5.2 Normal Density Curve
The normal density curve characterizes the normal distribution, which is the most widely used probability distribution for continuous variables. The normal distribution is symmetric and bellshaped (for this reason it is often referred to as the “bell curve”). The normal density function has two parameters: the mean [latex]\mu[/latex] and the standard deviation [latex]\sigma[/latex]. The parameter [latex]\mu[/latex] controls the centre (location) of the distribution and [latex]\sigma[/latex] controls the shape of the distribution. When [latex]\sigma[/latex] is larger, the curve appears shorter and fatter; when [latex]\sigma[/latex] is smaller, the curve appears taller and slimmer.
Figure 5.2 shows three normal density curves–[latex]N(0, 2), N(0, 1)[/latex] and [latex]N(4, 1)[/latex]. [latex]N(0,1)[/latex] and [latex]N(4, 1)[/latex] have the same standard deviation; therefore, they have the same shape; if you shift the location of [latex]N(0, 1)[/latex] to the right by 4, the two distributions are exactly the same. [latex]N(0,1)[/latex] and [latex]N(0, 2)[/latex] have the same mean; therefore, they center at the same location. [latex]N(0, 2)[/latex] has a larger standard deviation; therefore, the density curve is shorter and fatter.
If a random variable [latex]X[/latex] follows a normal distribution with mean [latex]\mu[/latex] and standard deviation [latex]\sigma[/latex], we write [latex]X \sim N(\mu, \sigma)[/latex]. The probability density function of a normal random variable [latex]X[/latex] is given by:
[latex]f(x) = \frac{1}{\sqrt{2 \pi} \sigma} e^{\frac{(x \mu)^2}{2\sigma^2}}, \infty < x < \infty \text { with } \pi \approx 3.142 \text { and } e \approx 2.718[/latex].
The normal density curve has the following properties:
Key Facts: Properties of a Normal Density Curve
 The curve extends from negative infinity [latex](\infty)[/latex] to positive infinity [latex](\infty)[/latex], i.e., the entire real line.
 The total area under the curve is 1. This is a common property for all density curves.
 The curve is bellshaped, unimodal, and symmetric at the mean [latex]\mu[/latex].
 Empirical rule (68.395.499.7 rule) for a normal curve:

 68.26% of the observations are within the interval [latex][\mu  \sigma, \mu + \sigma][/latex] (one standard deviation to either side of the mean), i.e., [latex]P(\mu  \sigma \leq X \leq \mu + \sigma) = 0.6826.[/latex]
 95.44% of the observations are within the interval [latex][\mu  2\sigma, \mu + 2\sigma][/latex] (two standard deviations to either side of the mean), i.e., [latex]P(\mu  2\sigma \leq X \leq \mu + 2\sigma) = 0.9544.[/latex]
 99.74% of the observations are within the interval [latex][\mu  3\sigma, \mu + 3\sigma][/latex] (three standard deviations to either side of the mean), i.e., [latex]P(\mu  3\sigma \leq X \leq \mu + 3\sigma) = 0.9974.[/latex]