What are the grand mean, the UCL and the LCL of a control chart for the mean? Why do you suppose + (with underline) standard errors are used in control charts and not two standard errors or even one standard error?

To understand the terms you mentioned, let's break it down into two parts: the control chart for the mean and the terms associated with it.

1. Control Chart for the Mean:
A control chart for the mean is a statistical tool used in quality control to monitor and analyze the stability and variation of a process over time. It is commonly used to track the mean (average) value of a variable or characteristic of interest.

2. Terms Associated with Control Charts:
a) Grand Mean:
The grand mean refers to the overall average of the data being monitored. In a control chart for the mean, it represents the central value around which the data points should fluctuate.

b) Upper Control Limit (UCL):
The upper control limit is a statistical boundary that establishes the upper limit of acceptable variation in the mean of the process data. Any data point above this limit suggests a potential problem or an out-of-control condition that needs investigation.

c) Lower Control Limit (LCL):
The lower control limit is the statistical boundary that sets the lower limit of acceptable variation in the mean of the process data. Any data point below this limit also indicates a potential problem or an out-of-control condition.

Now, let's address your second question regarding the use of "+ standard errors" in control charts.

In control charts, the standard error is a measure of the estimated standard deviation of the sample mean. The standardized value for each data point is calculated as the deviation of the observed sample mean from the grand mean, divided by the standard error. These standardized values are plotted on the control chart.

The choice of using "+ standard errors" in control charts, instead of other values such as one or two standard errors, is based on statistical principles and the desired sensitivity of the control chart.

By using "+ standard errors," control charts emphasize smaller shifts or variations in the process mean. This approach helps in early detection of changes that may indicate a process going out of control. The specific number of standard errors chosen varies based on industry standards, historical data analysis, and the level of sensitivity required in that particular process.

Overall, control charts are based on the principle of detecting unusual or out-of-control conditions using statistical bounds. The choice of control limits and the number of standard errors determines the sensitivity and effectiveness of the chart in monitoring process variation.