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Vijay Tomar has provided the best answer to this question by elaborating the IMR chart and also providing an example of working with Control Chart for transformed data.

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The assumption of normality in IMR (Individuals and Moving Range) control charts is indeed a common practice. However, it is true that in the real world, data often deviates from a perfect normal distribution. While transformed data can be used to address non-normality to some extent, its usefulness in checking process stability depends on various factors, including the nature of the transformation applied and the specific characteristics of the data.   Transforming data involves applying mathematical operations to the original dataset to achieve a more desirable distribution or meet certain assumptions. Some common transformations include logarithmic, square root, and Box-Cox transformations. These transformations can help make the data more symmetrical and reduce skewness or variability.   To illustrate the usefulness of transformed data for checking process stability, let's consider an example with 20 data points. Suppose we have collected data on the time it takes for a process to complete a certain task in min. Here are the analysis original data points:   AHT1     25          30          32          31          29          27          30          28          33              26          24          28          34          40          45          42          50          38              35          30   First, lets perform the normality test and create an IMR control chart using the original data. Control chart was plotted using these data points and analyze the process stability based on the control limits and the presence of any out-of-control points             Next, let's apply Logarithmic transformation to the data, perform the normality test and create a new control chart. The transformed data points are as follows:       LOG AHT1          1.40       1.48       1.51       1.49       1.46       1.43       1.48              1.45       1.52       1.41       1.38       1.45       1.53       1.60       1.65              1.62       1.70       1.58       1.54       1.48               Interpretaion:   By analyzing both control charts, we can assess the process stability based on the control limits and patterns in the data points. If the data points fall within the control limits, show no significant patterns, and exhibit randomness, the process is considered stable.   In this example, both the raw and transformed data control charts show data points with same pattern. Therefore, we can conclude that there is no change in the process stability, even though the raw data was more skewed than the transformed data. The transformation allowed us to assess the process stability accurately. In this examples , though the moving range suggests process stability, one needs to understand the reason for shift in pattern along with the one data point that failed at point 17 in the individual chart.   The point to remember is that  the appropriateness of the transformation and the resulting interpretation depends on the specific context and understanding of the underlying process. It's always recommended to consult with subject matter experts and consider additional analyses if needed.   In order to utilize an I-MR Chart, there is no mandatory requirement for  the data to be always normal distributed, however   extremely skewed data can cause some unwanted outcome including high false-alarm rates. If data appears skewed, we can investigate to see whether that is an indication of an out of control process or as per expectations for this  type of process, and if expected, we can look for transformation of the data.

An I-MR Chart  is a control chart which is used when data is in continuous category and is collected once at a time. It consists of two charts placed one over above, I Chart which is individual chart and MR chart is plotted for moving range which is absolute value of the difference between two consecutive points. Data following normal distribution is an assumption while drawing I-MR chart however in practical or real-world problem data doesn’t follow normal distribution all the time hence Process stability follows major role. I-MR charts are very sensitive to Normality of the data. Non-Normal data if considered as normal data can cause unexpected behaviors including false alarm rates and difficulty in identifying the special cause variation. If data is not normal, it is always advisable to do transformation using Box-Cox or Johnsons transformation to avoid the false alarm and get the right behavior of data for stability and control. A normal distribution may have the value from minus to plus infinity.  In the real-world example this doesn’t occur physically very often.  For example, Cycle time cannot be in negative numbers.   Following is the Example for drawn I-MR charts when data is considered as normal however data is not normal and respective I-MR charts using data transformation: - Cycle time in Minutes: - Sl.No Cycle Time (In Minutes) S.No Cycle Time (In Minutes) S.No Cycle Time (In Minutes) 1 3196 11 267 21 322 2 241 12 302 22 147 3 372 13 518 23 774 4 42 14 554 24 185 5 481 15 566 25 556 6 6081 16 900 26 555 7 131 17 158 27 361 8 26 18 109 28 556 9 1445 19 167 29 898 10 363 20 51 30 170 Table 1 Probability Plot of dats Using Mini-tab Normality test for data in Table 1: - Normality test is done to illustrate whether data is normal or non-normal.   I-MR Charts drawn in Minitab for Table 1 mentioned assuming data following Normal distribution: - The chart clearly illustrates that process is out of control, however out of control points are trigged due to false alarms I-MR Chart drawn in Minitab for Table 1 mentioned after transforming days using Box-Cox transformation: - The Chart clearly illustrates that process is in control, our of control data points mentioned earlier were due to false alarm in wrong assumption of data being normal.        

In real world, the data rarely follows a normal distribution. Data is affected by outliers and measurement errors, because of which IMR chart may not be effective in detecting changes, as these charts assume that the data is normal.   In such cases, we can use charts for non normal data or we can transform the data, depending on the scenario and the data type. Data transformation techniques, such as the Box-Cox or Johnson transformation, can help stabilize variances and make the data more symmetric, allowing for more accurate interpretations and applications of control charts.   Data can also be split into rational sub groups and then buliding control charts on each split data.  Checking for process stability using transformed data helps to identify potential problems in a process. By transforming the data, we can highlight any trends or patterns that might not be visible in the raw data. This can help us to identify potential causes of variation. Then corrective actions can be identified to improve the stability of the process and prevent any problems from re-occurring.   Example: A call center will monitor the waiting times in the queue. If the data is not normal due to variability, then the distribution will be a non normal distribution and IMR chart may not accurately detect the actual problem or the trends. If the wait time data can be transformed, then the analysis on the data will be more accurate and will help the call center to take effective decisions.  

If the data is not normal and is transformed, then that impacts the accuracy and validity of the control charts if in case they are to be monitored. This makes it important to handle the transformation of data in a more cautious way. There are no particular method of transformation that fits a particular scenario. Different method of transformation have different benefits. Some of the key considerations while choosing transformational techniques are, extent of non-normality prevalent in the data, how easy it is to apply the method, how much better is the interpretability of the data, how well the transformation supports the process performance and other factors. Its worthwhile to give due consideration of non-parametric method before considering the data transformation. Median charts, run charts, Weibull distributions are few methods that fit into non-parametric methods. Data smoothing is an approach that can assist in removal of noise from the data, clustering, binning and regression methods are part of noise removal approach. Data generalization is another approach of data transformation, where low level attributes are transformed to high level attributes. This is suitable in case if there are finite number of values in data that have large number of distinct values. Like in case of data set that has data for city, street, country and state, we can define hierarchy of these attributes by ordering them with respect to their relations. Order would be country, state, city and street. These are couple of customized data transformation as needed for specific scenario.