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# Continuous Data and SPC

I-MR (Individual Moving Range) is a control chart that is used to detect the presence of special causes with continuous data that are individual observations.

Xbar-R (Mean and Range) is a control chart that is used to detect the presence of special causes with continuous data that are in subgroup with size 8 or less.

Xbar-S (Mean and Standard Deviation) is a control chart that is used to detect the presence of special causes with continuous data that are in subgroup with size 9 or more.

There is no winner for this question. Applause for all the respondents - Ramjanam Singh, Partho Karmakar, Suresh Kumar Gupta, Raghavendra Rao Althar, Amit Simon.

## Question

Q 560. Imagine a 24x7 water bottling plant. The work is done in 3 shifts. QC team wants to check the consistency of the level of water being filled in each bottle. For this they pick 1 bottle every hour from the production line and check the water level. This implies, they pick 8 bottles every shift and 24 bottles every day. Which control chart (I-MR chart, Xbar-R chart, Xbar-S chart) should be used here for checking process stability? Support your answers with relevant illustrations.

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In this business case, I would prefer to use Xbar-S chart as the sample size is more than >10. And data is collected at different times. This chart will help in study the process variation. Generally, this control chart is used to study the standard deviation and process mean. S chart depicts width of sub-group data.

As we know, standard deviation is a good scale of variation over range because here entire data is considered, not only the lowest and highest values. Xbar-S chart will be used here to measure the process stability.

 Hourly Sample Shift 1 Shift 2 Shift 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Total 8 8 8 Grand Total 24

After computing the X bard and S values, we shall determine the control limits. First sub-group will be used to determine the process mean and standard deviation. Then, after plotting the X Bar and S chart, with the help of result values, we will plat the Sigma and X bar charts.

This graph will show the statistical process stability and control limits.

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In this scenario, the appropriate control chart to use would be the Xbar-R chart. The Xbar-R chart is used to monitor the stability of the process when the sample size is less than or equal to 10 and the subgroup size is constant. In this case, the subgroup size is 3 as there are three shifts in a day, and the sample size is 8 bottles per shift, which is less than 10.

To create an Xbar-R chart, the QC team would need to take measurements of the water level in each of the bottles they sample. They would then calculate the average water level (Xbar) and the range of the measurements (R) for each set of 8 bottles sampled during each shift.

Once they have collected enough data, they can plot the Xbar and R values on the chart to look for any patterns or trends in the data. The chart will show the average water level and range for each set of 8 bottles sampled during each shift and over time.

If the chart shows that the process mean and variation are stable over time and within control limits, then the process is considered to be under control. If the chart shows any patterns or trends that indicate the process mean or variation is changing, then the process may be out of control and require further investigation.

Here is an example of an Xbar-R chart for water level measurements in the bottling plant for QC done for consecutive 5 days  (Data is taken considering the fact that measurement is same for every day for sake of this example):

Xbar-R chart example:

 Day Shift Bottle 1 Bottle 2 Bottle 3 Bottle 4 Bottle 5 Bottle 6 Bottle 7 Bottle 8 1 1 20.1 20.3 20.0 20.2 20.1 20.3 20.2 20.0 1 2 20.2 20.2 20.0 20.1 20.1 20.0 20.2 20.1 1 3 20.1 20.2 20.2 20.3 20.0 20.1 20.1 20.3 2 1 20.1 20.3 20.0 20.2 20.1 20.3 20.2 20.0 2 2 20.2 20.2 20.0 20.1 20.1 20.0 20.2 20.1 2 3 20.1 20.2 20.2 20.3 20.0 20.1 20.1 20.3 3 1 20.1 20.3 20.0 20.2 20.1 20.3 20.2 20.0 3 2 20.2 20.2 20.0 20.1 20.1 20.0 20.2 20.1 3 3 20.1 20.2 20.2 20.3 20.0 20.1 20.1 20.3 4 1 20.1 20.3 20.0 20.2 20.1 20.3 20.2 20.0 4 2 20.2 20.2 20.0 20.1 20.1 20.0 20.2 20.1 4 3 20.1 20.2 20.2 20.3 20.0 20.1 20.1 20.3 5 1 20.1 20.3 20.0 20.2 20.1 20.3 20.2 20.0 5 2 20.2 20.2 20.0 20.1 20.1 20.0 20.2 20.1 5 3 20.1 20.2 20.2 20.3 20.0 20.1 20.1 20.3

In this example, the Xbar values are plotted in the top chart and the R values are plotted in the bottom chart. The horizontal axis shows the time period over which the data was collected, and the vertical axis shows the average water level and range, respectively.

The chart shows that the process mean for the water level is relatively stable over time, as the Xbar values are mostly within the control limits (the dashed lines on the chart). The chart also shows that the process variation, as measured by the R values, is also stable over time.

Therefore, based on this chart, the QC team can conclude that the water bottling process is under control and producing consistent water levels in each bottle.

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To decide which control chart should be used to check the consistency of water level in the bottles, we need to consider the type of data and the sample size.

Since the QC team picks only one bottle every hour, the sample size is one. This means that the individual measurement chart (I-MR chart) should be used for monitoring the water level.

The I-MR chart consists of two charts - the individual chart (I-chart) and the moving range chart (MR-chart). The I-chart is used to monitor the mean of the process, and the MR-chart is used to monitor the variability of the process.

Here is an illustration of an I-MR chart:

The I-chart shows the individual measurements of water level in each bottle, and the centerline represents the average water level. The upper and lower control limits (UCL and LCL) are calculated based on the average range of the data.

The MR-chart shows the moving range between consecutive measurements, and the centerline represents the average range. The UCL and LCL are calculated based on the control limits of the MR-chart.

If the data points fall within the control limits, the process is considered stable and under control. If there are any data points outside the control limits or any non-random patterns in the data, it indicates that the process is unstable and needs further investigation.

Therefore, by using the I-MR chart, the QC team can monitor the consistency of the water level in each bottle and take corrective actions if necessary to maintain process stability.

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Outcome to measure here is “water level up to mark”, metric will be actual water level measures. Data type is continuous. Sample size is fixed at 1 bottle per hour. I-MR chart is the best one. Individual Moving Range chart is used when continuous data is to be monitored with 1 sample. We need to compute the control limits for this process of water bottling. Compute the average of entire data set that is available at given point of time. Select the period of data such that in that period there has not been any major change in the processes. Idea is to make sure that we are looking at the set of data that is from common processes. Moving range is another component of I-MR chart, moving range is absolute difference between two consecutive points. Calculate average of moving range values. This information will help us to compute Upper Control Limit (UCL) and Lower Control Limits (LCL) for Individual chart and Moving Range chart. Using these initial values, we can set up process stability monitoring for the process.

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In this scenario, I think, the most relevant chart to check process stability is the Xbar-R chart, because an Xbar-R chart is used to plot the process average (Xbar) and variability (R) when subgroup size is constant (i.e. 8 bottles per shift). This is also due to the small sample size and continuous form of data. Both these factors are also ideal for an Xbar-R chart.

an I-MR chart is used for only one observation per subgroup. Since QC team checks 8 bottles every shift, I-MR chart is not the best choice. An Xbar-S chart uses (S) standard deviation instead of (R) range. However, Xbar-S should not be used as the data is not normally distributed.

Key use of X-Bar R chart

1. Data must be continous

2. Sample size (subgroup) must be at least two and should not be more than 10 to 12

3. Sample size cannot vary

4. Data should be collected in a random manner

5. at least 20 to 25 sample groups must be collected to ensure adequate measure of process variation has been taken.

Here is a chart plotted with the given example. Measurements are random.

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Very different perspectives to the same situation. Respondents have covered all the three control charts as the most appropriate to be used in this situation.

However, what is missing is the rationale for the choice and hence there is no winner for this question.

The choice is of the time period or the frequency at which the control chart should be plotted. It is one of the inputs that goes into deciding the right control chart and it is a balance between cost of sampling and testing versus cost of not detecting shifts in Mean or variation.

Sampling less frequently will not allow for sufficient detection of shifts in the process and hence the customer suffers.

Sampling too frequently will incur additional cost and hence the organization suffers.

Given a choice, do not be afraid to sample more frequently to begin with and then reduce the frequency once the process seems to be stable. As a rule of thumb, I-MR chart should always be used first to understand the shifts and drifts in individual points and will allow for better process control. Once the process is reasonably stable, you could then move to either Xbar-R or Xbar-S.

Illustration, I generated some random numbers for 10 days. All 3 charts given below

I-MR Chart shows that process is unstable.

Xbar-R chart (shift wise) shows that the process is stable.

Xbar-S chart (day wise) shows process is stable.

Now if we directly start using Xbar-R or Xbar-S chart, then we will never realize that the process is going out of control within a shift.

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