Q.Adil, a young man, shows aggressive actions that threaten or cause harm to people or animals and indulges in serious rule violations. What are these symptoms indicative of? Mention two other types of aggressive behaviour.

Abnormality Definition — First Principles

Imagine you are looking at a set of data points — say, the heights of all students in your class. Most heights cluster around some average value. A few students are taller, a few shorter, but almost everyone falls within a certain range. Now suppose one student is 8 feet tall. That student's height is abnormal — it lies far outside the pattern of the rest of the group.

That is the core intuition behind abnormality in statistics: an observation that deviates so much from the rest of the data that it stands out as unusual.

The Precise Definition

In formal terms, an abnormality (often called an outlier) is a data point that lies at an abnormal distance from other values in a random sample from a population. There is no single universal cutoff — it depends on the context — but the most common working definition in Indian exam syllabi (especially for statistics and probability) is:

This is the 3-sigma rule. The reasoning: for a normal (bell-shaped) distribution, about 99.7% of all data lies within three standard deviations of the mean. Anything beyond that has less than a 0.3% chance of occurring naturally — so we call it abnormal.

Why 3-Sigma? The Intuition

The number 3 is not arbitrary. In a normal distribution:

• Within 1σ of μ: ~68% of data
• Within 2σ of μ: ~95% of data
• Within 3σ of μ: ~99.7% of data

A point beyond 3σ is so rare that it is more likely to be a mistake, a special cause, or a genuinely exceptional case — hence "abnormal."

A Concrete Example

Suppose the mean height of Indian adult males is 165 cm with a standard deviation of 7 cm. A person who is 190 cm tall:

$z = \frac{190 - 165}{7} = \frac{25}{7} \approx 3.57$ …