Q 576. Laney U' Chart is an advanced variation of the traditional C and U charts. Compare the three different control charts with their advantages and disadvantages. What are the specific conditions under which you will prefer to use a Laney U' chart. Provide an example to support your answer.

 

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Control Chart is a graph used to study how a process changes over time with data plotted in time order. Both C & U charts are used to showcase attribute data to check the stability of a single unit, which might have one or more defects. The main distinguishing factor between the two is that the C chart is used when the sample size is fixed, and the U chart is used if the sample size is not fixed. 
They also work as a health checkup for the process and enables us to take necessary preventive action for causes that make a process unstable or out of control. The Laney U' chart is like a traditional U chart however it is more accurate in identifying the variation between common & special cause. 
Example in manufacturing number if defects per batch is monitored but the sample size varies because some batches are bigger than others. Using U chart with constant control limit will not reflect the process performance accurately, as the sample sizes affect the variability. In this case, the Laney U' chart would be appropriate. It would adjust the control limits based on the varying sample sizes, providing a more accurate indication of process stability and allowing for effective monitoring and control

The Laney’s U chart is based on the working principle of a u chart used to monitor number of defects per unit– the difference being that it adjusts for very large sample sizes(usually > 5000), to provide an accurate assessment of control points than in a u chart. Here in this case there may more than 1 defect in the same unit and hence defects per unit is the measure .

 

 Laney’s U chart is used in case of over dispersion or under dispersion , to distinguish between common cause and special cause variation . Points on a U chart are shown out of control due to over dispersion. Laney’s sigma Z is a measure of over dispersion. While  U chart shows points out of control ,applying same data to plot Laney’s u chart shows points in control.

 

Practical use- case Example:

 

Let us take an example of an automatic toll deduction from vehicles in a toll plaza. Because the number of vehicles crossing the toll will be different at each day , the number of errors that occurs due to non-deduction of toll fees from vehicles also differs. The Toll booth manager is interested in seeing the number of errors at each toll counter of the toll plaza  to fix the issue .He can use the Laney u chart since the sample size is large and individual counters would have varying sub group size with error numbers also different. There may be more than 1 defect for the same vehicle passing through the toll Plaza.

Here the Error is Toll fee not deducted automatically from vehicle passing , due to number of reasons such as Scanning machine not able to read, scanning machine not consistent , scanning position of vehicle not consistent, different type of vehicle car, trucks etc., Here the control limits vary because the sub group size varies for each counter each day .Thus a Laney u chart can accurately distinguish the defects at each counter accounting to varying sub group size and adjusting the over dispersion. By monitoring those points out of control and u value – Toll plaza can work on the counter which is having more errors than the control limits and using Laney u chart, the Toll Plaza manager can set right the loss of revenue issue in the Toll booth.

Steps to do Laney U chart:

 

Control charts are the graphs plotted used to study how the process changes over time. Control chart has an upper line, lower line and the central line. Upper line defines upper control limit (+ 3 sigma away from central line) and lower line defines the lower control limit (-3 sigma away from central line) and the central line is the average. These charts also display the common cause variations and the special cause variations. Common cause is the natural variation that exists within any process. Special cause is the variation occurred due to a special event or change within the process and not natural. Studying the control charts, one can understand if the process is consistent with the variations or is unpredictable. Basis the stability of the process, changes are made to the process to make the process consistent.

 

Preferable conditions where Laney U chart can be used –

1.       It can be used when there is a large variation in the sample size. The traditional U chart is not effective when the variation is large.

2.       If the data shows non normality then the Laney U chart can be used as it considers mean values instead of median to study in the variations in the sample size.

 

Control Chart	Laney U Chart	U chart	C  chart
Definition	Laney U chart is an attribute chart used to measure the number of defects produced per unit. This chart is used when the sample displays over dispersion and under dispersion and when the sample size is large (> 5000) and assumptions are not met.	u-chart is an attribute control chart used to measure "count"-type data where the sample size is greater than one, typically the average number of nonconformities per unit.	c chart is an attribute chart used to measure the number of defects (count). It is generally used to monitor the number of defects in constant size units.
Pros	Laney U chart can distinguish between common-cause variation and special-cause variation when the data shows over dispersion or under dispersion.	Used to monitor the process stability over time and monitor the effects of before and after process improvements and when the sample size varies.	Identifies if the process is stable and predictable and also monitors the effects of before and after process improvements. c chart is especially used when there are high opportunities for defects in the subgroup, but the actual number of defects is less.
Cons	Projecting the chart for sampler sample sizes is difficult and has limited applicability.	Not effective when the sample size is very large and sensitive when the variation in the means is huge.	Not effective when the sample size is large and also when the data is variable.
Data type & Sample Size	Discrete data and the sample size should be large (>5000)	Discrete data and calculate defects when the sample size is variable	Discrete data and calculate defects when the sample size is constant
Data Assumption	Poisson distribution assumption for the data is not valid.	Poisson distribution for the data is valid.	Poisson distribution for the data is valid.

 

No.	Laney’s U chart	U chart	C Chart
Descrp	A Laney U' chart is a control chart which is used to monitor data when the sample size is very large and the assumptions are not met. Laney’s U chart is used for number of defects per unit for variable set of data.	U chart is a control chart for data which has defects. It helps in identifying Special causes when the data has defects and has variable Sample size. The data follows Poisson distribution	C chart is a control chart for data which has defects. It helps in identifying Special causes when the data has defects and has Fixed Sample size. The data follows Poisson distribution
	It uses a weighted average of the defect rate across various subgroups to boost its sensitivity to minor changes in the process.	When the sample size varies there is a variation in the control limits as well.	C Chart should not be used for Variable sample size because it makes it difficult to understand.
Pros	It helps in giving more accuracy in distinguishing Special cause variation and common cause variation.	U chart is used to evaluate how many errors there are per unit of measurement to determine whether the process is under control. The control limit is fixed when the samples are the same size or when average sample size is considered.	C charts determines if the number of defects in each group are in control or not. It helps to identify the variations in each process and understand the problem indicators to take corrective measures leading to an improvement in the process.
	It helps is reducing false alarms and early detection of issues as it is sensitive to small changes in the process which further results in better accuracy and flexibility	U chart helps to show the patterns in defects data and predict future performance. It helps in tracking the production capacity by monitoring the number of defects per unit measurement and verifying the production levels are meeting the demand	C chart monitors the number of defects overtime and helps in identifying trend which help in determining the root cause of certain processes, eg Quality control as it provides a clear and easy approach to identify issues and take corrective measures
Cons	It includes calculating weighted average of the subgroup defect rates and a weighted standard deviation which sometimes is difficult to understand and perform leading to complexity in the process. It requires large set of data which consumes a lot of time. It is not useful for fixed sample sizes	It requires breaking the data down into smaller groups, which might take time and may not be possible for all Processes.	It is based on the assumption that the process is consistent and the defect rate is stable throughout. It may not offer accurate information if the process is unstable. The chart can be expensive and hard to create because it involves a lot of data and time

 

Conditions:

·       Laney’s U chart should be used when the Sample size is very large, >5000

·       When the data follows Poisson Distribution

·       Data follows Variable sample size

·       The probability of Defects occurring the process is constant

	Laney U’ Chart	U Chart	C Chart
Purpose	Laney P’ chart is useful in the situation when process data has large sub-group size and exhibit over dispersion or under dispersion (wider or narrower control limits).	U chart is used to monitor average number of defects per sample unit (average defect rate per unit). U chart is more useful when subgroup sizes are different.	C chart is used to monitor number of defects in one subgroup and compare with other subgroups. Similar, to np except that it counts for defect rather than defectives.
Example	For example, if the volume of sample subgroup is varying and very large, it would be appropriate to use Laney u chart. Ex. track the number of errors in the bills in the hospital billing department	For example, in a varying sample subgroup unit, different count of defects, DPU of one subgroup can be compared to another subgroup	For example, in a fixed sample subgroup of 50 units, defects in one subgroup can be compared to another subgroup
Appropriateness	Laney u' chart is widely used in healthcare quality monitoring that has a very large sample size (n > 5,000).	The u control chart is used if the area did not stay constant. The plotted values are fraction of the subgroup sizes.	The c control chart is used if the area stayed constant from sample to sample
Over/ Under Dispersion	The calculations for the Laney attributes charts include Sigma Z, which is an adjustment for overdispersion or under dispersion.	A low number of samples in sample subgroup make the band between high and low limits wider resulting in all data points falling inside of the control limits. (Under dispersion). Similarly larger sample size resulting in over dispersion.	A large number of samples in sample subgroup make the band between high and low limits narrower resulting in the data points falling outside of the control limits. (Over dispersion). Similarly, lower sample size resulting in under dispersion.
Control limits	Depending on Sigma Z value, control limits are adjusted for under or over dispersion 1.       If Sigma Z = 1, control limits are same as on traditional u chart 2.       If Sigma Z > 1, control limits are wider than traditional u chart to adjust for over dispersion 3.       If Sigma Z < 1, control limits are narrower than traditional u chart to adjust for under dispersion	Centerline = u bar = Sum of all defects / Sum of units inspected Avg. number of samples in a subgroup = N bar = Sum of units inspected / Number of subgroups Control limits = u bar +/- 3 * sqrt (u bar / n bar)	Centerline = c bar = Sum of all defects / Number of subgroups Control limits = c bar +/- 3 * sqrt(c bar)

While all the published answers are excellent, Ankur's, Avishi's and Sarala's responses stand out. Ankur is the winner for this question as he has explained the crucial adjustment aspect for over-dispersion and under-dispersion. 

A reminder to other respondents that the AI-generated answers are not being approved.