The Normal Distribution

This lesson covers the normal distribution as required by the Edexcel A-Level Mathematics specification (9MA0), Paper 3 Section A -- Statistics. You need to understand the properties of the normal distribution, standardise using the Z formula, and use the standard normal distribution to find probabilities.

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What is the Normal Distribution?

The normal distribution is a continuous probability distribution described by a symmetric, bell-shaped curve. It models many real-world variables such as heights, weights, IQ scores, and measurement errors.

Notation

X ~ N(mu, sigma²) where mu is the mean and sigma² is the variance.

Note: the second parameter is the variance, not the standard deviation.

Example

Heights of adult women: X ~ N(164, 49) (mean 164 cm, SD 7 cm, since 7² = 49).

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Properties of the Normal Distribution

1. Bell-shaped and symmetrical about the mean mu.
2. Mean = median = mode (all equal to mu).
3. Asymptotic to the x-axis -- extends infinitely but never touches it.
4. Total area under the curve = 1.
5. Completely defined by mu (location) and sigma² (spread).

The 68-95-99.7 Rule