Normal DistributionJEE Main

Concept

The normal (Gaussian) distribution is the symmetric bell curve that arises whenever many small independent effects add up. It is fully described by its mean $\mu$ (centre) and standard deviation $\sigma$ (spread), and obeys the famous 68-95-99.7 rule.

Key formula

Derivation

The exponent $-(x-\mu)^2/2\sigma^2$ makes the curve peak at $x=\mu$ and fall off symmetrically; the prefactor $1/(\sigma\sqrt{2\pi})$ normalises the total area to 1.

Standardising with $z = (x-\mu)/\sigma$ turns any normal into the standard normal, so probabilities depend only on how many $\sigma$ away you are.

Integrating gives $P(|X-\mu|<k\sigma) = \operatorname{erf}(k/\sqrt2)$: about 68% for $k=1$, 95% for $k=2$, 99.7% for $k=3$.

Scenarios to explore

• Normal Distribution — Bell curve: mean, variance and the 68-95-99.7 rule.

Real-world applications

• Measurement errors and the Central Limit Theorem.
• Grading on a curve and standardised test scores.
• Quality control and process capability (six sigma).

JEE exam tips

• Total area under the curve is always 1, regardless of $\mu$ and $\sigma$.
• Standardise to $z$-scores before using a normal table.

Common mistakes

• Confusing the standard deviation $\sigma$ with the variance $\sigma^2$.
• Assuming all bell-shaped data are exactly normal.
• Reading the peak height as a probability — it is a density, not a probability.

Exam traps to avoid

• Increasing $\sigma$ lowers the peak (area is conserved).
• The density $f(x)$ can exceed 1 when $\sigma$ is small — that is fine for a density.