Who thinks the wrong formula has been used in the article "Bandwidths for dummies" can Google yourself or read the justification below.
Click on the link below for Google results.
https://www.google.com/search?q=kleine+steekproef+uit+een+grote+populatie+opvatten+als+met+of+zonder+terugleggen
Accountability
The formula we used for summing is used for uncorrelated probabilities. Compare it to a vase with 8 red and 8 black balls. If you have to take a ball out of there blindly 10 times, it is important whether you do that with a return or without a return. Every time you take out a red one, the chance of red decreases and the chance of white increases. For example, if you draw red 8 times, the chance of white balls in the last two draws becomes 100%. Then the events are correlated: each result affects the next opportunity.
If, on the other hand, you put the ball back in the vase after each draw, the starting situation remains the same for each draw: these are non-correlated probabilities. You can draw balls of any color endlessly, and with every draw, there is always an equal chance of both outcomes.
The formula used in this article to make the curve (summing probabilities) is indeed always good for "with return" but not always for "without putting back". Yet we have chosen this formula, despite the fact that people only die once and there can be no question of recovery from the initial situation. Strictly speaking, you could argue that with each death, the following odds are affected. After all, deceased people are no longer in the running for the next opportunity, just like the ball that is not put back in the vase. Thus, a different, more complex formula could be used, which takes into account the "no back" effect. At least, according to some statisticians who put their reputation at risk with this, because I think this is really elementary statistics. An example of woo-woo: pure bluff to stop sound reasoning from claiming superior knowledge. Which is either not there, or is deliberately concealed. The most important thing is that terms are used that make others think "that's fine, he knows about it, at least he says that himself all the time".
Below we explain our choice for the formula without return. First of all, there is a basic rule in statistics, but we have more considerations – because you also have to understand why and when you apply a rule.
- For small samples from large populations, it is recommended to use "with recall". We find this in every handbook of statistics. See the following passage from a syllabus of the VU.
If you take 10 balls out of a vase with ten thousand balls, how much effect will that have on the next chance? Or 300 balls from a vase with 17 million balls? This has a negligible impact on the chances of follow-up.
- The population decreases with every (excess) mortality and we do not take that into account. In this objection, the population is therefore seen as that vase with a finite number of balls. But the opposite is true: the population is actually growing, so the increase due to births and immigration is greater than the loss of mortality. Even if this effect were to be settled, any differences will only become visible far beyond the decimal point.
- Our bandwidth now shows a situation in which people COULD die more often. This makes the band around the expected mortality (one microfraction) wider. So the band is theoretically a bit too wide. If we make the bandwidth narrower, the excess mortality will only increase. In any case, the differences between the two methods will be negligible.
In short: the method has been applied correctly. The statisticians who try to undermine "attic room calculators" should be able to put forward more substantive arguments than just "Leave this to real statisticians" on the basis of a rule that does not apply.
