The chapter discusses the 3σ rule, a statistical principle that states the probability of a random variable deviating from its mean by more than three standard deviations is at most 5%. This rule is derived from the Vysochanskii–Petunin inequality, which is based on the Gauss inequality. The 3σ rule is particularly useful for unimodal distributions, where the probability of deviating from the mode by more than three standard deviations is bounded by \(\frac{4}{81}\). The chapter begins by introducing the Bienaymé–Chebyshev inequality, which provides a general bound on the probability of a random variable deviating from its mean. It then introduces the concept of unimodality, where the distribution of the random variable has a single mode, and shows that this assumption leads to a tighter bound of \(\frac{4}{9} \left( \frac{\sigma^2}{r^2} \right)\) for \(r > 1.63\sigma\). The Gauss inequality, which bounds the probability of deviation from the mode, is presented with three proofs: one using the inverse function, another using the original distribution function, and a third using Cramér's approach. The Vysochanskii–Petunin inequality extends the 3σ rule to any arbitrary center, not just the mean, and provides a more general bound for deviations from any point. The proof of the Vysochanskii–Petunin inequality involves distinguishing between two cases: when the mode is near the origin and when it is far away. The inequality is derived using inequalities involving the area under the density function and the expected squared deviation from the center. The chapter concludes with a comparison of the bounds provided by the Gauss inequality, the Vysochanskii–Petunin inequality, and the Bienaymé–Chebyshev inequality, highlighting their relative strengths under different conditions.The chapter discusses the 3σ rule, a statistical principle that states the probability of a random variable deviating from its mean by more than three standard deviations is at most 5%. This rule is derived from the Vysochanskii–Petunin inequality, which is based on the Gauss inequality. The 3σ rule is particularly useful for unimodal distributions, where the probability of deviating from the mode by more than three standard deviations is bounded by \(\frac{4}{81}\). The chapter begins by introducing the Bienaymé–Chebyshev inequality, which provides a general bound on the probability of a random variable deviating from its mean. It then introduces the concept of unimodality, where the distribution of the random variable has a single mode, and shows that this assumption leads to a tighter bound of \(\frac{4}{9} \left( \frac{\sigma^2}{r^2} \right)\) for \(r > 1.63\sigma\). The Gauss inequality, which bounds the probability of deviation from the mode, is presented with three proofs: one using the inverse function, another using the original distribution function, and a third using Cramér's approach. The Vysochanskii–Petunin inequality extends the 3σ rule to any arbitrary center, not just the mean, and provides a more general bound for deviations from any point. The proof of the Vysochanskii–Petunin inequality involves distinguishing between two cases: when the mode is near the origin and when it is far away. The inequality is derived using inequalities involving the area under the density function and the expected squared deviation from the center. The chapter concludes with a comparison of the bounds provided by the Gauss inequality, the Vysochanskii–Petunin inequality, and the Bienaymé–Chebyshev inequality, highlighting their relative strengths under different conditions.