TE Taylor Enterprises, Inc.
www.variation.com
Quality and Statistics
Books, Software, Training and Consulting


Search variation.com

Enter keywords: 

Exact Match Search

 

Site Map

Products (HOME)

Books

Software

Courses

Consulting


Expertise

Acceptance Sampling

Process Validation

CAPAs and Trending of Quality Data

FMEA

Measurement Systems Analysis

Spec Setting, Tolerance Analysis and Robust Design

General Statistics

Statistical Process Control

Design of Experiments

Six Sigma


Store  

What's New

Technical Library

FAQ


  Contact Info

Chairman
Dr. Wayne A. Taylor
President
Ann Taylor
Telephone
1 (847) 367-1032
FAX
1 (847) 367-1037
Postal address
5510 Fairmont Rd.
Libertyville, IL 60048
USA
Electronic mail
info@variation.com
Web
www.variation.com

 

Subscribe to our Web Site

By entering your e-mail address and clicking the Subscribe button, you will automatically be added to our mailing list.  You will receive an e-mail when new versions of our software or books are available as well as other significant announcements.  (privacy policy).

E-mail address to send notifications to:

    

 

Copyright © 2017 by Taylor Enterprises, Inc., All Rights Reserved.

              Normalized Individuals (IN) Control Chart          

Dr. Wayne A. Taylor

Abstract: The only commonly used control chart that cannot be normalized is the Individuals (I) chart.† A procedure, called a Normalized Individuals (IN) chart is provided for normalizing data associated with an I chart.† The IN chart works nearly identical to the Laney Uí and Pí charts for count data.† The IN chart has certain theoretical advantages as the estimates of the standard deviation remain unbiased in all situations where the process is stable.† The IN chart also has the advantage that it can be used for other applications not involving counts.

1.0††† Introduction

-S, U, P, Laney Uí and Laney Pí control charts all allow the charts to be normalized based on the sample size or number of opportunities.† The only commonly used control chart that cannot be normalized is the Individuals (I) chart.† A procedure, called a Normalized Individuals (IN) chart is provided for normalizing data associated with an I chart.† This chart has been implemented in Taylor (2017c), including an Excel spreadsheet.

Donald Wheeler (2011) recommends an I chart for handling count data, which he refers to as an XmR chart:

ďIn contrast to this use of theoretical models which may or may not be correct, the XmR chart provides us with empirical limits that are actually based upon the variation present in the data. This means that you can use an XmR chart with count based data anytime you wish. Since the p-chart, the np-chart, the c-chart, and the u-chart are all special cases of the chart for individual values, the XmR chart will mimic these specialty charts when they are appropriate and will differ from them when they are wrong.Ē

Richard Laney (2002) points out that the I chart cannot be normalized to account for differences in sample size or opportunities, resulting in constant control limits.† He provides the Laney Uí and Pí charts that address this issue for count data.† The IN chart works nearly identical to the Laney Uí and Pí charts for count data, so is equally effective at addressing the constant limits concern.† The IN chart has certain theoretical advantages as the estimates of the standard deviation remain unbiased in all situations where the process is stable.

The IN chart also has the advantage that it can be used for other applications not involving counts.† This includes control charts of lots with between lot variation and unequal sample sizes.† It also includes control charts of stability data for out of trend values with unequal time periods.

The †and IN control charts handle most needs, simplifying the selection of a control chart.


2.0††† Individuals (I) Chart for the Normal Distribution

The I chart is the basis for the other procedures provided.† Assume the values are represented by , , Ö, , where the †are independent normal with common standard deviation σ.† They may have different means.

The averages †are subject to 1 or more shifts.† This means †in most cases except possibly for a small number of instances where a mean shift occurs.

ESTIMATING THE STANDARD DEVIATION

Because the average may shift 1 or more times, the total standard deviation of the †may overestimate σ.† A more robust estimator of σ is based on , which has distribution .† In most cases , so:

As a result, †has the standard half-normal distribution with:

Mean: ††††††††††††††††††††††† †††††††††††

Median:††††††††††††††††††††† †††††††††††

Standard Deviation:†††††††††††

When , this results in the following unbiased estimates of the standard deviation σ:

where †and the constant .† This results in the following estimate of the standard deviation:

†† where††

is an unbiased estimate of σ so long as no shifts occur.† If shifts occur some of the †are biased.† A more robust, but slightly less powerful, estimate of σ is:

CONTROL LIMITS FOR THE INDIVIDUALS (I) CHART

Control limits are the average plus and minus 3 standard deviations of the values being plotted.† For an I chart the values †are plotted.† The estimated average of the †is:

Using the estimates of σ from the previous section:

CONTROL LIMITS FOR THE MOVING S CHART

For the Moving S chart †are plotted.† †has:

††††††††††††††††††

†can be substituted for †in the above equation.

  

Table 1:† Control Limits for Individuals (I) Chart

Type of Chart

Estimator

Formulas

Individuals Chart:

 

 

Plot of values

 

 

 

Uses:

 

Moving S Chart:

 

 

Plot of

 

Moving Range Chart:

 

 

Plot of

 

 

Note that the Moving S and Moving R charts differ by a factor of d2 and provided essentially the same information.† However, the estimates and †are handier for other applications including estimating process capability, so the Moving S chart is preferred.


CONTROL LIMITS FOR THE MOVING R CHART

For the Moving R chart †are plotted.† The control limits are similarly scaled by :

The lower control limits of the moving S and R charts are negative.

  

3.0††† Normalized Individuals (IN) Chart for the Normal Distribution

The Normalized Individuals (IN) chart assumes instead:

The I chart is a special case of the IN chart with .† The †represent the sample size or number of opportunities that the †are based on.† The relationship between the average and standard deviation above is based on the effect of addition.† Assume .† Assuming the Yís are independent, regardless of the distribution of the Yís, †and .† †in most cases, except possibly for a small number of instances where a mean shift occurs.

The normalized values are then:

The normalized values †are plotted on the IN chart.

ESTIMATING THE STANDARD DEVIATION

Because the average may shift 1 or more times, the total standard deviation of the †will overestimate σ if there are shifts in the average.† A more robust estimate of σ is based on:

In most cases , so:

†has the standard half-normal distribution

When , this results in the following unbiased estimates of the standard deviation σ:

This results in the following estimate of the standard deviation:

†is an unbiased estimate of σ so long as no shifts occur.† If shifts occur some of the †are biased.† A more robust, but slightly less powerful, estimate of σ is:

CONTROL LIMITS FOR THE NORMALIZED INDIVIDUALS (IN) CHART

Control limits are the average plus and minus 3 standard deviations of the statistic being plotted.† For the IN chart the values †are plotted.† The estimated average of the †is:

Using the estimates of σ from the previous section the control limits for the ith point are:

††††††† or††††††††

CONTROL LIMITS FOR THE NORMALIZED MOVING S CHART

For the Normalized Moving S chart †are plotted.† †has:

††††††††††††††††††

†can be substituted for †in the above equation.† The lower control limits of the normalized moving S chart are negative.† Based on:

†with distribution function

Exact control limits using †= 0.0013498980316301 and †= 0.99865010196837 percentiles are:

†can be substituted for †in the above equation.

  

4.0††† Comparison to Laney Uí Chart

The Laney Uí chart has control limits:

The resulting estimate of the standard deviation is:

Compare this to the estimate of the standard deviation for the IN chart:

 

Table 2:† Control Limits for Normalized Individuals (IN) Chart

Type of Chart

Estimator

Formulas

Normalized Individuals Chart:

 

Plot of

 

 

Uses:

Normalized Moving S Chart:

 

Plot of

 

  

When the †are all equal, the two estimates are equivalent.

The two estimates differ with how they handle a changing number of opportunities.

†is an unbiased estimate, as previously shown.† can be biased.† This is due to the fact that the correction factor †assumes the two items subtracted are independent of each other.† They are not independent because they include the common term .† For larger sets of data, the estimate †is more precise and the two parts are nearly independent.

Table 3 shows the performance of the following two estimators for the case where the standard deviation is 1.† Difference combinations of , †and †are shown.† †is number of opportunities for all the other data points combined.

Laney:†† †††††††††††††† Taylor:†††

 

Table 3: †Comparison of Laney and Taylor Estimators
of the Moving Standard Deviation

 

 

 

O1

O2

O3

Average

SD

Average

SD

1

1

1

1.000

0.756

1.000

0.756

1

10

1

0.897

0.658

1.000

0.756

1

1

10

1.000

0.756

1.000

0.756

1

10

10

0.943

0.712

1.000

0.756

1

1

100

1.000

0.756

1.000

0.756

1

10

100

0.989

0.748

1.000

0.756

1

1

1000

1.000

0.756

1.000

0.756

1

10

1000

0.999

0.755

1.000

0.756

  

Table 1 was generated using simulations of 100,000,000 trials each, giving 4 digits of precision.† Only 3 digits are shown.† Table 1 confirms:

  • is unbiased
  • †= , and thus unbiased when .
  • †is biased when .† However, this bias is 1% or less when .

While is the theoretically better estimator, for all practical purposes the two estimators perform the same.† Either can be used.

One nice feature of the Laney Uí chart is that is a useful measure of over dispersion.† A value close to 1 suggests a U chart could be used.† For an IN chart, †could be defined as below for applications involving normalized count data:

It is common practice to use a pair of charts to show the average and variation (-S, I-MSD).† Similarly, a Laney Uí or Pí chart can be paired with a moving †chart.

 

5.0††† Examples of Applications

COMPLAINT DATA

The first set of data is the complaint data shown in Table 4.† There are 20 values.† The sales volume, representing the number of opportunities, steadily increases.† The data is over dispersed relative to the Poisson distribution with about half the points falling outside the control limits on a U-chart.

  

Table 4:† Example Complaint Data

Month

Complaints (Xi)

Sales Volume (Oi)

Normalized Complaints Rates (Ni)

1

426

90000

0.004733333

2

543

110000

0.004936364

3

428

90000

0.004755556

4

67

40000

0.001675

5

303

60000

0.00505

6

481

70000

0.006871429

7

304

90000

0.003377778

8

718

120000

0.005983333

9

681

150000

0.00454

10

1030

210000

0.004904762

11

704

190000

0.003705263

12

1062

250000

0.004248

13

1085

220000

0.004931818

14

1311

210000

0.006242857

15

1309

230000

0.005691304

16

1342

220000

0.0061

17

1740

310000

0.005612903

18

1468

330000

0.004448485

19

1364

320000

0.0042625

20

1824

330000

0.005527273

  

Figure 1 shows the Laney Uí Chart and Figure 2 shows the IN chart of this data.† They are nearly identical and result in the same conclusion that the complaint rate is unchanged.† SigmaZ values are also shown, which are similar.† The two charts use different estimators of the standard deviation, so there will be slight differences between the 2 charts for each individual set of data. †The IN chart can be used anytime a Laney Uí or Pí chart can be used.

A Normalized Moving S chart is shown in Figure 3.† A Moving †chart can be added to a Laney Uí chart.† Figure 4 shows a moving chart for the complaint data.† It looks nearly identical to the Normalized Moving S chart in Figure 3, except for the scale.† The IN chart also has an option of using the median rather than average to estimate the standard deviation.† This option can also be extended to the Laney Uí chart.

  

Figure 1:† Laney Uí Chart of Complaint Data

  

Figure 2:† IN Chart of Complaint Data

  

Figure 3:† Normalized Moving S Chart of Complaint Data

  

Figure 4: †Moving SigmaZ Chart of Complaint Data

  

BETWEEN/WITHIN LOT VARIATION

While for count data the Laney Uí and IN charts are interchangeable, there are many other applications of the IN chart to non-count data where only the IN chart is applicable.

For example, the †chart assumes there is a single source of variation.† An alternative model better fitting some processes is the between/within lot variation model.† This model assumes there are two sources of variation, one for the lot averages and one for individual units within a lot around the lot average.† An I chart of the lot averages is recommended in this case.† However, if the sample size varies from lot-to-lot, an IN chart is more appropriate.† Table 5 shows an example set of data.† Figure 5 shows an IN chart of lot averages where some averages are based on 5 samples and other 13.

   

Figure 5: IN Chart of Lot Averages with Unequal Sample Sizes

  

Table 5:† Example Between Lot Variation Data

Lot

Value (Sum)

N

Normalized Value (Average)

1

1265.983327

13

97.38333286

2

1265.494062

13

97.34569709

3

1234.095432

13

94.93041783

4

467.0938368

5

93.41876736

5

480.5207487

5

96.10414975

6

1263.362522

13

97.18173242

7

471.6089798

5

94.32179597

8

483.3272231

5

96.66544462

9

1247.482101

13

95.96016158

10

1252.155511

13

96.31965468

11

475.8327811

5

95.16655622

12

1244.424527

13

95.72496359

13

479.2992493

5

95.85984986

14

1225.883257

13

94.29871211

15

1246.284863

13

95.86806642

16

1247.863834

13

95.98952569

17

476.4285921

5

95.28571841

18

480.9507213

5

96.19014427

19

1246.33594

13

95.8719954

20

1244.075692

13

95.69813015

  

Table 6:† Example Stability Data

Month

Value

0

100.8281

3

99.8719

6

99.16905

9

98.08248

12

98.12253

18

94.95572

24

93.00887

36

88.96801

48

85.21958

 

LINEAR TRENDS WITH UNEQUAL INTERVALS

Another application is when there is a linear trend over time but values are collected at unequal intervals.† An example is the detection of out of trend (OOT) values during a stability study where data is collected at times 0, 3, 6, 9, 12, 18, 24, 36 and 48 months.† Table 6 shows an example set of data.

Figure 6 shows a linear regression of the data.† The twelve-month data point appears to be higher than expected, however, falls within the 95% prediction interval.† Flagging a point outside the 95% prediction interval is a poor approach to detecting OOT values.† It would result in false signal for around 5% of the stability points.† This translates to close to 50% of stability studies signaling an OOT point.† Further the OOT value has widened the prediction interval so it does not fall outside them.† Robust regression estimators could solve the second issue but not the first.

  

Figure 6: Linear Regression of Stability Data

 

Before trending the data on an IN chart, the differences between consecutive values must be calculated as shown in Table 7.† When this is done, the normalized values are the slopes.

   

Table 7:† Changes and Slopes of Example Stability Data

Month

Change

Length Interval

Normalized Change (Slope)

3

-0.65563532

3

-0.218545106

6

-0.85848342

3

-0.286161138

9

-1.17796488

3

-0.392654961

12

0.193200231

3

0.064400077

18

-3.09996878

6

-0.516661463

24

-1.98809354

6

-0.331348924

36

-3.72314215

12

-0.310261846

48

-4.31874831

12

-0.359895693

   

Figure 7 shows the resulting IN chart.† The median standard deviation estimator was used to avoid any OOT point inflating the estimated variation and widening the control limits.† It shows the 12-month point is OOT.

  

Figure 7: IN Chart of Changes to Detect Out-Of-Trend Point

 

It is clear that the assumption †applies to the complaint and between lot data.† Both are additive sets of data for which this assumption is assured to be met.† It is not as clear it is met for the stability data.† There are numerous sources of variation including measurement error, variation in the starting values for each unit and variation in the slopes for each unit.† Some of these are constant and some grow linearly with time.† Combining all these sources of variation results in something somewhere in between, making the assumption reasonable.

  

6.0††† Conclusions

Based on the methods and comparisons presented, the following recommendations are made relative to control charting practice:

  • The IN chart handles count and pass/fail data where a Laney Uí or Pí chart might be used.† It also handles many other situations involving non-count data where a Laney Uí or Pí chart do not apply.
  • The †and IN charts handle most needs, simplifying the selection of a chart.† These are the only 2 charts needed in most cases.† The decision between them is based on whether there are multiple values per time period or not.
  • The exception to this rule is nonnormal data.† One such case is when counts are low and follow the binomial or Poisson distributions.† In this case U and P charts with adjusted control limits should be used as described in Taylor (2017a, b).
  • If the Laney Uí or Pí chart is used, consider accompanying it with a moving †chart.† It is inconsistent to show -S, and I-Moving S charts, but not to do the same for a Laney Uí or Pí chart, as they are all based on a time ordered series of estimates of the standard deviation.
  • S and moving S charts are preferable to R and Moving R charts because the estimates and †are useful for other applications including estimating process capability.

  

7.0††† References

Laney, David (2002), Improved Control Charts for Attributes, Quality Engineering, 14(4), 531Ė537.

Wheeler, Donald (2011), What About p-Charts?, Quality Digest, www.qualitydigest.com/inside/quality-insider-article/what-about-p-charts.html.

Taylor, Wayne (2017a), Adjusted Control Limits for U Charts, Taylor Enterprises, Inc., www.variation.com/techlib/brief2.html.

Taylor, Wayne (2017b), Adjusted Control Limits for P Charts, Taylor Enterprises, Inc., www.variation.com/techlib/brief3.html.

Taylor, Wayne (2017c), Statistical Procedures for the Medical Device, Taylor Enterprises, Inc., www.variation.com/procedures.


Copyright © 1997-2017 Taylor Enterprises, Inc.
Last modified: September 08, 2017