In this example, point 47 is equal to 41, which is just above the LCL. It affects the final LCL and UCL, yet falls within the control limits.

Although the standard deviation is not robust in the mathematical sense -- i.e. bounded no matter how large the outlier becomes -- it is rather robust in the practical sense -- not greatly affected, if the outlier is not "too" large. Just exclude the data point and convince yourself that this is the case.

How do we decide which points should be treated as a special cause and excluded

Special causes are not excluded, but they are investigated. This is the key idea behind spc charts. If the responsible person is not willing to investigate and thus optimise the process, there is not point of generating spc charts.

In my experience it is helpful to use such a point as 47 for the first investigation. Talking to the people who perform the measurement and the operators who change the input parameters on the manufacturing machines. You might learn that the real process differs from the one you expected. You might need to optimise your sampling theme.

Is outlier test a good idea?

No. SPC performs this test by using the 3 Sigma control limits. Using a second method, will just provide a second result. You end up discussing which result is "correct". Thus, you spend your time performing calculations and discussing them. However, you should spend your time optimising the processes.

SPC-charts work without histograms. Thus, it is not

*necessary* to plot the dataset in a histogram. However, it might help to understand the underlying process. Thus, my advices are:

a) Plot the dataset in several different ways (histograms, boxplot, multiple regression and qq plots of the residuals etc.) as each method highlights different components/aspects of the dataset.

b) Don't get lost in performing the statistical analysis, but focus your attention onto the practical conclusion and/or counter action.