Why is the Uncertainty of Counting Phenomena Equal to the Square Root of N?

In summary: This means that the uncertainty in the measurement is equal to the square root of the number of measurements. For example, if 50 radioactive decays are measured in 1 second, the uncertainty would be 7 decays per second. This is because the measurement follows a Poisson distribution, and the standard deviation can be calculated as the square root of N. This concept is important in understanding the concept of "uncertainty" in measurements.
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My physics teacher told me once about Arthur Eddington's famous observation that the ratio of scaling factors of the electromagnetic and gravitational forces was the same order as the uncertainty of N where N is the number of particles in the universe. From then on, I wondered what "uncertainty" could possibly mean. He couldn't really explain it to me, or I didn't understand it (or both).

For inherently random phenomena that involve counting individual events or occurrences, we measure only a single number N. This kind of measurement is relevant to counting the number of radioactive decays in a specific time interval from a sample of material. It is also relevant to counting the number of Lutherans in a random sample of the population. The (absolute) uncertainty of such a single measurement, N, is estimated as the square root of N. As example, if we measure 50 radioactive decays in 1 second we should present the result as 50±7 decays per second. (The quoted uncertainty indicates that a subsequent measurement performed identically could easily result in numbers dif- fering by 7 from 50.)

I can see that for "random" or acausal (stochastic) phenomena, the "absolute" uncertainty in a measurement is equal to the square root of that quantity of measurements. What is that about? I think I understand square roots and squaring just fine, and am looking for a little help in explanation.
 
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In example of radioactive decay, you can calculate the probability for each number of decays within one second, it follows a poisson distribution. And you can evaluate the standard deviation - it is sqrt(N).

Many phenomena which involve counting have an uncertainty of sqrt(N).
 

Related to Why is the Uncertainty of Counting Phenomena Equal to the Square Root of N?

What is "Uncertainty as √N — why?"

Uncertainty as √N is a scientific concept that explains the relationship between the number of observations or measurements (N) and the level of uncertainty in those measurements. It states that the uncertainty decreases as the square root of the number of observations increases.

Why is this concept important in science?

This concept is important in science because it helps scientists understand the level of precision and accuracy in their measurements. It also helps in determining the significance of experimental results and in making predictions based on data.

How does this concept apply to real-world situations?

This concept applies to real-world situations in various fields such as physics, chemistry, and biology. For example, in medical research, the uncertainty in clinical trial results is reduced as the number of participants increases. In weather forecasting, the uncertainty in predicting the exact location and intensity of a storm decreases as more data is collected.

What are the limitations of this concept?

While this concept is a useful tool in understanding uncertainty, it has its limitations. It assumes that all measurements are independent and follow a normal distribution, which may not always be the case in the real world. It also does not take into account systematic errors that may affect the measurements.

How can scientists use this concept to improve their research?

Scientists can use this concept to improve their research by increasing the number of observations or measurements in their experiments. This can help reduce the uncertainty and increase the precision and accuracy of their results. Additionally, understanding this concept can help scientists determine the appropriate sample size for their experiments to achieve statistically significant results.

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