The Normal Curve and Its Properties

A random variable $X$ is said to follow a normal distribution with mean $\mu$ and variance $\sigma^2$ — written $X \sim N(\mu, \sigma^2)$ — if its probability density function is

A Std-12 Commerce student is never asked to use this formula directly — it is stated here only so the shape and the two parameters ($\mu$, $\sigma$) that fully determine the curve are clear. Every practical numerical question is solved using the standard normal variate and the area table from Sections 3 and 4.

Key properties of the normal curve (frequently examined):

1. Bell-shaped and symmetric. The curve is perfectly symmetric about the mean $\mu$; the left half is the exact mirror image of the right half.
2. Mean = Median = Mode. Because of perfect symmetry, all three measures of central tendency coincide at the centre of the curve.
3. Unimodal. The curve has a single peak, located exactly at $x = \mu$.
4. Asymptotic to the x-axis. The two tails of the curve extend indefinitely in both directions and approach (but mathematically never touch) the horizontal axis.
5. Total area under the curve = 1, i.e. the area represents the total probability, and because of symmetry, exactly 0.5 of the area lies on each side of the mean.
6. Points of inflection at $\mu \pm \sigma$ — this is where the curve changes from concave-down (near the peak) to concave-up (in the tails).
7. Shape depends only on $\sigma$; position depends only on $\mu$. A larger $\sigma$ produces a flatter, more spread-out curve; a smaller $\sigma$ produces a taller, narrower curve. Shifting $\mu$ slides the whole curve left or right without changing its shape.
8. The Empirical (68–95–99.7) Rule — a practical shortcut used constantly in quality-control and business applications: