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I am attempting to calculate the average number of days it takes for an order to go through our process - from receipt of order to delivery. When calculating the average days, how should I handle those orders that have not yet completed the cycle? Do I remove those entirely or apply today's date as the completed date?

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You have a survival analysis situation. Those orders that have not yet been completed are called 'censored'; they constitute partial information. For example, if a given order has been in process for five days but has not yet been completed, you know that the length of time is >5 even if you don't know the exact duration. Survival analyses are designed to accommodate censoring naturally. If your data fit a known distribution, such as the Weibull, you can get predicted means quite conveniently. Most often data do not follow any known distribution very well, however, so people tend to use the Cox proportional hazards model to analyze such data and use the median as the representative value instead of the mean.

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