A set of test scores is normally distributed with \mu = 75 and \sigma = 5. What is the z-score for a student who scored 85?

2.0 z = (85 - 75) / 5 = 10 / 5 = 2.0. This means the score is 2 standard deviations above the mean.

Normal Distribution Explorer

Adjust mean and standard deviation to see how the normal distribution bell curve shifts and stretches. Shade probability regions to compute areas under the curve and connect z-scores to percentiles.

WHAT IS THE NORMAL DISTRIBUTION?

The normal distribution, often called the "bell curve," is a symmetric, unimodal distribution defined by its mean () and standard deviation (). In AP Statistics, it is the cornerstone of inference. Many real-world variables—like heights, exam scores, and errors in measurement—tend to follow this pattern. A key property is the **Empirical Rule** (68-95-99.7 rule), which states that approximately 68% of data falls within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs.

HOW TO USE THIS VISUALIZATION

1. **Shift the Mean**: Drag the center of the curve to see how the entire distribution slides along the x-axis without changing shape. 2. **Adjust Standard Deviation**: Increase to see the curve flatten and spread out, or decrease it to make the curve taller and narrower. 3. **Calculate Probabilities**: Use the boundary markers to shade an area under the curve. The shaded region represents the proportion of data or the probability . 4. **Standardize**: Toggle the Z-score mode to see the Standard Normal Distribution ().

CORE FORMULAS

Z-score formula for standardizing values

Notation for a normally distributed variable

AP EXAM CONNECTION

Unit: Unit 1: Exploring One-Variable Data (Topic 1.10)
Learning Objective: VAR-2.B

COMMON MISCONCEPTIONS

Thinking all symmetric distributions are normal.
Forgetting that the total area under the density curve must always equal 1.
Confusing the mean () with the median (though they are equal in a perfectly normal distribution).

KEY TAKEAWAYS

The normal distribution is perfectly symmetric and unimodal.
Z-scores tell you how many standard deviations a value is from the mean.
The total area under the curve represents 100% of the data or a total probability of 1.