Welcome back to Professor Baker’s Math Class! Today, we are moving into Section 6-2: The Normal Distribution. If you have ever heard the term "grading on a curve" or looked at data regarding heights or test scores, you have likely encountered the famous Bell Curve. This section is pivotal because it helps us make sense of how data clusters around an average.

What is the Normal Distribution?

The normal distribution is a specific type of bell-shaped graph that is symmetric. This means if you draw a line down the middle, the left side is a mirror image of the right. Here are the key characteristics found in your class notes:

  • The Mean and Median are the same.
  • Most data is clustered near the mean (the peak of the bell).
  • The Standard Deviation determines the shape. A large standard deviation creates a wide, flat bell, while a small standard deviation creates a tall, steep bell.

The 68-95-99.7% Rule

One of the most useful tools in statistics is the empirical rule for normal distributions. It tells us exactly how spread out the data is:

  • 68% of data lies within 1 standard deviation of the mean.
  • 95% of data lies within 2 standard deviations of the mean.
  • 99.7% of data lies within 3 standard deviations of the mean.

Understanding Z-Scores

How do we compare two different data points from different datasets? We use a z-score (or standard score). The z-score tells you exactly how many standard deviations a specific data point is above or below the mean.

The formula we use is:

$$ z = \frac{\text{Data point} - \text{Mean}}{\text{Standard deviation}} $$

If the z-score is positive, the data is above the mean. If it is negative, it is below the mean. For example, a z-score of 2.0 means the value is significantly higher than average (in the 97th percentile!).

The Central Limit Theorem

Finally, we touch upon a cornerstone of statistics: The Central Limit Theorem. This theorem states that if we take many samples of the same size from a population, the percentages obtained from those samples will form a normal distribution.

To find the standard deviation for these sample percentages, we use the following formula, where $p$ is the percentage and $n$ is the sample size:

$$ \sigma = \sqrt{\frac{p(100 - p)}{n}} $$

Make sure to download the attached PDF for detailed examples, including the distribution of apple weights and newborn birth weights. These examples will help you visualize these formulas in real-world scenarios.

Next Steps: Review the slides attached below, watch the lecture video, and complete the Section 6-2 Quiz. You can do this!