Measuring average time

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I have a dataset of times (time to execute a task) and I'm now interested in calculating the average time. I might just be mixing stuff up and doing it more complicated than it is but.. The time's have a distribution that is lognormal, is it appropriate to use the arthmetic mean?

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The arithmetic mean corresponds to most people's intuitive sense of average for reasonable distributions. When you have a large peak with far outliers in one direction, people tend to like the median better for many purposes. Whether the mean is appropriate for your use depends on your use. Whatever average you use, you should be clear about what it is.

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To be clear, are you trying to compute the average duration of a task competion? Or trying to compute the average start time?

And what do you intend to do with the result? That answer would help settle the question of mean vs. median.

Depedending on what you intend to do, one stat may be much better suited to your needs than the other.