The Three Sigma Rule is a statistical bound that states that for a unimodal distribution, the probability of a random variable deviating from its mean by more than three standard deviations is at most 5%. This result is derived from the Vysochanski–Petunin inequality, which is based on the Gauss inequality. The Gauss inequality provides a tighter bound for deviations from a mode of a unimodal distribution. The Vysochanski–Petunin inequality extends this result to any center, not just the mean. The proof of the Gauss inequality involves using calculus and properties of unimodal distributions. The Vysochanski–Petunin inequality is proven by considering two cases: one where the mode is close to the center and another where it is far away. The inequality provides a bound that is tighter than the Bienaymé–Chebyshev inequality for unimodal distributions. The three sigma rule is a special case of the Vysochanski–Petunin inequality when the center is the mean and the deviation is three standard deviations. The proof of the Vysochanski–Petunin inequality involves using calculus and properties of unimodal distributions to derive the bound. The result is significant because it provides a tighter bound for unimodal distributions than the general Chebyshev inequality. The three sigma rule is a well-known result in statistics, and its derivation from the Vysochanski–Petunin inequality highlights the importance of unimodal distributions in statistical theory.The Three Sigma Rule is a statistical bound that states that for a unimodal distribution, the probability of a random variable deviating from its mean by more than three standard deviations is at most 5%. This result is derived from the Vysochanski–Petunin inequality, which is based on the Gauss inequality. The Gauss inequality provides a tighter bound for deviations from a mode of a unimodal distribution. The Vysochanski–Petunin inequality extends this result to any center, not just the mean. The proof of the Gauss inequality involves using calculus and properties of unimodal distributions. The Vysochanski–Petunin inequality is proven by considering two cases: one where the mode is close to the center and another where it is far away. The inequality provides a bound that is tighter than the Bienaymé–Chebyshev inequality for unimodal distributions. The three sigma rule is a special case of the Vysochanski–Petunin inequality when the center is the mean and the deviation is three standard deviations. The proof of the Vysochanski–Petunin inequality involves using calculus and properties of unimodal distributions to derive the bound. The result is significant because it provides a tighter bound for unimodal distributions than the general Chebyshev inequality. The three sigma rule is a well-known result in statistics, and its derivation from the Vysochanski–Petunin inequality highlights the importance of unimodal distributions in statistical theory.