Welcome back to class! In Section 6-2, we are diving into one of the most important concepts in statistics: The Normal Distribution. You likely recognize this shape as the "bell curve." It appears everywhere in nature and social sciences, describing everything from the heights of adult males to the weights of apples in a fall harvest.

Key Characteristics of the Bell Curve

As we discussed in the lecture, a normal distribution isn't just any curve. It has specific properties that make it incredibly useful for quantitative literacy:

  • Symmetry: The curve is a mirror image on both sides of the center.
  • Central Tendency: The mean (average) and the median are the exact same number.
  • Clustering: Most of the data points are clustered near the mean, creating the "hump" of the bell, with fewer data points appearing as you move toward the "tails."

The 68-95-99.7% Rule

One of the most powerful tools we have for normal distributions is this rule of thumb. If your data is normally distributed:

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

Understanding Z-Scores

How do we compare data points from different data sets? We use the 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. A positive z-score means the value is above average, while a negative z-score means it is below average.

The formula we use is:

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

The Central Limit Theorem

Toward the end of the lesson, we touched on a more advanced concept: the Central Limit Theorem. This theorem tells us that if we take many samples of the same size from a population, the percentages obtained from those samples will form a normal distribution.

For a population percentage $p$ and a sample size $n$, the standard deviation for this distribution is calculated as:

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

Note: In this specific formula, we use $p$ as a whole number percentage (e.g., 30 for 30%), not a decimal.

Class Resources

I have attached the slides and the assignment below. The assignment (Section 6-2) includes multiple-choice questions that will test your ability to calculate z-scores and apply the Empirical Rule to real-world scenarios like fingerprint ridge counts and yearly temperatures.

Good luck with the study material, and remember: statistics is the art of thinking between the lines!