Normal Distribution, Z-Scores & Central Limit Theorem

Reading Time: 8 Mins | Level: Intermediate

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1. The Normal (Gaussian) Distribution

The Normal Distribution is the bell-shaped probability distribution foundational to all inferential statistics. It is governed by two parameters: the Mean ($\mu$) and the Standard Deviation ($\sigma$).

The 68-95-99.7 Empirical Rule:

• $68.27\%$ of data falls within $\mu \pm 1\sigma$.
• $95.45\%$ of data falls within $\mu \pm 2\sigma$ (precisely $\pm 1.96\sigma$ for 95.0%).
• $99.73\%$ of data falls within $\mu \pm 3\sigma$.

2. The Central Limit Theorem (CLT)

The Central Limit Theorem is arguably the most powerful concept in statistics:

Standard Error of the Mean:

$$SE = \frac{\sigma}{\sqrt{n}}$$

As your sample size $n$ increases, your uncertainty ($SE$) shrinks by the square root of $n$.