# Stand-up Maths: Do these scatter plots reveal fraudulent vote-switching in Michigan?

Another group of people using statistical methods have shown that Biden has stolen Trump's votes! To be more precise, they have misused statistical methods to prove their point. Well as you know there are lies, damned lies and statistics.

The "proof" relies on a fact that multiple elections were held at the same time (at least in some states). The people who have conducted the analysis have calculated the difference between the fraction of votes for Trump (in presidential elections), $$v_t$$ and the fraction of votes for Republican candidates in other elections, $$v_r$$:

\begin{equation} \Delta = v_t - v_r . \end{equation}

They have found that $$\Delta$$ decreases with $$v_r$$. Their claim is that with larger $$v_r$$ we should observe larger $$v_t$$, therefore $$\Delta \approx 0$$. Which appears logical from the first glance, but is actually false.

Let us imagine that there is some non-zero probability of defecting (voting for president representing another party), $$p_d$$. Assuming that there were $$1 - v_r$$ votes cast for Democrats (equivalent to assumption that there are no third parties), fraction of votes cast for Trump will be given by:

\begin{equation} v_t = v_r (1-p) + (1-v_r) p . \end{equation}

Fraction of votes cast for Trump is a sum of votes from non-defecting Republican voters and votes from defecting Democrat voters. Let us now obtain the expression for difference:

\begin{equation} \Delta = \left[ v_r (1-p) + (1-v_r) p \right] - v_r
= p ( 1 - 2 v_r ) . \end{equation}

So, it appears qualitatively the same as this simplistic model predicts. And we have no election rigging built into the model.

We invite you to watch a video by Matt Parker from Stand-up maths, which explores this topic.