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:**$≈ 10.56%$.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.