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WAMath MethodsSyllabus dot point

What are the defining properties of the normal distribution and the empirical 68-95-99.7 rule?

Describe the normal distribution, its symmetry and parameters, and apply the empirical 68-95-99.7 rule

WACE Year 12 Mathematics Methods Unit 4 the normal distribution: the bell shape, symmetry about the mean, the role of mu and sigma, and the empirical 68-95-99.7 rule, with worked SCSA-style examples.

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  1. What this dot point is asking
  2. Defining properties
  3. The empirical rule
  4. Comparing distributions

What this dot point is asking

SCSA Unit 4 makes the normal distribution the central continuous model. This dot point asks you to describe its shape and parameters and to use the empirical 68-95-99.7 rule for quick probability estimates, examined in both sections, with the rule especially useful in the calculator-free section.

Defining properties

The mean, median and mode of a normal distribution all coincide at μ\mu because of the symmetry. The notation N(μ,σ2)N(\mu,\sigma^{2}) uses the variance σ2\sigma^{2} as the second parameter, not the standard deviation.

The empirical rule

For any normal distribution the proportion of values within a fixed number of standard deviations is the same, giving the 68-95-99.7 rule.

By symmetry, each tail beyond 2σ2\sigma holds about 2.5%2.5\%, and beyond 3σ3\sigma about 0.15%0.15\%. These let you answer many questions without a calculator.

Comparing distributions

Because σ\sigma sets the width, two normal distributions with the same mean but different standard deviations look very different: the smaller σ\sigma concentrates values near the mean. The empirical rule helps compare them: a value 2σ2\sigma above the mean is in the top 2.5%2.5\% of either distribution, regardless of the actual numbers.