Q.There are some people who are hostile, egocentric and antisocial. They are said to be high on ________ .

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:

Since $|z| > 3$, this height is abnormal by the 3-sigma criterion. It is not impossible — just extremely rare.

When Abnormality Matters

In Indian exam contexts, you will encounter abnormality in:

• Statistics: Identifying outliers before calculating mean/median
• Probability: Using Chebyshev's inequality (which works for any distribution, not just normal)
• Quality control: In Six Sigma, a process is "out of control" if a point falls beyond 3σ from the mean

The One-Sentence Takeaway

Abnormality = an observation that lies more than three standard deviations away from the mean, under the assumption of a normal distribution — a statistical red flag that something unusual is happening.