List the properties of the normal density curve.

Verified step by step guidance

1. Understand that the normal density curve, also known as the normal distribution curve, is a symmetric, bell-shaped curve that describes the distribution of many types of data.
2. Note that the curve is symmetric about the mean \(\mu\), meaning the left and right sides are mirror images.
3. Recognize that the total area under the curve is equal to 1, representing the total probability of all outcomes.
4. Observe that the curve approaches, but never touches, the horizontal axis (the x-axis), extending infinitely in both directions.
5. Identify that the mean \(\mu\), median, and mode of the distribution are all located at the center of the curve, and the spread of the curve is determined by the standard deviation \(\sigma\).

Key Concepts

Shape of the Normal Density Curve

The normal density curve is symmetric and bell-shaped, centered around the mean. It is unimodal, meaning it has a single peak at the mean, and tails that extend infinitely in both directions without touching the horizontal axis.

Mean, Median, and Mode Equality

In a normal distribution, the mean, median, and mode are all equal and located at the center of the curve. This equality reflects the perfect symmetry of the distribution, where half the data lies on each side of the mean.

Area Under the Curve and Standard Deviation

The total area under the normal curve equals 1, representing the entire probability space. The spread of the curve is determined by the standard deviation; about 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.