Using Normal vs. Standard Normal Distribution

**Real-Life Example: Employee Salaries**

**Scenario: Analyzing Employee Salaries**

1. **Normal Distribution**:

   - **What**: A normal distribution describes the spread of salaries with a specific mean (μ) and standard deviation (σ).

   - **When to Use**: Use this distribution to model and understand the general distribution of salaries. For example, if the average salary is $60,000 with a standard deviation of $5,000, you represent this with a normal distribution N(60000, 5000²). This helps in understanding where most salaries fall and how spread out they are.

2. **Standard Normal Distribution**:

   - **What**: This distribution is used for standardizing values. It converts raw data into Z-scores, which tell how many standard deviations a value is from the mean.

   - **When to Use**: Use the standard normal distribution to make comparisons or calculate probabilities. For instance, to find the percentage of employees earning more than $70,000, convert $70,000 into a Z-score using the formula Z = (X - μ) / σ. For $70,000, this is Z = (70000 - 60000) / 5000 = 2. You then use the standard normal distribution to find the probability for a Z-score of 2, indicating the percentage of employees earning more than $70,000.

**In Summary**:

- Use the **normal distribution** for modeling and understanding the distribution of raw salary data.

- Use the **standard normal distribution** for standardizing data, making comparisons, or finding probabilities.