AP Statistics (AP Stats): Density Curves and Median/Mean Location

AP Statistics – Density Curves and Median/Mean Location

AP Statistics: Density Curves and Median/Mean Location – Exam-Ready Study Guide

What This Is

Density curves model the overall shape of a distribution, smoothing out the bumps in histograms to reveal patterns. The mean (?) and median are key measures of center, but their locations shift depending on the curve’s shape (symmetric vs. skewed). This topic is essential on the AP exam because it underlies probability calculations, normal distributions, and inference. Real-world example: A factory tests battery lifespans. If the density curve is right-skewed (most batteries last a long time, but a few die early), the median (typical lifespan) will be less than the mean (average lifespan), which is pulled higher by the long-lasting outliers.

Key Terms & Formulas

• Density curve: A smooth curve that models the distribution of a quantitative variable; the area under the curve = 1 (100% of the data).
• Mean (?): The balance point of the density curve (like the fulcrum of a seesaw). For skewed data, the mean is pulled in the direction of the tail.
• Median: The point that divides the area under the curve into two equal halves (50% on each side).
• Symmetric density curve: Mean = median (e.g., normal distribution).
• Right-skewed (positively skewed) density curve: Mean > median (tail pulls mean to the right).
• Left-skewed (negatively skewed) density curve: Mean < median (tail pulls mean to the left).
• Normal density curve: Bell-shaped, symmetric; defined by ? (mean) and ? (standard deviation). Use normalcdf(lower, upper, ?, ?) on TI-84 to find probabilities.
• Empirical Rule (68-95-99.7): For normal distributions, ~68% of data falls within 1? of ?, ~95% within 2?, and ~99.7% within 3?.
• Percentile: The value below which a given percentage of the data falls. Use invNorm(area to left, ?, ?) on TI-84 to find percentiles.
• Uniform density curve: A flat (rectangular) curve where all outcomes are equally likely (e.g., rolling a fair die).

Step-by-Step / Process Flow

How to analyze a density curve on the AP exam (FRQ-style):

1. Sketch the curve (if not provided) and label the mean (?) and median. Note symmetry/skew.

2. Example: If the curve is right-skewed, draw the mean to the right of the median.

3. Compare mean and median based on the curve’s shape.

4. Example: "The density curve is left-skewed, so the mean is less than the median."

5. Calculate probabilities or percentiles using the curve’s properties.

6. For normal curves, use normalcdf (probabilities) or invNorm (percentiles).
7. For uniform curves, use area = (length) × (height).

8. Example: For a uniform curve from 0 to 10, P(X < 4) = 4 × (1/10) = 0.4.

9. Interpret results in context.

10. Example: "There is a 34% chance a randomly selected battery lasts between 5 and 7 hours."

11. Check conditions if using normal calculations (e.g., "The problem states the distribution is approximately normal").

Common Mistakes

• Mistake: Assuming the mean and median are always equal.

• Correction: Only true for symmetric curves. For skewed curves, the mean is pulled toward the tail. Why? Outliers in the tail "drag" the mean in that direction.

• Mistake: Confusing normalcdf and invNorm.

• Correction:
  • normalcdf(lower, upper, ?, ?)-probability (area under the curve).
  • invNorm(area to left, ?, ?)-value (percentile).

• Why? normalcdf gives the % of data between two values; invNorm gives the value for a given %.

• Mistake: Forgetting to label-and-on a normal curve sketch.

• Correction: Always label the mean (center) and standard deviation (distance from center to inflection point). Why? AP graders deduct points for missing labels.

• Mistake: Misinterpreting the median’s location in skewed data.

• Correction: The median is the "equal areas" point, not the peak. For right-skewed data, the median is left of the mean. Why? The tail pulls the mean right, but the median stays at the 50% mark.

AP Exam Insights

1. FRQs often ask you to:
2. Sketch a density curve and label mean/median based on skew.
3. Calculate probabilities or percentiles for normal/uniform distributions.

4. Compare mean vs. median in context (e.g., "Explain why the mean salary is higher than the median salary for this company").

5. Tricky distinctions:

6. Mean vs. median: The mean is sensitive to outliers; the median is resistant. AP loves to test this in skewed distributions.

7. Normal vs. non-normal: Always check if the problem states the distribution is normal before using normalcdf/invNorm. If not, you can’t assume normality!

8. Calculator pitfalls:

9. Forgetting to include-and-in normalcdf/invNorm. Defaults are ?=0, ?=1 (standard normal), but most problems use different values.

10. Mixing up "area to left" vs. "area to right" in invNorm. Tip: Draw a quick sketch to visualize the area.

11. Common FRQ setup:

12. A density curve is given (e.g., right-skewed). You’re asked to:
    1. Label the mean and median.
    2. Estimate the % of data above/below a value.
    3. Explain why the mean is greater than the median.

Quick Check Questions

1. Multiple Choice: A density curve is left-skewed. Which of the following is true? (A) Mean > Median (B) Mean < Median (C) Mean = Median (D) The relationship cannot be determined.

Answer: (B) Mean < Median. Explanation: In left-skewed data, the tail pulls the mean left of the median.

1. FRQ Part: The lifetimes of a certain brand of lightbulbs are approximately normally distributed with a mean of 1,000 hours and a standard deviation of 100 hours. (a) What proportion of lightbulbs last more than 1,150 hours? (b) What lifetime corresponds to the 25th percentile?

Answer: (a) normalcdf(1150, 1E99, 1000, 100) = 0.0668 (6.68%). (b) invNorm(0.25, 1000, 100) = 932.55 hours. Explanation: (a) Use normalcdf for P(X > 1150); (b) use invNorm for the 25th percentile.

1. Multiple Choice: For a uniform density curve from 0 to 20, what is P(5 < X < 15)? (A) 0.25 (B) 0.50 (C) 0.75 (D) 1.00

Answer: (B) 0.50. Explanation: Area = (15 – 5) × (1/20) = 10/20 = 0.5.

Last-Minute Cram Sheet

1. Mean (?): Balance point; pulled by skew.
2. Median: Equal-areas point; resistant to outliers.
3. Symmetric curve: Mean = median.
4. Right-skewed: Mean > median (tail pulls mean right).
5. Left-skewed: Mean < median (tail pulls mean left).
6. Normal curve: Bell-shaped; use normalcdf/invNorm with-and ?.
7. Empirical Rule: 68% (1?), 95% (2?), 99.7% (3?).
8. Uniform curve: Area = (length) × (height).
9. Always label-and-on normal curve sketches.
10. For invNorm, input "area to left" (not right).