normal distribution song

Definition of a Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric and bell-shaped, describing how values of a random variable are distributed. The normal distribution is also known as a Gaussian distribution. It is characterized by its mean (?) and standard deviation (?), where the mean defines the center of the distribution and the standard deviation determines the spread or width.

Mathematically, the probability density function (PDF) of a normal distribution is given by:

Where:

• is the variable
• is the mean (center)
• is the standard deviation
• is the variance

Properties of a Normal Distribution

Symmetry: The normal distribution is perfectly symmetric about its mean, meaning the left and right halves of the distribution are mirror images.

Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution (?).

Bell Shape: The graph of a normal distribution is bell-shaped, with the highest point at the mean. The tails of the distribution extend infinitely but never touch the horizontal axis.

68-95-99.7 Rule (Empirical Rule):

• Approximately 68% of the data falls within 1 standard deviation (?) of the mean ().
• About 95% of the data falls within 2 standard deviations.
• About 99.7% of the data falls within 3 standard deviations.

Asymptotic: The tails of the distribution approach the horizontal axis but never touch it, meaning that values far from the mean are possible, though increasingly rare.

Unimodal: A normal distribution has a single peak, meaning it is unimodal (only one mode).

Area Under the Curve: The total area under the normal distribution curve is equal to 1, representing the total probability of all outcomes.

Standard Normal Distribution: A special case of the normal distribution where the mean (?) is 0 and the standard deviation (?) is 1. It is often used for standardizing data (z-scores). The formula for the z-score is:

This converts any normal distribution to a standard normal distribution.

Importance of Normal Distribution:

Many natural phenomena follow a normal distribution (e.g., heights, IQ scores, measurement errors), which illustrates the tendency of these attributes to cluster around a central value and create a symmetrical bell-shaped curve, reflecting the probability of values occurring within a certain range.