Question

Explain five properties of normal distribution. (Greek letters
are mu and sigma.)

Answer 1

The normal distribution is characterized by five main properties: a symmetric bell-shaped curve centered around the mean, total area under the curve equals to one, mean equals to median and mode, the variability described by standard deviation, and data distribution percentages known as the Empirical Rule.

Step-by-step explanation:

The normal distribution is a fundamental concept in statistics, often represented by a continuous random variable X with a specific mean (μ) and standard deviation (σ). Here are five key properties of the normal distribution:

1. The graph of the normal distribution is symmetric and bell-shaped, centered around the mean μ.
2. The total area under the normal distribution curve equals one, signifying that it represents a complete probability distribution.
3. The mean, median, and mode of the distribution are all equal and located at the center of the distribution curve.
4. The standard deviation (σ) determines the width of the bell curve: larger values of σ result in a wider and flatter distribution, while smaller values lead to a narrower and taller curve.
5. Approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and around 99.7% falls within three standard deviations (this is known as the Empirical Rule).

When we have a normal distribution with a mean of 0 and a standard deviation of 1, it is called the standard normal distribution, denoted by the letter Z.

For example, if a college entrance exam score has a normal distribution with μ = 52 points and σ = 11 points, we can say the scores are distributed according to X ~ N(52, 11).