Need Clarifications about Normal Data

1) Dan’s advice is good.
2) A 1-sided distribution is naturally non-normal, so there is no reason to normalize it.  To analyze process variability, you need special statistical tools, one of which is a Pearson Analysis.  Certain software packages can do this for you.  However, I don’t think that’s necessary for you, just follow Dan’s advice.
3) Remember that when X = A + B + C +…, the VARIANCE of X equals the sums of the Variances of A, B, C, etc.  Variance is Sigma squared, so Sigma(X) is the square root of the sum.  In other words, you can’t add Sigmas in a series, but you can add variances.

The question for me why actual data is normal but data is not. Did you check it with control chart? Is there any shift in process or some outlier? (like meintance or breakdown)

More Clarification! 
 
All the steps in your process appear to be the times required to complete a process step.  That does not produce a one‑side distribution.  The time needed to complete an event is equivalent to the measurement of a physical dimension.  Both measurements, assuming random error, will end up normally distributed.  With time you have a one‑side specification limit.   You might want a two‑side specification limit when errors are generated when someone is working too fast. 
 To the question of why your data is not normal.  Perhaps you do not have enough data points.  How many data points in the population?  Did you use the same person to take all the data?  Are the subgroup normal, etc., etc? You may find that answering these questions can add insight into how to shorten the process time.  For example, if several people were involved in the study  you need to examine the data for each person.  Perhaps one person need training!

I have run into the same issue evaluating efficiency problems. Typically, the distributions will be skewd one way due to machine constraints. Bimodal distributions or distributions with outliers often point toward method or man issues. Some of the lean tools such as process mapping, pace studies and work sampling often provide good insight into actual capabilities.

And if at all you wish to calculate the limits for your average, using Chebychev’s inequality might be of help.
 

Hi linga,
Seeing your operational definition of the CTQ and your goal statement.. I’ll say that though you have one sided specification , your distribution should be two tailed and a normal and not one tailed.
Secondly one of the reason that you are not getting a normal distribution may be the discrimination you are choosing to measure the time… If you are choosing minutes, and your actuall process width is not more than 10 minute, take seconds. I expect u’ll get a normal distribution. Further I can comment only on seeing the histogram and normality graph ….on the potential causes of you not getting normality.
I hope this gives u a direction to think..
vivek