the number, date and time as appropriate. c) decide on a subgroup size. A large subgroup will introduce a greater smoothing effect. A small subgroup will react more quick… d) calculate the sum of the first and second measurements and place in the second column of the CX row. Repeat for the second and third measurements, etc. e) calculate the mean of each of the CX entries and place the result in row. f) calculate the absolute difference between the first and second measurements and place the result in the second column of the Rmoving row. Repeat for the second and third measurements, etc. g) bearing in mind the highest and lowest values for the mean x and R, select appropriate scales for the moving mean chart and the moving range chart and enter in the template. h) plot the moving mean () and the moving range (Rmoving) values and construct a tally chart of individual measurements in the grid marked \u201cDistribution of X\u201d. i) mark in on the chart any significant process events. j) calculate the overall mean () of all the moving means () and the mean of all the moving ranges (). Enter in the appropriate control box of Figure B.2. <\/td>\n<\/tr>\n | ||||||
| 33<\/td>\n | k) check visually that the tally chart for the individuals is approximately normal. If this is not the case refer to Clause 9 for information on methods to handle non-normal data. l) calculate the centre lines and estimated plus\/minus three standard deviation control limits using the data in Table 5 and Table 6. m) plot the centre lines and control limits on the template. Table 5 Key data for constructing a moving average mean and moving range chart Table 6 Key constants for control limits 8.2.5 Interpret and act for control 8.2.6 Interpret and act for performance capability and improvement <\/td>\n<\/tr>\n | ||||||
| 34<\/td>\n | 8.3 Case study (manual method) Table 7 % Silicon content provisional data <\/td>\n<\/tr>\n | ||||||
| 35<\/td>\n | 8.4 Case study (computer generated approach) Figure 10 Spreadsheet-based moving mean and moving range control chart for silicon content <\/td>\n<\/tr>\n | ||||||
| 36<\/td>\n | 9 Construction and use of median control charts 9.1 Purpose a) the sample size for plotting is restricted to 1; b) a robust control method is required that makes no assumptions about the pattern of variation of the data, namely whether it is normal or not; c) a simple method of application is required demanding the minimum of calculation; d) only a small number of results are available for the construction of provisional control limits. The minimum number is 7. How… e) it is required to limit the distortion effect on the value of the control limits caused by having extreme readings in small a… 9.2 Method of construction (manual) a) Take at least 7 measurements, at the predetermined frequency, of the value of the characteristics to be plotted. Place in order of measurement in column 1 of a table. b) In column 2 of the table rank the results of column 1 in descending order. c) Determine and record the median, M, thus: 1) if the total number of measurements is odd, select the mid value in column 2; 2) if the total number of measurements is even, calculate the mean of the two values that straddle the centre of the rankings in column 2. d) Insert the upper half of the data in column 4. <\/td>\n<\/tr>\n | ||||||
| 37<\/td>\n | e) Determine, and record in column 5, the median of the upper half, Q3, thus: 1) if the total number of measurements is even, the overall median, calculated in c)2), is not included in this determination; 2) if the total number of measurements is odd the overall median, selected in c)1), is included in this determination. f) Repeat d) for the lower half of the data in column 6. g) Determine the median of the lower half, Q1, and place in column 7: 1) if the total number of measurements is even, the overall median, calculated c)2), is not included in this determination; 2) if the total number of measurements is odd the overall median, selected in c)1), is included in this determination. h) Plot the data, and calculate and draw the control limits using Table 8. Table 8 Key data for constructing a median control chart 9.3 Case study (manual method) Table 9 Derivation of the medians for spend against budget a) the upper control limit is given by: Q3 + 1.5 (Q3 – Q1) = 19.8 + 1.5(19.8 + 18.6) = 19.8 + 57.6 = 77.4 b) the lower control limit is given by: Q1 – 1.5 (Q3 – Q1) = 18.6 – 1.5(19.8 +18.6) = 76.2 <\/td>\n<\/tr>\n | ||||||
| 38<\/td>\n | Figure 11 Median control chart for spend versus budget Figure 12 Conventional individuals chart for spend versus budget <\/td>\n<\/tr>\n | ||||||
| 39<\/td>\n | 10 Construction and use of basic cusum charts 10.1 Purpose a) Essentially it is a special type of moving mean chart. Moving means are used to smooth volatile data and detect underlying pe… b) It is sensitive to changes in the mean of the process. c) Any change in the mean, and the extent of the change, is indicated visually by a change in the slope of the chart: 1) A horizontal slope indicates an on-target or reference value. 2) A downwards slope indicates a mean less than the reference or target value; the steeper the slope the bigger the difference. 3) An upwards slope indicates a mean more than the reference or target value; the steeper the slope the bigger the difference. d) It can be used as a running basis for control, for example, using simple V-masks (see 11.1). e) It can be used retrospectively in performance improvement projects for investigative and diagnostic purposes. 10.2 Method of construction (manual) a) Choose a reference, target or preferred value. The mean of historical results provide the best discrimination. b) Order the individual results in a meaningful (e.g. chronological) sequence. Place these in column 1. c) Subtract the reference value from each result and place into column 2. d) Progressively sum the values obtained in b). Place in column 3. This is the column to be plotted as a cusum chart. 10.3 Method of construction (computer generated approach) <\/td>\n<\/tr>\n | ||||||
| 40<\/td>\n | 10.4 Case study (manual method) Table 10 Example of construction of cusum chart for % silicon content Figure 13 Basic cusum chart for % silicon content <\/td>\n<\/tr>\n | ||||||
| 41<\/td>\n | a) A horizontal chain line indicates that the process performance is running at the overall mean level of 2.23. This level was c… b) The first change in primary slope occurs at measurement 12. The slope of the line up to this point is sharply negative indica… c) From measurement 13 to 20 the slope is slightly positive, indicating a process mean just above the reference value of 2.23. The estimated value of the mean over this period is 2.24. d) From measurement 21 to 36 the slope of the line indicates that the process level is constant at a higher level than previously. The estimated value is now 2.30, the optimum value. e) Arising from this visual interpretation, the volatility of the original data is removed to reflect the various estimated changes in level. f) The individuals chart in Figure 8 can now be refined further to detect, and reveal the results of, the second adjustment of the process (see Figure 14). g) Using the cusum chart in diagnostic mode revealed the need for a second adjustment to the process to bring it to the desired … Figure 14 Refined individuals control chart for % silicon content <\/td>\n<\/tr>\n | ||||||
| 42<\/td>\n | 11 Construction and use of individual measurement cusum charts for process control 11.1 Purpose 11.2 Case study (manual method) Figure 15 Cusum chart for % silicon content in control mode <\/td>\n<\/tr>\n | ||||||
| 43<\/td>\n | 12 Construction and use of individual and moving range charts for attribute data 12.1 Purpose 12.2 Data classification 12.2.1 Classified data 12.2.2 Count data <\/td>\n<\/tr>\n | ||||||
| 44<\/td>\n | 12.2.3 Issues to consider <\/td>\n<\/tr>\n | ||||||
| 45<\/td>\n | 12.3 Case studies 12.3.1 Case study 1 Figure 16 Cusum chart for nonconforming deliveries <\/td>\n<\/tr>\n | ||||||
| 46<\/td>\n | Figure 17 np control chart for nonconforming deliveries Figure 18 X control chart for nonconforming deliveries <\/td>\n<\/tr>\n | ||||||
| 47<\/td>\n | 12.3.2 Case study 2 Figure 19 Cusum chart for fabric nonconformities <\/td>\n<\/tr>\n | ||||||
| 48<\/td>\n | Figure 20 Cusum chart for fabric nonconformities per roll Figure 21 c chart for fabric nonconformities per roll 12.3.3 Case study 3 a) Reference to BS 5701-3 shows that the standard deviation used to calculate standard control limits for count data is equal to the square root of the mean. For a standard count, c, chart: b) Count data is discriminated to the nearest integer. <\/td>\n<\/tr>\n | ||||||
| 49<\/td>\n | Table 11 Count of system outages per week over a period of one year Figure 22 c chart of outages per week Table 12 Conversion of outage count per month to monthly outage rate <\/td>\n<\/tr>\n | ||||||
| 50<\/td>\n | Figure 23 Individuals chart of system outage rate per month <\/td>\n<\/tr>\n | ||||||
| 51<\/td>\n | Annex A (informative) Case studies for typical applications A.1 General a) steering a process to the most desirable level; b) improving process consistency by standardizing on objective decision rules for detecting out of control situations; c) creating a mind-set for improving process performance; d) indicating whether or not an improvement, or deterioration, is real or imaginary; e) showing the general applicability of the method; f) gaining knowledge about the process; g) presenting information in a standardized and readily assimilated manner. A.2 Food and drink, environment A.2.1 Scenario <\/td>\n<\/tr>\n | ||||||
| 52<\/td>\n | A.2.2 Strontium-90 case study Table A.1 Strontium-90 in cows\u2019 milk <\/td>\n<\/tr>\n | ||||||
| 53<\/td>\n | Figure A.1 Individuals chart for strontium-90 in cows\u2019 milk in UK location a) The value of strontium-90 peaked at 1.36 Bq\/g of calcium in 1964. b) The progressive decrease shown reflects a decline in atmospheric fall-out following the partial above-ground nuclear weapons test ban treaty in 1963. c) Judged against the latest stable mean the chart is seen to be out-of-control until 1986. d) In 1986, in spite of the Chernobyl reactor incident, and unlike the caesium-137 spike in Figure A.2, strontium-90 stabilized … e) It is noted that the values have peaked at 1.36. This is over 26 times higher than the latest stable readings. It is pertinen… f) From 1986 the control chart is seen to be in-control at a base level of 0.051 Bq\/g of calcium. As such concentrations in milk… <\/td>\n<\/tr>\n | ||||||
| 54<\/td>\n | A.2.3 Caesium-137 case study Table A.2 Concentrations of caesium-137 in cows\u2019 milk in a UK location Figure A.2 Individuals chart for caesium-137 in cows\u2019 milk a) A maximum of 1.9 was recorded at the first monitored reading in 1966. b) A progressive decrease in caesium-137, due to the partial cessation of above-ground nuclear weapons testing with the USA\/UK\/USSR treaty of 1963, continued until 1986. c) In 1985, just prior to Chernobyl, the caesium-137 level was beginning to stabilize at 0.05 Bq\/l. This value is much less than that experienced in 1966. <\/td>\n<\/tr>\n | ||||||
| 55<\/td>\n | d) A 220-fold jump in caesium-137 level then took place as a direct result of the Chernobyl nuclear incident in 1986. e) A further progressive decrease in caesium-137 took place after Chernobyl and started to stabilize seven years later, in 1993. This stabilization is seen to be at a level of around 0.10, still twice that which was experienced prior to Chernobyl. f) The European Community Food Intervention Levels (EC\/686\/95) have set post-accident activity levels for food at which interven… A.3 Healthcare A.3.1 Diabetes Table A.3 Daily fasting blood sugar levels (mg\/dl) Figure A.3 Individuals chart of daily fasting blood sugar levels <\/td>\n<\/tr>\n | ||||||
| 56<\/td>\n | A.3.2 Asthma Table A.4 Project desirables and associated means to improve asthma care Table A.5 Daily readings of peak expiratory flow rate (PEFR): Days 1-10 <\/td>\n<\/tr>\n | ||||||
| 57<\/td>\n | Figure A.4 Individuals chart for peak expiratory flow rate Table A.6 Daily readings of peak expiratory flow rate (PEFR): Days 11-20 <\/td>\n<\/tr>\n | ||||||
| 58<\/td>\n | Table A.7 Daily readings of peak expiratory flow rate (PEFR): Days 21-30 A.4 Finance A.4.1 London Stock Exchange (LSE) Trades Figure A.5 Cusum chart of the number of trades executed at the LSE per month <\/td>\n<\/tr>\n | ||||||
| 59<\/td>\n | Figure A.6 Individuals chart of the number of trades executed at the LSE per month A.4.2 London Stock Exchange all share index <\/td>\n<\/tr>\n | ||||||
| 60<\/td>\n | Figure A.7 FTSE all share index by month A.5 Conclusion <\/td>\n<\/tr>\n | ||||||
| 61<\/td>\n | Annex B (informative) Control chart templates Figure B.1 Individual and moving range control chart template <\/td>\n<\/tr>\n | ||||||
| 62<\/td>\n | Figure B.2 Moving mean and moving range control chart template <\/td>\n<\/tr>\n | ||||||
| 63<\/td>\n | Annex C (normative) Relationship between Cpk and percentages and parts-per-million (ppm) out-of-specification Table C.1 Proportion of values expected outside of an upper and lower limit in terms of values of capability indices for a normally distributed characteristic <\/td>\n<\/tr>\n | ||||||
| 64<\/td>\n | Bibliography <\/td>\n<\/tr>\n<\/table>\n","protected":false},"excerpt":{"rendered":" Guide to statistical process control (SPC) charts for variables – Charts for individual values<\/b><\/p>\n |