Normal Distribution

This lesson covers the normal distribution, the most important continuous probability distribution at A-Level. The normal distribution is used to model a wide range of natural phenomena and is central to statistical inference and hypothesis testing.

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Properties of the Normal Distribution

The normal distribution is a continuous probability distribution with a characteristic bell-shaped curve. If $X$X follows a normal distribution with mean $\mu$μ and variance $\sigma^2$σ2, we write:

$X \sim N(\mu, \sigma^2)$X∼N(μ,σ2)

Key properties:

• The curve is symmetrical about the mean $\mu$μ.
• The mean, median, and mode are all equal (at $\mu$μ).
• The total area under the curve is 1.
• Approximately 68% of data falls within $\mu \pm \sigma$μ±σ.
• Approximately 95% of data falls within $\mu \pm 2\sigma$μ±2σ.
• Approximately 99.7% of data falls within $\mu \pm 3\sigma$μ±3σ.

Region	Percentage of data
$\mu \pm \sigma$μ±σ	$\approx 68\%$≈68%
$\mu \pm 2\sigma$μ±2σ	$\approx 95\%$≈95%
$\mu \pm 3\sigma$μ±3σ	$\approx 99.7\%$≈99.7%