Himanshu.Sharma

Background: Quantitative data is divided into two categories – Discrete and continuous data. As No. of water drops is Discrete but volume of water is continuous.
 So when data is countable and can take certain value it is called Discrete data, whole no is used to write Discrete data, Example – Number of student in a class, Number of player in cricket team, Number of Defect per day, Number of books in Book shelf etc. Following type of Graph can be used to represent summery of discrete data:
1-   Frequency table
2-   Bar Chart
3-   Pie Chart
4-   Pareto Chart
In Frequency table values are represented thorough tally mark representing frequency of data. Below example shows defect monitoring in injection moulding process. Tally mark helps in live data monitoring to avoid end of shift reconciliation. Summery of data can be used for different type of bar chart.

 
Bar chart use to do following:
1-   Compare counts by category:
2-   Display a frequencies on y axis
3-   Understand relationship between categorical variable.
 In Below Example: Bar chart shows Summary of defects on X axis and defect frequencies on Y axis - help to visualize defect wise frequency as shown in below example:

 
In Minitab and Excel many option are available for bar charts based on requirement. Below options are available in Minitab:

 
Cluster Bar chart showing comparison between Male and Female and preferences for ice cream flavour as shown below:

 
Another Example:  Pizza delivery service promise to deliver pizza in specified time. Delivery status compared with time as peak and non-peak.
Below possible outcome can be seen using bar graph that peak time having more late deliveries compares to non-peak time.

 
Bar chart for Ordinal data:  Service quality compared on 5 point Likert scale for customer feedback. As below:
1-   Very Good
2-   Good
3-   Average
4-   Bad
5-   Very Bad

 
Pie Chart: We can also make Pie Chart as shown below for defects to see frequency or proportion distribution for of each defect:
 

 
Pareto Chart is combination of line and bar graph and used to prioritize focus area for project.Below pareto chart shows Flow mark contribute maximum as 41.9% and flow chart and sink mark contribute 69.7%. It shows frequency of defects on primary axis and cumulative percentage on secondary axis. in Below Example X Axis and Primary Y axis using discret data and secondery Y axis having variable data.
 

 
 
Above example can be used to present graphical summery of discrete data and required inferance can be drawn.
 

Short term capability is sometimes calculated as Zlt +1.5 – Mostly when data is discrete because discrete data is almost every time long term or long term capability is sometime calculated as Zst -1.5. More often we have short term data and we calculated short term capability and to calculated long term capability we subtract 1.5 as Zst-1.5, because long term variation is more than short term variation. If focus on Y=f(X) , ie Output is function of Input than in short term there is very less variation in input factors as Man, machine, material , method, Mother Nature (Environment) and we get better Sigma level or capability but in long term input factor changes and effect output Y as well. So in long term 1.5 sigma level shift considered. Below is the Example of long term variation:
                                                                                   Y=f(X)
                        Output Y = f( man, machine, Material, Method, Measurement, Mother Nature)
 
Man – If we see man variation on long term is very significant because a man may be consistent for a month but for a year or six month he takes leaves, or leave the job then new man join or time to time change in his thoughts, mood and efficiency & focus, leads to more variation compare to short term. Although manpower training and competency process is to take care this factor but still variation will more in long term.
Machine – Machine may be consistent in short term means we get less variation in output  of a hour or a shift compare to variation  in output of six month or one year. Machine wear tear increase with the time, inadequate preventive maintenance, machine breakdown etc leads to more variation in long term. Machine Setting is also important aspect if machine setting change on every setup then process shift and more variation in long term. Companies do preventive maintenance and machine accuracy checks but still as machine gets older, consistency in output reduces and results in more long term variation.
Material – Input material consistency also change with the time for example – a sheet metal coil having less variation within coil but more variation between coils. Means one coil is not same as other coil. Another factor is Batch to Batch variation so batch to batch variation is more than within batch variation. Within coil, Coil to Coil, batch to batch variation is more than output may not be consistent in long term.
Method - To control method we have defined SOP and training to operator but if manual process and lack of poka yoke then some inconsistency can take place which can affect output. Ineffective root cause analysis also affects methods, which may lead to wrong countermeasure and unintended change in methods. Adherence of SOP by operator is also important if sop is available, displayed but operator does not follow consistently then variation will be more in long term.
Measurement – Consistency in measurement is also important in long run. In long run instrument or gauge wear tear take place so time to time Gage R&R and gage Stability study require. If our measurement is not consistent in long term then variation will be more in long run.
 
Mother Nature – Everyone likes nature variation as different weather, day night, morning evening but sometimes it leads to inconsistency in process and effect output. For Example – Investment casting process require stable environment as less temperature and humidity. So Investment Casting industry located in Nashik, INDIA because of more suitable environment compare to other part of INDIA. We all know long term variation in always more than short term variation in Environment which lead to high variation in long term.
 
So from above examples we understood that no process can be static over the time, even excellent process. By Convention, this long term variation is defined as 1.5σ Correction in short term α level. A 6σ short term process is considered 4.5α long term process. One more way to look at It:
 

 

 
 
·         Variance Within each subgroup can be pooled to determine an average of the within subgroup standard deviations.
 

 
·         Total Standard deviation is calculated from all of the data without regard to subgroup.
·         Pooled standard deviation does not account for between subgroup variation. Total standard deviation does.
·         Pooled Sigma is best estimate of within group variation.
Short term and Long term Data Collection
Short term
·         Gathered over limited number of cycle of interval.
·         Gathered over limited number of machine or operator.
·         Almost Always Continuous variable data
Long Term
·         Gathered over many cycles, intervals, equipment, operator.
·         May be discrete or continuous
·         Discrete data is Almost always long term.
 
 
Below is the short term and long term Sigma Level and respective yield:
Sigma Level Short Term
Sigme Level Long Term
% Yield
2
0.5
69.15
3
1.5
93.32
4
2.5
99.38
5
3.5
99.98
6
4.5
99.99966
 
So it is a valid assumption that consider 1.5σ shift in long term.