Data Analysis › Statistics

Long-Tail Distribution

A few huge values and a very long tail of small ones, where the average misleads.

Also known as: long tail, heavy-tailed distribution, long-tailed data

A long-tail distribution has a small number of very large values and a very long stretch of small ones. Revenue by customer, plays per song, file sizes, page views per URL: a handful of rows carry most of the total, and a huge mass of rows carry almost none.

The consequence for reporting is that the mean sits above almost every value in the dataset. Suppose, as an illustration, that five of three hundred customers account for most of the month’s revenue. A sentence like “average revenue per customer is 340” then describes none of the three hundred. Reporting the median and a high percentile such as p90 is more honest, and a Pareto chart shows where the concentration actually sits.

Three places this bites:

• Comparison. Tests on long-tailed metrics need far more data to reach the same sensitivity as binary metrics, because a few units decide the answer. Settle how much data you need before you start, not after.
• Targets. A target to “raise average order value” on a long-tailed metric can be met by one large account. Tie it to the median or to an order count as well.
• Outlier handling. A long tail is not the same thing as outliers: if the extreme rows are real customers, they are the point of the metric, not a defect to remove.

The trade-off is this: a mean over a long-tailed metric is unstable, because it moves with whichever big account happened to fall inside the window. A median is stable but throws away the fact that a few rows matter enormously. Report both, and say which decision each one supports.

One last possibility to rule out: a long tail can be two simple distributions glued together. Combining segments with genuinely different behaviour produces one tall-and-thin mixture, so splitting the data often reveals two easy shapes where there seemed to be one exotic one.