Normal Bell Curve N(μ, σ²)

The probability density is $\dfrac{1}{\sigma\sqrt{2\pi}}\exp\!\big(-\dfrac{(x-\mu)^2}{2\sigma^2}\big)$, parameterised by mean $\mu$ and standard deviation $\sigma$. It is symmetric about $\mu$, with about $68\%$ of area within $\mu\pm\sigma$ and $95\%$ within $\mu\pm 2\sigma$.

Slide $\mu$ and $\sigma$. The lab plots the bell, marks the peak height $1/(\sigma\sqrt{2\pi})$ and the inflection points at $\mu\pm\sigma$. Try $\mu = 0$, $\sigma = 1$ (standard normal) and see the classic $0.4$ peak.

The Central Limit Theorem: sums of many small independent random effects tend to be normal, no matter the underlying distribution. Heights of Indian students, exam scores, measurement errors, and stock-return logs all look approximately bell-shaped because they're sums of many tiny influences.

Class XII statistics uses it for confidence intervals and standard scores ($z$-scores). Quality control (six sigma standards in factories), psychology (IQ testing), and finance (Black-Scholes option pricing) all rest on normal-distribution assumptions. Knowing the bell intuitively pays off across subjects.