Control Charts in Lab and Trend Analysis

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USE OF CONTROL CHARTS IN LABS AND TREND ANALYSIS
BY YAMINI BHARDWAJ SIGMA TEST RESEARCH
CENTRE Email Mail_at_sigmatest.org
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Why control charts and trend analysis

• According to ISO/IEC 170252017 clause 7.7.1
  Ensuring the Validity of Results.
• The laboratory shall have a procedure for
  monitoring the validity of tests results.
• The resulting data shall be recorded in such a
  way that trends are detectable and, where
  practicable, statistical techniques shall be
  applied to review the results.

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CONTROL CHARTS

• The control chart chart on which some
  statistical measure of a series of sample is
  plotted in a timely order to steer the process
  with respect to that measure and to control and
  reduce variation.
• By comparing current data with existing control
  charts, one can draw conclusions about whether
  the process variation is consistent (in control)
  or is unpredictable (out of control, affected by
  special causes of variation). 

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Theoretical basis of control chart

• The control chart is a graphical display of data
  from process which allow a visual assessment of
  the process variability.
• At defined intervals, subgroup of items of a
  specified size are obtained and value of
  characteristic or feature of the item is
  determined. The data obtained is summarized
  through use of statistics and these statistics
  are plotted on control chart.
• A control chart consists of -
• Central line it reflects the level around which
  plotted statistics are expected to vary.
• Warning Limits it reflect that increased
  attention to be paid to the process when the
  point of observations fall outside the warning
  limit but inside the control limits.
• Control Limits/ Action Limits these are placed
  on both side of the central line defining the
  band within which the statistic can be expected
  to lie randomly when process is in control.

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Theoretical basis of control chart
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Benefits of using control charts

• A very powerful tool for internal quality control
• Changes in the quality of analysis can be
  detected very rapidly
• Easier to demonstrate ones quality and
  proficiency to clients and auditors.
• Indicate if the process is stable or not
• Estimate the magnitude of the inherent
  variability of the process.
• Identify, investigate and reduce the effect of
  special causes of variability.
• Identification of patterns of variability such as
  trends, cycle, runs etc.
• Assist in the assessment of the performance of a
  measurement system.

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Types of control charts
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• Shewhart control chart Control chart with
  shewhart control limits intended to distinguish
  between variation in a plotted measure due to
  random causes and that due to special causes. It
  is a graph of the values of a given subgroup
  characteristic versus the subgroup number.
• The control limits used are 3-sigma control
  limits.
• Control charts with no pre specified control
  limits It is used to detect any lack of control
  in RD stages, or in earlier pilot trials or
  initial studies.
• Control charts with pre specified control limits
  it is based on adopted standard values
  applicable to statistical measures plotted on the
  chart. The standard values are based on
• Prior representative data.
• Desired target value defined in specification.
• An economic value derived from consideration of
  needs of service and cost of production.
• Acceptance control chart Control charts intended
  to evaluate whether or not the plotted measure
  can be expected to satisfy specified tolerance.

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Shewhart Control chart
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Types of Shewhart control charts
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Variable Control Chart

• Control charts for variables is a means of
  visualizing the variations that occurred in the
  central tendency and mean of a set of
  observation.
• Benefits-
• Most of the process have the output
  characteristic that are measurable. So
  applicability is broad.
• The measurement value contains more information.
• The performance of the process can be analysed
  without regard to the specification.
• The subgroup size of variables are much smaller
  than that of attribute charts so are more
  efficient.

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Control limit formulae for shewhart variables
control charts
STATISTIC NO STANDARD VALUES GIVEN NO STANDARD VALUES GIVEN STANDARD VALUES GIVEN STANDARD VALUES GIVEN
CENTRAL LINE UCL AND LCL CENTRAL LINE UCL AND LCL
X X x A2R or x A3s Xo or µ X0 As0
R R D3R , D4R R0 or d2s0 D1s0 , D2s0
s s B3s , B4s s0 or d4s0 B5s0 , B6s0
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Mean charts

• This type of chart graphs the means (or averages)
  of a set of samples, plotted in order to monitor
  the mean of a variable.
• Mainly for precision check
• This graph shows changes in process and is
  affected by changes in process variability.
• It shows erratic and cyclic shifts in the
  process.
• It can also detect steady process changes like
  equipment wear.

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Range charts (r-chart)

• An R-chart is a type of control chart used to
  monitor the process variability (as the range)
  when measuring small subgroups (n 10) at
  regular intervals from a process.
• It is important for repeatability precision
  check.
• For better understanding of the trend and
  variation in the process -R charts are used
  together.

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-range control charts

• CASE -1 No standard values given.
• Table 1 shows measurement of outside radius of a
  plug. Four measurements are taken every half an
  hour for a total of 20 samples. And the specified
  tolerance are 0.219 dm and 0.125 dm.

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Subgroup No. Radius Radius Radius Radius Mean Range (R)
Subgroup No. X1 X2 X3 X4 Mean Range (R)
1 0.1898 0.1729 0.2067 0.1898 0.1898 0.0338
2 0.2012 0.1913 0.1878 0.1921 0.1931 0.0134
3 0.2217 0.2192 0.2078 0.1980 0.2117 0.0237
4 0.1832 0.1812 0.1963 0.1800 0.1852 0.0163
5 0.1692 0.2263 0.2066 0.2091 0.2028 0.0571
6 0.1621 0.1832 0.1914 0.1783 0.1788 0.0293
7 0.2001 0.1927 0.2169 0.2082 0.2045 0.0242
8 0.2401 0.1825 0.1910 0.2264 0.2100 0.0576
9 0.1996 0.1980 0.2076 0.2023 0.2019 0.0096
10 0.1783 0.1715 0.1829 0.1961 0.1822 0.0246
11 0.2166 0.1748 0.1960 0.1923 0.1949 0.0418
12 0.1924 0.1984 0.2377 0.2003 0.2072 0.0453
13 0.1768 0.1986 0.2241 0.2022 0.2004 0.0473
14 0.1923 0.1876 0.1903 0.1986 0.1922 0.0110
15 0.1924 0.1996 0.2120 0.2160 0.2050 0.0236
16 0.1720 0.1940 0.2116 0.2320 0.2024 0.0600
17 0.1824 0.1790 0.1876 0.1821 0.1828 0.0086
18 0.1812 0.1585 0.1699 0.1680 0.1694 0.0227
19 0.1700 0.1667 0.1694 0.1702 0.1691 0.0035
20 0.1698 0.1664 0.1700 0.1600 0.1666 0.0100
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R-Chart R-Chart R-Chart
Central line R 0.0287
UCL D4R 2.282 x 0.0287 0.0655
LCL D3R 0 X 0.0287 0 (since nlt 7)
X-chart X-chart X-chart
Central line X 0.1924
UCL X A2R 0.1924 (0.729 x 0.0287) 0.2133
LCL X-A2R 0.1924 - (0.729 x 0.0287) 0.1715
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• On examination the chart reveal that last
  three points are out of control and it indicate
  that some cause of variation may be operating.
• At this point remedial action is required and
  charting is continued by establishing revised
  control limits by discarding the out of control
  points.

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• Revised control limits

Revised X ? X /k 3.3454/17 0.1968 Revised X ? X /k 3.3454/17 0.1968 Revised X ? X /k 3.3454/17 0.1968
Revised R ? R /k 0.5272/17 0.0310 Revised R ? R /k 0.5272/17 0.0310 Revised R ? R /k 0.5272/17 0.0310
R-Chart R-Chart R-Chart
Central line R 0.0310
UCL D4R 2.282 x 0.0310 0.0707
LCL D3R 0 X 0.0287 0 (since nlt 7)
X-chart X-chart X-chart
Central line X 0.1968
UCL X A2R 0.1968 (0.729 x 0.0310) 0.2194
LCL X-A2R 0.1968 - (0.729 x 0.0310) 0.1742
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• CASE 2 - Standard values given.
• The tea importer wants to control his packaging
  process such that the mean weight of packages is
  100.6 g and based on previous packaging processes
  the standard deviation is 1.4g.
• Table 2 shows the subgroup average and subgroup
  average of 25 samples of size 5.

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Subgroup No. Subgroup average Subgroup Range
1 100.6 3.4
2 101.3 4.0
3 99.6 2.2
4 100.5 4.5
5 99.9 4.8
6 99.5 3.8
7 100.4 4.1
8 100.5 1.7
9 101.1 2.2
10 100.3 4.6
11 100.1 5.0
12 99.6 6.1
13 99.2 3.5
14 99.4 5.1
15 99.4 4.5
16 99.6 4.1
17 99.3 4.7
18 99.9 5.0
19 100.5 3.9
20 99.5 4.7
21 100.1 4.6
22 100.4 4.4
23 101.1 4.9
24 99.9 4.7
25 99.7 3.4
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R-Chart R-Chart R-Chart
Central line d2s0 2.326 X 1.4 3.3 g
UCL D2s0 4.918 x 1.4 6.9 g
LCL D1s0 0 X 1.4 0 (since nlt 7)
X-chart X-chart X-chart
Central line X 100.6
UCL X As0 100.6 (1.342 x 1.4) 102.5
LCL X - As0 100.6 - (1.342 x 1.4) 98.7
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Control charts for individuals, X and moving
range R

• Control charts for individuals are plotted when
  there is no rational subgroup possible to provide
  inter batch variability or when cost required for
  measurement is high so that repeated observations
  are not possible.
• Moving range is the absolute difference between
  successive pair of measurements in a series.

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Moving Ranges, R Moving Ranges, R
Central line R
UCL D4R
LCL D3R
Individuals, X Individuals, X
Central line X
UCL X E2R
LCL X - E2R
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• Cautions while preparing moving range charts-
• This chart is not sensitive to process change as
  mean and range chart.
• Care should be taken in interpretation if the
  process distribution is not normal
• This chart does not isolate piece-to-piece
  repeatability of a process.

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• Case study -
• The table shows the result of laboratory analysis
  of moisture samples of 10 successive lots of
  skim milk powder. As the sampling variation is
  negligible, so it was decided to take only one
  observation per lot.

Lot No. 1 2 3 4 5 6 7 8 9 10
X moisture 2.9 3.2 3.6 4.3 3.8 3.5 3.0 3.1 3.6 3.5
R Moving Range 0.3 0.4 0.7 0.5 0.3 0.5 0.1 0.5 0.1
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Moving Ranges, R Moving Ranges, R Moving Ranges, R
Central line R 0.38
UCL D4R 3.267 x 0.38 1.24
LCL D3R 0 x 0.38 0
Individuals, X Individuals, X
Central line X 3.45
UCL X E2R 3.45 (2.66 x 0.38) 4.46
LCL X - E2R 3.45 - (2.66 x 0.38) 2.44
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Recovery control charts

• These are charts created using a blank matrix
  that has been spiked with a known concentration
  of analyte.
• We chart the percent recovery of the spike. As
  long as the results fall within specified
  criteria, the QC passes.
• A typical acceptance for matrix spikes is 70
  120, but for large screens with many analytes,
  often 50 150 is acceptable

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• The following data were obtained for the
  repetitive spike recoveries of field samples.

Sample recovery Sample recovery Sample recovery
1 94.6 8 96.2 15 101.5
2 93.1 9 73.8 16 74.6
3 100.0 10 104.6 17 108.5
4 122.3 11 123.8 18 104.6
5 120.8 12 93.8 19 91.5
6 93.1 13 80.0 20 83.1
7 117.7 14 99.2 21 100.8
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Control chart for Duplicate samples

• An effective method for determining the precision
  of an analysis is to analyze duplicate samples.
• Duplicate samples are obtained by dividing a
  single gross sample into two parts
• We report the results for the duplicate samples,
  X1 and X2, by determining the standard deviation
  and relative standard deviation, between the two
  samples

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ILLUSTRATION
Consider the following analysis data of duplicate
samples
Months Measurements Measurements Mean Std . Dev. CV
Months Y1 Y2 Mean Std . Dev. CV
1 10.22 10.9 10.56 0.481 4.55
2 10.25 10.37 10.31 0.085 0.82
3 10.27 11.05 10.66 0.552 5.17
4 10.35 9.28 9.815 0.757 7.71
5 10.28 11.08 10.68 0.566 5.30
6 10.36 10.23 10.30 0.092 0.89
Grand Average Grand Average Grand Average 10.39 0.422 4.07
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• Determination of central line and control limits.
• Central line Std. Dev ( s )/Grand Average X
  100
• Upper control limit (UCL) (UCL)s/ X 100
• (UCL)s B4 s
• Lower control limit (LCL) (LCL)s/ X 100
• (LCL)s B3 s
• Here, B4 is the function of number of observation
  in subgroup. (n)
• Here, n2 so from table B4 3.267

Central line 4.07
Upper Control Limit 13.27
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UCL
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Applications in testing laboratory
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• Estimation of Measurement Uncertainty.
• Results from the control charts can, together
  with other data be used for calculating the
  measurement uncertainty, it may give a realistic
  estimate of the measurement uncertainty.
• Method Validation /Verification
• When the method has been changed only slightly,
  or if a standard method is adopted in the
  laboratory, control charts can be used to
  complement that the process is still under
  control.
• Performance of equipment.
• Equipment control charts can be drawn to monitor
  the bias, changes due to ageing, wear, drift
  noise.

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• Method Comparison
• By plotting control charts for two methods in
  parallel, it is easy to compare important
  information
• spread (from the standard deviation or from the
  range)
• bias (if a CRM is used)
• matrix effects (interferences), if spiking or a
  matrix CRM is used
• robustness, i.e. if one method is more
  sensitive to temperature shifts, handling etc.
• Method Blank and Reagent blank Monitoring.
• The control chart drawn for matrix blank/reagent
  blank can help to monitor the contamination
  occurring in a process due to cross
  contamination, gradual build-up of the
  contaminant, procedure failure or instrument
  instability.

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• Person comparison or qualification
• Control charts are helpful in comparing the
  performance of different persons in the
  laboratory. control charts can be employed during
  training and qualifying new staff in the
  laboratory. It is a powerful tool to estimate
  inter-analyst variation.
• Environmental parameters checks.
• The control charts give a very simple graphical
  presentation of any trends or unexpected
  variation that might influence the analyses.
• Control charts can also help to identify the
  effect of matrix on the recovery of the analyte.

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process control
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Westgard rules

• Westgard Rules are multirule QC rules to help
  analyze whether or not an analytical run is
  in-control or out-of-control.
• It uses a combination of decision criteria,
  usually 5 different control rules to judge the
  acceptability of an analytical run.
• The advantages of multirule QC procedures are
  that false rejection can be kept low while at the
  same time maintaining high error detection. This
  is done by selecting individual rules that have
  very low levels of false rejection, then building
  up the error detection by using these rules
  together

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• Rule 1-2s
• Definition The 1-2s Control Rule indicates one
  control result has exceeded the established mean
  /- 2SD range. This is a warning rule, which
  does not indicate an out-of-control condition,
  but is intended to initiate further testing.
• Interpretation If no other control rule is
  violated, then the warning is attributed to
  normal random error. Patient results are
  acceptable.
• Corrective Action No corrective action is
  required. However, the warning suggests a
  system error may be in the development. A
  comprehensive check of the routine maintenance
  schedule and review of the quality control
  handling and sampling technique is recommended.

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• Rule 1-3s
• Definition The 1-3s Control Rule indicates one
  control result has exceeded the established mean
  /- 3SD range. This is a rejection rule, which
  is sensitive to random error.
• Interpretation Excessive random error exists.
  The analyzer is out-of-control. The results are
  not acceptable and should be re-analyzed after
  corrective actions have solved the problem.
• Corrective Action Rerun the quality control
  level that is in question, emphasizing proper
  technique. If the repeated level is within /-
  2SD range then the problem can be attributed to
  random error. If the repeated level exceeds the
  /- 2SD range, then further corrective action
  should be conducted. The following are probable
  causes
• Inadequate or wrong /- 2SD range.
• Improper storage temperature correction of
  quality control results.
• Improper technique when handling the quality
  control. Change of quality control batch.
• Inadequate maintenance of the instrument.

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• Rule 2-2s
• Definition The 2-2s Control Rule indicates that
  two consecutive control results have exceeded the
  same mean /- 2SD limit. This is a rejection
  rule, which is sensitive to systematic errors.
• Interpretation A systematic error exists. The
  analyzer is out-of-control. This may be an
  early indicator for a shift in the mean value.
  Patient results are not acceptable and should be
  re-analyzed after corrective action has solved
  the problem.
• Corrective Action To resolve systematic errors,
  corrective action should be conducted to address
  the following probable causes
• Inadequate or wrong /- 2SD range.
• Improper technique when handling the quality
  control.
• Improper storage temperature correction of the
  quality control results.
• Change of the quality control batch.
• Inadequate maintenance of the instrument.

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• Rule R-4s
• Definition The R-4s Control Rule indicates that
  one result has exceeded the mean - 2SD limit and
  the adjacent result has exceeded the mean 2SD
  limit. This is a rejection rule, which is
  sensitive to random error.
• Interpretation Excessive random error exists.
  The analyzer is out-of-control. The results are
  not acceptable and should be re-analyzed after
  corrective action has solved the problem.
• Corrective Action As per Rule 1-3s

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• Rule 4-1s
• Definition The 4-1s Control Rule indicates four
  consecutive control results have exceeded the
  same mean /- 1SD limit. This is a rejection
  rule, which is sensitive to systematic errors.
• Interpretation A systematic error exists. The
  analyzer is out-of-control. This may be an
  early indicator for a shift in the mean value.
  The results are not acceptable and should be
  re-analyzed after corrective action has solved
  the problem.
• Corrective Action As per Rule 2-2s.

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• Rule 8-x,9-x,10-x,12-x
• Definition These Control Rule indicates eight,
  nine, ten or twelve consecutive control results
  have fallen on the same side of the mean. This is
  a rejection rule, which is sensitive to
  systematic errors.
• Interpretation A systematic error exists. The
  analyzer is out-of-control. The results are not
  acceptable and should be re-analyzed after
  corrective action has solved the problem.
• Corrective Action As per Rule 2-2s.

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• Rule 7-t
• Definition reject when seven control
  measurements trend in the same direction, i.e.,
  get progressively higher or progressively lower.
• Interpretation More than one process present
  (e.g. shifts, machines, raw materials)

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• 2 of 32s
• Definition reject when 2 out of 3 control
  measurements exceed the same mean plus 2s or mean
  minus 2s control limit.
• Interpretation represent sudden, large shifts
  from the average.  These are often fleeting a
  one-time occurrence of a special cause.

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Illustration of rules
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• Day 21, 22, 24, 26, 27, 30, 31, 33, 34, 36-44
  in control
• Day 23, 28, 29 12s
• Day 25 - 13s
• Day 32 22s
• Day 35 - R4s

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references

• ASTM manual on presentation of data and control
  chart analysis.
• FAO Internal Quality Control Of Data
  http//www.fao.org/docrep/w7295e/w7295e0a.htm

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• Thank you!

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