A** normal density curve **is an idealized representation of a normal distribution in which the area under the curve is defined to be 1.

- The points at which the curve changes from being concave up to being concave down are called the
*inflection points* - On a normal density curve, these inflection points are always exactly one standard deviation away from the mean

**The Empirical Rule for Normal Distributions**

One way to calculate the area under a density curve up to a certain point is by using the **Empirical Rule**.

The **Empirical Rule** states that

- 50% of all data points are above the mean and 50% are below
- Approximately 68% of all data points are within 1 standard deviation of the mean
- Approximately 95% of all data points are within 2 standard deviations of the mean
- Approximately 99.7% of all data points are within 3 standard deviations of the mean

**Z-Score**

If the data point you are interested in is not a whole number standard deviation away from the mean, you use a **z-score**.

A z-score measures how many standard deviations a score is away from the mean. The z-score of the term \begin{align*}x\end{align*} in a population distribution whose mean is \begin{align*}\mu\end{align*} and whose standard deviation is \begin{align*}\sigma\end{align*} is given by: \begin{align*}z=\frac{x-\mu}{\sigma}\end{align*} .

Use a **z-score probability table** to find a decimal expression of the percentage of values that are less than \begin{align*}x\end{align*}.