# What's a good measure of spread when using a weighted average

It's often useful to display a measure of spread when presenting an average.

Example:

• the average revenue per customer per transaction is \$100, meaning that (assuming a distribution roughly centered around its mean) the typical customer spends around$100 for each transaction he makes

• the standard deviation of revenue per customer per transaction is \$10 This way, we have a pretty clear understanding of the overall distribution, i.e. we know that the revenue per customer per transaction typically fluctuates in a range of around 10% of the average revenue per transaction. But now, consider this other example: • the weighted average of revenue by customer is \$150, when using the count of transactions for each customer as a weighting parameter (e.g. a customer with 100 transactions would count for twice as much as a customer with only 50 transactions)

• what would then be a good measure to give a notion of the spread?

In that scenario you would use the weighted standard deviation: $$s=\sqrt{\frac{1}{\sum_iw_i}\sum_i w_i(x_i-\bar{x})^2}$$ where $$x_i$$ is the $$i$$-th data point and $$w_i$$ the weight on that data point, and $$\bar{x}$$ is the (weighted) sample mean. Like the regular standard deviation, this also has a bias-corrected version: $$s=\sqrt{\frac{N}{(N-1)\sum_iw_i}\sum_i w_i(x_i-\bar{x})^2}$$ where $$N$$ is the number of data points in the sample. Or, if any of the weights are 0, then it is the number of non-zero weights (the number of data points that actually contribute to the sample statistics).