The normal distribution is a fundamental concept in statistics and a key model for continuous random variables. Its importance stems from its natural occurrence in many real-world phenomena and its central role in the Central Limit Theorem. This theorem suggests that the distribution of sample means approximates a normal distribution, regardless of the population's original distribution, provided the sample size is large.

Understanding the Normal Distribution

• Symmetry Around the Mean: The curve is symmetrical around the mean, where mean, median, and mode are equal.
• Defined by Mean and Variance: The mean (μ) sets the center; variance (σ²) defines the spread.
• Asymptotic: The tails extend indefinitely, indicating all values are possible but increasingly unlikely.

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Standard Normal Distribution

Centralized around zero, standard deviation of one.

Applications

• Measurement Errors: In scientific experiments.
• Biological Attributes: Like population heights.
• Financial Models: Like stock market returns.

Using Normal Distribution Table

• Standardization: Convert to Z-score using $Z = \frac{X - μ}{σ}$.
• Lookup: Find Z-score in the table for the probability.

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Sketching Normal Curves

• Draw Axis: Mark variable range.
• Indicate Mean: Center curve at the mean.
• Shape Curve: Symmetrical, bell-shaped.

Example: Widget Weights

• Given: Mean = 100 grams, Standard Deviation = 15 grams.
• Find: Probability of > 120 grams.
• Solution: $Z = \frac{120 - 100}{15} = 1.33$.
• Probability: $≈ 9.12$.

Example: Test Scores

• Given: Mean = 70, Standard Deviation = 8.
• Find: Probability of < 60.
• Solution: $Z = \frac{60 - 70}{8} = -1.25$.
• Probability: $≈ 10.56$.