+1 - this misuse of the scales is so blatant, it makes you wonder about how credible the person putting the graphs together want to be

@Henry It's so blatant it makes me wonder why the person didn't just lie and invent the numbers. It takes 5 seconds to glance at the Y-axis as opposed to taking an hour to identify a source and curate the numbers yourself.

@ShmuelNewmark Using false data is an obvious lie. Messing with the graph axes to present the data in a way favorable to your argument is not strictly lying (it is lying). It has plausible deniability. It can be used as a technicality to work around rules. And its a trick most people are not aware of, which is why its so useful.

@ShmuelNewmark The problem is that you need to know to loo at the Y-axis. The vast majority of people will assume, when shown two graphs of similar statistics, that the axes are scaled identically because that's what you do when you're displaying data honestly. We need to remind ourselves that not everyone argues in good faith.

The bottom graphs peak to peak is the same. Sometimes you do need to scale the graphs to show relevant data changes. The graphs do show difference, since the time/resolution/precision/x axis is the same. You can infer the scale from data anyway.

It would be helpful to see the error bars on the data. That is, to compute the expected mean and confidence levels of the second-digit statistics, given some reasonable assumptions about the expected second digit distributions, given the sample sizes (and other factors if possible)

What about the first chart? It appears that all the axes are the same

The fact that this, (so far) most upvoted answer simply side-steps the question and instead resorts to some kind of a straw man argument about "a second chart" (which, let's be honest, should not have been present in the question at all) is a good example of how political preferences easily overpower any claims of skepticism.

@KT agreed. I think it should be flagged as "not an answer". It only addresses a small part of the claim and the rest is speculation/ad hominem.

To have a more authentic answer to the question "do the votes violate bedford's law?" The obviously straw man crappy excel plot should be deleted from the question, and this answer (which appears to be to be blatantly farming political points from people who aren't actually interested in the question) should be deleted. Relying on "bad faith" for a fact of reality is the opposite of what a real skeptic does.

Looking at the combined chart, Biden's line seems to deviate further from expectations than Trump's does, overall (with exceptions at 4, 5, 7, and 9). This suggests that the Y axis was chosen to better illustrate the two (Trump's has higher correlation, so zooming out draws more attention to the two primary discrepancies; Biden's has lower correlation, so zooming in better communicates the individual data points), and that the issue here is that there's no note highlighting the different scales. Personally, I would've suggested the Y be from 7.5 to 12.5 with 0.5 increments, to fit both.

@StevenSagona The burden of proof lies on the claimant. We do not have to chase their claim, only show their claim is not supported. The question included those graphs as support. Those graphs are manipulated. Addressing that is part of the answer. Points are a tool to sift answers, not an end nor a metric. Claims on Skeptics are often many layered and require many skills to address them. It's ok to address just one layer, the answers together form a whole. I'm not a statistician. I hadn't heard of Benford's Law before this. But many claims rely on manipulating framing, and that I know.

@Henry The scaled Y axis might have been an artifact from the tool scaling the Y axis differently since the max values are different. If it was left in default state due to time constraints to perform the task we should not infer malicious intent. Remember to apply "the golden rule", "the silver rule" and "Hanlon's razor" in situations concerning morality.

@StevenSagona The odd thing is that Bedford's law predicts a graph curving down from the left, which they all do. "Biden's fit, but aren't quite as smooth" appears to be the entire claim. The graphs are distributions of the digits 1-9 in the vote counts. Someone making up numbers would use digits evenly, giving a flat line. Bedford's is the crazy observation that real-world numbers have more lower-digits than high.

@Schwern, "The burden of proof lies on the claimant. We do not have to chase their claim, only show their claim is not supported." Completely ridiculous. The question is "Is Benford's law violated?" You should focus on answering the question, which is what a person in "good faith" would interpret what this person is asking. As you can see with the other answers which actually attempt to answer the question, it's obvious that this law IS violated. An authentic extended answer would try to answer why this violation does or doesn't imply fraud.

@OwenReynolds, it is obvious from the first plot that Benfords law is violated and while I agree the second plot is misleading, this is completely besides the point.

@StevenSagona The other answers show Benford's law is being misapplied and violation is moot. A good faith claim would have relied on statistical analysis to show that Biden's results both deviate significantly from Benford's law and are greater than Trump's own deviation. I'm not qualified to do that, someone is welcome to. Instead they invite the reader to eyeball graphs manipulated to support their claim; I pointed out and removed their manipulation. They would also use a broader set of data to avoid cherry picking; and Henry points out Chicago is an odd choice.

@StevenSagona We can rephrase the question as a statement to be refuted: "A violation of Benford's law shows fraud in the 2020 election." That has two parts: 1) a violation of Benford's law shows election fraud, and 2) the presented evidence shows a violation in the 2020 election. If we only address the violation, we have failed to connect it to voter fraud and we've fallen for a straw man. Both must be true for the statement to be true. Others addressed #1. I went for a glaring portion of #2. And shout out to SUNY Stony Brook, I took my first CS class there.

@N4v If I write a Physics question that starts, "Given Einstein's famous equation, E=mc^3, why is it that..." - do you bother reading the rest of the question to see if there's anything to answer?

"Put them all together in one graph with the same scale and they don't look so different anymore." Maybe this is in a reasonable margin of error, but even on the same scale I observe a stronger correlation with Trump to the Benford line than with Biden. Can you show trend lines for both sets of data and explain the differences observed?

Can I recommend the use of datatheif for reverse-engineering graphs datathief.org

@MindaugasBernatavičius - the full text of that law (which, by the way, is "Heinlein's Razor", named for the science fiction author Robert Heinlein) is "Never attribute to malice that which can be adequately explained by stupidity, but don't rule out malice". Emphasis mine...

@BobJarvis-ReinstateMonica - that is fine and agreement worthy (regarding Heinlein / Hanlon). With regards to Trumps numbers - it's not the fact that the numbers go above/below the line, but that the distribution of the shape get's closer to a normal distribution. Although just from Benford's law we can't prove malicious activities - law is empirical work (gathering evidence).

@KT.: So you're saying that the evidence used to back up a claim should not be verified or discussed, when discussing the validity of said claim? Because that's what happening here. The article depends on those graphs to back up its story. The graphs are provably altered to sow confusion (without blatantly showing false numbers), and when the alteration is undone, the alleged evidence that backs up the story outright dissipates. Schwern is correct that this is a perfect example of How To Lie With Statistics 101: imply everything you can to make people draw the wrong conclusion.

@KT.: Your advice is to further cherry-pick the data that is already being cherry-picked by the article authors? And you don't see a problem with that? This question is asking for validation of the statistical data. Literally. Because Benford's law is a statistical law. Without the statistical data, you can't even begin to verify whether it is or isn't violated. So if an article is built on shoddily constructed data that highly suggests willful manipulation to confuse the conclusions made from the data, that taints the validity of the conclusions in the article that are drawn from that data.

@KT.: Furthermore, your "let's just focus on the first chart" argument completely misses the point that statistics are broadly scoped. If you fixate on a cherry-picked subset of the data, it's highly likely that you're going to find something abnormal. This is no different than stopping a person of your choice in the street, measuring their height, and nothing that their exact height is not exactly the average person's height. That's not what an average is or represents. The entire article focuses on 2 out of 3000 counties. That's in no way representative data, and clearly cherry-picked.

@Flater: Putting words into my mouth which I did not say ("So you're saying...", "Your advice is ...") is a fun rhetorical technique, but please try to avoid it. Whether and how much cherrypicking (a.k.a. multiple testing) taints the validity of the particular hypothesis is a valid question, yet it is not addressed in this answer. Instead, the answer literally cherrypicks a straw man chart to refute, thus precisely missing the broader scope. It is the most upvoted for a reason probably other than "skeptical answer quality" and I find this amusing.

+1 for mentioning How to Lie with Statistics. That book gave me huge trust issues with numbers.

@KT. 0) Please reference my earlier comments about narrow answers and claims. 1) It's a cheap and eye-catching answer because it is a cheap and eye-catching trick; I didn't choose the trick. 2) Other answers cover why Benford's Law is moot; the whole thing is a straw man. 3) I agree there's other anomalies in the claim. I'm not qualified to address them. But if you'd like to answer, please do! Cherry picking is an obvious one; in a country of 300 million people you will find anomalies. And why the switch to 2nd digits only for Allegheny County?