A definition of what is typical would be something which happens 95 times out of every 100 occasions. And, by definition, events that are atypical or unusual occur 5 or less times per 100 (5% of the time or less). When something which is variable distributes in a normal way, the center is the mean value and frequency is bell-shaped:
In this distribution of IQ scores, values rise as you go right, the mean value is 100, and a standard deviation away from the mean value is worth 15 points. 95% of all the scores show up within 2 standard deviations of the mean value of 100 — so that 95 of every 100 people who take an IQ test end up with IQ scores ranging from 70 up to 130.
But not all things which vary will vary according to this bell-shaped model. While being over 2 standard deviations from the mean is always unusual for IQ tests, in some distributions you have more than a 5% chance of being more than 2 standard deviations from the mean. Finding out what is unusual may require more work.
But there is a rule which works for every distribution of values: Chebychev’s theorem. In Chebychev’s theorem, there is a proof stating that the minimum portion of all values in a given population that is within k standard deviations of the mean will always — not sometimes, always — be at least as much as
With Chebychev theorem, it’s always the case that 5 standard deviations from the mean is unusual (i.e., <5% probable) and always the case that 20 standard deviations from the mean is so unusual that it demonstrates a “special cause” (variation due to common causes cannot explain values sitting 20 standard deviations from a mean).
Real World Use
A real world application for this theorem would be to examine the historic reporting rate of serious adverse events in the VAERS system for a mean reporting rate per million doses, and a standard deviation of the reporting rate.
Year … Net Doses (M) … serious AER … rate/M
1991 …… 127.818932 …………… 1,301 ………… 10.18
1992 …… 152.522129 …………… 1,441 ………… 9.45
1993 …… 170.365246 …………… 1,452 ………… 8.52
1994 …… 190.238536 …………… 1,545 ………… 8.12
1995 …… 145.786858 …………… 1,466 ………… 10.06
1996 …… 147.702015 …………… 1,455 ………… 9.85
1997 …… 166.13962 ….……….… 1,588 ………… 9.56
1998 …… 187.758677 …………… 1,673 ………… 8.91
1999 …… 202.176571 …………… 2,037 ………… 10.08
2000 …… 210.608388 …………… 2,006 ………… 9.52
2001 …… 202.266286 …………… 2,332 ………… 11.53
The mean rate of serious adverse event reports is 9.62 per million net doses delivered. The standard deviation of the rate of reports is 0.92 per million net doses delivered. The 99% upper bound on the standard deviation is 1.97 per million net doses delivered.
When the serious adverse event reporting rate for Moderna is compared with these historic figures, the rate is so high (182 serious adverse event reports per million doses) that — even using the 99% upper bound standard deviation — the Moderna serious adverse event reporting rate is 87 standard deviations above the mean.
Because Chebychev’s theorem proves that a 5-standard-deviation shift is always unusual, and a 20-standard-deviation shift demonstrates a special cause (common cause variation cannot explain shifts of 20 standard deviations or more; regardless of underlying distribution of values), data indicate that Moderna is especially harmful.
251.85 million doses by March 2023
45,728 Serious Adverse Event Reports by March 2023
Reference
CDC. Surveillance for Safety After Immunization: Vaccine Adverse Event Reporting System (VAERS) --- United States, 1991--2001. https://www.cdc.gov/mmwr/preview/mmwrhtml/ss5201a1.htm
OurWorldInData. COVID-19 doses administered by manufacturer. https://ourworldindata.org/covid-vaccinations
MedAlerts VAERS Search Tool. https://medalerts.org/vaersdb/index.php
Thank. Your response makes perfect sense. I don’t have any URLs with anything better, but even if I could find some it would in all likelihood require piecing together limited information from different sources. You are so right that taking everything from one source is much more difficult to criticize or refute.
I love this theorem and this analysis. I’m curious why you used the 1991-2001 time frame to calculate the mean and standard deviation instead of using the 1991-2015 figures.