Normal Distribution:

📊 What is the Normal Distribution?

The normal distribution, also known as the Gaussian distribution, is a symmetric probability distribution centered around the mean. It's characterized by its bell shape and is fully described by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$).

🔑 Key Properties

• Symmetry: The distribution is symmetric around the mean, meaning the left and right sides are mirror images.
• Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal.
• Empirical Rule: Approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

🧮 Formula for the Normal Distribution

The probability density function (PDF) of the normal distribution is given by:

f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}

Where:

• $x$ is the value of the variable
• $\mu$ is the mean
• $\sigma$ is the standard deviation
• $e$ is Euler's number (approximately 2.71828)
• $\pi$ is Pi (approximately 3.14159)

📏 Standard Deviation and Mean

• Mean ($\mu$): The average value of the data set. It determines the center of the distribution.
• Standard Deviation ($\sigma$): A measure of the spread or dispersion of the data. A larger standard deviation indicates a wider distribution, while a smaller one indicates a narrower distribution.

🔢 Z-Scores

A z-score (also called a standard score) indicates how many standard deviations an element is from the mean. It is calculated as:

z = \frac{x - \mu}{\sigma}

Z-scores are used to standardize normal distributions, allowing for easy comparison of data points from different distributions. A z-score of 1.5 means the data point is 1.5 standard deviations above the mean.

✍️ Example: Calculating Probabilities

Suppose we have a normal distribution with a mean ($\mu$) of 50 and a standard deviation ($\sigma$) of 10. We want to find the probability that a value is less than 60.

1. Calculate the z-score:

z = \frac{60 - 50}{10} = 1

2. Find the probability: Using a z-table or calculator, we find the probability that $z < 1$ is approximately 0.8413.

Therefore, the probability of a value being less than 60 is approximately 84.13%.