Why is the detection limit 3 times the signal-to-noise ratio

The detection limit and quantitation limit are important attributes of analytical methods. However, in actual scientific research, many colleagues only know that the detection limit is 3 times the signal-to-noise ratio but do not know the reason why. This article attempts to explain this in simple language. First, let's review some basic mathematical statistics knowledge from university, that is, the 'normal distribution' and the powerful killer app, the central limit theorem (don't skip it... I promise no mathematical formulas).

Alright, since math formulas can be tough, let's just take a look at a normal distribution graph of human intelligence.

Human intelligence follows a typical normal distribution curve, with an average IQ of 100. Around 70% of the population falls within the IQ range of 85 to 120. Individuals like Mozart and Einstein had IQs greater than 160. From the graph, I am confident with over 99% certainty that Mozart and Einstein fundamentally differ from the average person. They are outliers.

Now, what does this have to do with our topic today?

Imagine conducting chromatographic analysis. You run 100 blank samples and obtain 100 signal values at the retention time of the target compound. These signals appear random, yet they seem to follow a certain pattern. They don't infinitely increase or decrease but fluctuate around a mean value. Through statistics, you realize these seemingly random signals converge around a mean value, adhering to the central limit theorem.

By statistical analysis of these noise data, you will find that the probability of noise values falling within u (mean) ± 1σ (standard deviation) is 68.2%, the probability of noise values falling within u ± 2σ is 95.5%, and the probability of noise values falling within u ± 3σ reaches 99.7%.

The standard deviation indicates the magnitude of data fluctuations. One standard deviation represents the same distance from the mean as one standard deviation. Two standard deviations, three standard deviations represent distances from the mean that are two and three times the standard deviation, respectively. In other words, if I have a sample now and its signal exceeds 3 times the standard deviation, I have a 99.7% confidence that this signal is different from the noise. In instrument analysis, we consider this as a signal distinct from noise, indicating the detection of something.

Let's make a simple analogy: noise is like the average intelligence level of people in society, while the detection limit is akin to Einstein's intelligence. When the detection limit differs from the noise, it's similar to how Einstein's intelligence stands out from the average. This should be easy to understand.

So, the true detection limit should be a concentration higher than the signal value by 3 times the standard deviation.

The normal distribution not only has wide applications in scientific experiments but also casts its shadow over social life and the vast world.

We've briefly discussed this, which could be considered a form of popular science education. For those who don't understand, they can revisit the relevant knowledge of probability theory.