Define the normal distribution.

The normal distribution, also known as the Gaussian distribution, is a type of continuous probability distribution for a real-valued random variable. Here are the steps to define it:

1. Shape: The normal distribution is symmetric and follows a bell-shaped curve. It is defined by its mean (μ) and standard deviation (σ). The mean determines the center of the distribution, and the standard deviation determines the spread of the distribution.

2. Mean: The mean of the normal distribution is the highest point of the curve (the peak) and it represents the most probable event. In a normal distribution, the mean is equal to the median and mode.

3. Standard Deviation: The standard deviation measures the dispersion of the distribution. If the standard deviation is small, the data points are close to the mean, and the curve is steep. If the standard deviation is large, the data points are spread out from the mean, and the curve is flat.

4. Area Under the Curve: The total area under the curve of a normal distribution is equal to 1, which represents the total probability of all possible outcomes.

5. Empirical Rule: In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. This is known as the empirical rule or the 68-95-99.7 rule.

6. Probability Density Function: The probability density function of a normal distribution is given by the formula:

   f(x) = (1 / √(2πσ^2)) * e^(-(x-μ)^2 / (2σ^2))

   where:

   • f(x) is the height of the curve at any point x.
   • e is the base of the natural logarithm (approximately equal to 2.71828).
   • π is a constant (approximately equal to 3.14159).
   • σ is the standard deviation.
   • μ is the mean.

7. Properties: Some properties of the normal distribution include:

   • It is completely determined by its mean and standard deviation.
   • It is symmetric about the mean.
   • The mean, median, and mode are all equal.
   • The curve approaches but never touches the x-axis.

In summary, the normal distribution is a fundamental concept in statistics and is used in various fields such as natural and social sciences to represent real-valued random variables whose distributions are not known.