Student Mathematics Analysis and Approaches (AA)…

What Are the Properties of the Normal Distribution Curve?

A normal distribution curve is continuous, bell-shaped, symmetric about its mean, and has a total area of 1 beneath it. Its position and spread are determined completely by the mean $\mu$ and standard deviation $\sigma$ .

For a normally distributed random variable, IB notation is

X \sim N(\mu,\sigma^2),

where $\mu$ is the mean and $\sigma^2$ is the variance.

Property	Meaning
Symmetry	The curve is symmetric about $x=\mu$ , so values equally far from the mean have equal probability density.
Central measures	The mean, median, and mode are all equal and occur at $x=\mu$ .
Total area	The total area under the curve is $1$ , representing total probability.
Continuous distribution	Probabilities correspond to areas over intervals. At any single value, $P(X=a)=0$ .

The normal distribution also follows the approximate 68-95-99.7 rule:

$P(\mu-\sigma<X<\mu+\sigma)\approx0.68$
$P(\mu-2\sigma<X<\mu+2\sigma)\approx0.95$

The standard score $z=\frac{x-\mu}{\sigma}$ measures how many standard deviations a value lies above or below the mean. This explains the percentages: the areas between $z=-1$ and , between and , and between and are approximately , , and . For example, if , about of values lie between and , one standard deviation from the mean.

A common misconception is that the curve itself gives probability at a point. It does not: probability is represented by area under the curve over an interval.

Exam technique: This is IB Mathematics AA SL, subtopic SL 4.9: Normal distribution and calculations. On Paper 2, use your GDC for normal and inverse-normal calculations, state the required probability or boundary clearly, and interpret the result in context. Remember that the GDC requires $\sigma$ , not $\sigma^2$ .

What Are the Properties of the Normal Distribution Curve?

For a normally distributed random variable, IB notation is

X \sim N(\mu,\sigma^2),

where $\mu$ is the mean and $\sigma^2$ is the variance.

Property	Meaning
Symmetry	The curve is symmetric about $x=\mu$ , so values equally far from the mean have equal probability density.
Central measures	The mean, median, and mode are all equal and occur at $x=\mu$ .
Total area	The total area under the curve is $1$ , representing total probability.
Continuous distribution	Probabilities correspond to areas over intervals. At any single value, $P(X=a)=0$ .

The normal distribution also follows the approximate 68-95-99.7 rule:

$P(\mu-\sigma<X<\mu+\sigma)\approx0.68$
$P(\mu-2\sigma<X<\mu+2\sigma)\approx0.95$

A common misconception is that the curve itself gives probability at a point. It does not: probability is represented by area under the curve over an interval.

What Are the Properties of the Normal Distribution Curve?

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What Are the Properties of the Normal Distribution Curve?