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What is the Normal Distribution?

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, normal distribution will appear as a bell curve.

Characteristics of Normal Distribution

• Bell-shaped curve: Symmetric about the mean
• Mean, median, and mode: All are equal and located at the center
• Empirical Rule: 68-95-99.7 rule (see below)
• Completely described: By its mean (μ) and standard deviation (σ)

The Empirical Rule (68-95-99.7 Rule)

For a normal distribution:

• About 68% of values fall within 1 standard deviation of the mean
• About 95% of values fall within 2 standard deviations of the mean
• About 99.7% of values fall within 3 standard deviations of the mean

Standard Normal Distribution

A special case of the normal distribution where:

• Mean (μ) = 0
• Standard deviation (σ) = 1
• Any normal distribution can be converted to standard normal using z-scores

Z-Scores

A z-score measures how many standard deviations an element is from the mean:

z = (X - μ) / σ

Where:

• X is the value from the original distribution
• μ is the mean of the original distribution
• σ is the standard deviation of the original distribution

Applications of Normal Distribution

The normal distribution is used in many real-world applications:

• Quality control in manufacturing
• Standardized testing scores
• Financial market analysis
• Biological measurements (height, blood pressure, etc.)
• Error measurement in scientific experiments

Testing for Normality

Before applying normal distribution methods, you should check if your data is normally distributed. Common methods include:

• Visual inspection (histogram, Q-Q plot)
• Statistical tests (Shapiro-Wilk, Kolmogorov-Smirnov)
• Checking skewness and kurtosis

Limitations

While extremely useful, the normal distribution has limitations:

• Not all data follows a normal distribution
• Can be misleading when applied to skewed data
• Outliers can significantly affect mean and standard deviation
• Real-world data often has "fatter tails" than the normal distribution predicts