Why 'proportion of network' pie charts are misleading

Saturday 21st February 2015

0. Introduction

There's often a great deal of discussion over who has the largest share of the network based on the number of blocks solved in a certain arbitrary period of time. However there is a significant amount of variance in this sort of data that is often ignored.

1. Sample mean vs  population mean

Determining the statistics of a population after being given sample data is a fairly common use of statistical analysis, yet it is almost never applied to block solve rates. Without understanding the difference between the sampled block rate and the actual block rate, conclusions about block solve rates can be incorrect.

For bitcoin mining block solve rates, the sample mean is the number of blocks solved by a block maker, and the population mean can be thought of as the sample mean if the time period was repeated an infinite number of times at the same hashrate. 

2. Confidence intervals for the population mean

The population mean is what we want to know, but data only provides the sample mean. Luckily, for Poisson distributed random variables (such as block solve rates approximately are) we can estimate the range in which the population mean might occur. Details for the method of estimation can be found here.

The confidence interval can be thought of as an indication of the confidence one has in the the estimate: a 75% confidence interval will include the population mean (unknown but actual block solve rate) 75% of the time, but the 99% confidence interval will contain the population mean 99% of the time.

In the plot below, the x axis represents the number of blocks solved in any arbitrary time period and the y axis the upper and lower bounds for the listed confidence intervals.

It's a little hard to read the confidence intervals when the solve rate gets large, so the plot below provides the same data, except the population mean confidence intervals are presented as a percentage of the sample mean.

3. Summary

• The variance in block solve rates means that confidence intervals must be used to provide information about the population mean (or actual but unknown hashrate / block solve rate).
• Pie charts of block solve rates cannot provide this information, and give as much weight to a block maker that solves one block as to a block maker that solves one hundred blocks, or the same block maker in one hundred times the time.
• So don't put too much, um, confidence in 'proportion of network' pie charts.

There is a follow up post here, which explains how to improve the accuracy of pie chart proportion of network estimates. 

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