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Scientific results evaluation: Rules of Thumb

10 Mar 2016  | Ransom Stephens

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Now, back to the Queen and me. If a million people show up at Windsor Castle, then statistics says—in a broad, rule-of-thumb sense—that the uncertainty in that number is 1,000,000 = 1000. Using Table 1, if we make the offer some other day, she'll have more than 1,001,000 people or less than 999,000 on 32% of those days; more than 1,002,000 or less than 998,000 on 5% of those days; more than 1,003,000 or less than 997,000 on less than but about half a per cent of those days, and so on.

Meanwhile, the statistical uncertainty in the number of people who visit me is 4 = 2 (unless mom's in town). The number of visitors I should expect is 4 ±2, which means that about 30% of the time, I'll get fewer than two (i.e., 1) or more than six visitors and about 5% of the time either no one comes over or more than eight people show up. My 4 ±2 means I have a 50% statistical uncertainty (again, in the rule-of-thumb sense, not the carefully calculated sense, which would change the numbers but not the conclusion) compared to the Queen's 0.1%.

Because it's safe to assume that I'm of average popularity (yes, I flatter myself), then I can attract four people over for a beer. That is, four is the background noise level.

To get the statistical significance of the queen's popularity, we take the number of her visitors, subtract the background to get the signal, Nsignal, and divide the signal by the uncertainty in the background, (Nbackground).

Rule of thumb 2: Statistical significance

The statistical significance of the queen's popularity is 1,000,000/4=500,000 σ. From table 1, we see that half a million, σ is off the chart. It is extraordinarily unlikely that a random fluctuation could bring more visitors to my house than the queen's in the lifetime of the universe, which is fine because I don't really want to share my beer unless you're in town and feel like seeing my collection of Raider propaganda.

The two rules of thumb combined with the numbers in Table 1 indicate the correspondence between the statistical uncertainty of a measurement and its significance. Be careful. It.s easy to think that two standard errors above the noise, 2-σ, or 5-ish% chance of a fluctuation, is sufficiently significant. Because so many measurements are published every day, I recommend the scale shown in table 2.

Table 2: Appropriate responses to measurements as a function of statistical significance, assuming low systematic uncertainty.

Statistical Significance Ransom's recommended responseHit snooze

1. Try not to remember these results

2. Intriguing

3. Worthy of discussion at cocktail parties

4. Solid evidence

5. Convincing but still needs confirmation by another experiment

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