The Normal Distribution and Its Applications

Statistics — Third Secondary

The normal distribution and its properties

The normal distribution is a continuous probability distribution whose curve is a symmetric bell shape centred on the mean μ.
It appears in many natural and social phenomena, such as people’s heights and weights and the scores of large groups in tests.
It has the following properties:

The curve is perfectly symmetric about the vertical line through the mean μ. The mean, the median and the mode are equal and lie on this line.
The total area under the curve equals 1, and every part of the area represents a probability.
The area to the right of the mean equals 0.5, and the area to the left of the mean equals 0.5.
The shape depends on the standard deviation σ. A larger σ makes the curve wider and flatter, and a smaller σ makes it taller and narrower.
Approximately 68% of the values lie between μ − σ and μ + σ, about 95% lie between μ − 2σ and μ + 2σ, and about 99.7% lie between μ − 3σ and μ + 3σ.

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