Unit 1 Review: Advanced Distribution Analysis

Comparing Distributions of Quantitative Data

One of the most frequent tasks on the free-response section of the AP Statistics exam is comparing two or more distributions based on graphs (such as parallel boxplots, back-to-back stemplots, or side-by-side histograms). To earn full credit, you must go beyond simply listing statistics; you must explicitly compare them using linking words.

The Comparison Framework: SOCS (or CUSS)

When describing or comparing distributions, always address four key characteristics. A helpful mnemonic is SOCS:

1. Shape: Is the distribution symmetric, skewed right, skewed left, or uniform? Is it unimodal or bimodal?
2. Outliers (or Unusual Features): Are there any specific outliers (calculated via the $1.5 \times IQR$ rule) or gaps in the data?
3. Center: Which measure of center is appropriate (mean or median)?
4. Spread: How variable is the data? Use standard deviation or Interquartile Range (IQR).

Rules for Comparison

To compare distributions effectively, you must follow these three golden rules:

1. Use Comparative Language: Do not just say "Group A has a median of 50 and Group B has a median of 60." You must say, "The median of Group B (60) is greater than the median of Group A (50)."
2. Include Context: Always reference the variable name and the units (e.g., "test scores in points," "height in inches") rather than just saying "the data."
3. Address All Four Aspects: Mention Shape, Outliers, Center, and Spread for the correct data visualization.

Choosing the Right Statistics

The shape of the distribution dictates which measures of center and spread you should compare.

Note: The Median and IQR are resistant measures, meaning they are not heavily influenced by extreme values or skewness. The Mean and Standard Deviation are non-resistant.

The Normal Distribution and the Empirical Rule

The Normal Distribution is a continuous probability distribution that describes many natural phenomena. It forms the backbone of inference later in the course.

Properties of the Normal Model

A Normal density curve is determined fully by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$).

• Shape: Symmetric, single-peaked (unimodal), and bell-shaped.
• Center: The mean, median, and mode are all located at the center of the curve.
• Notation: We denote a Normal distribution as $N(\mu, \sigma)$.

The Empirical Rule (68-95-99.7 Rule)

For any Normal distribution, the area under the curve (which represents proportion or probability) follows a specific pattern based on standard deviations from the mean.

• Approximately 68% of observations fall within $\pm 1\sigma$ of the mean.
  (\mu - \sigma) \text{ to } (\mu + \sigma)
• Approximately 95% of observations fall within $\pm 2\sigma$ of the mean.
  (\mu - 2\sigma) \text{ to } (\mu + 2\sigma)
• Approximately 99.7% of observations fall within $\pm 3\sigma$ of the mean.
  (\mu - 3\sigma) \text{ to } (\mu + 3\sigma)

Worked Example: Using the Empirical Rule

Scenario: The distribution of heights of adult men is approximately Normal with mean $\mu = 70$ inches and standard deviation $\sigma = 2.5$ inches.

Question: Between what two heights do the middle 95% of men fall?

Solution:

1. Identify the requirement: The middle 95% corresponds to $\pm 2$ standard deviations.
2. Calculate bounds:
   • Lower: $\mu - 2\sigma = 70 - 2(2.5) = 70 - 5 = 65$
   • Upper: $\mu + 2\sigma = 70 + 2(2.5) = 70 + 5 = 75$
3. Answer: The middle 95% of men are between 65 and 75 inches tall.

z-Scores and Percentiles

Not all observations fall nicely on the integer standard deviation lines ($1\sigma, 2\sigma$). To compare observations from different Normal distributions, or to find the location of any specific value, we use standardized scores (z-scores).

Definition of a z-Score

A z-score tells us how many standard deviations a particular data value ($x$) is from the mean ($\mu$).

Formula:
z = \frac{x - \mu}{\sigma}

• Positive z-score: The value is above the mean.
• Negative z-score: The value is below the mean.
• z = 0: The value is the mean.

The Standard Normal Distribution

The Standard Normal Distribution is a special case where the mean is 0 and the standard deviation is 1.

• Notation: $N(0, 1)$
• We can transform any Normal distribution into the Standard Normal distribution using the z-score formula.

Percentiles

A percentile describes the location of a value in a distribution. The $p$-th percentile is the value with $p$ percent of the observations less than or equal to it.

• Visualizing Percentiles: On a density curve, the percentile corresponds to the area to the left of the z-score.

Worked Example: Comparing Apples and Oranges (Literally)

Imagine you scored a 1350 on the SAT (Mean=1100, SD=200) and a 30 on the ACT (Mean=21, SD=5). Which score is relatively better?

1. Calculate z-score for SAT:
   z_{SAT} = \frac{1350 - 1100}{200} = \frac{250}{200} = 1.25

2. Calculate z-score for ACT:
   z_{ACT} = \frac{30 - 21}{5} = \frac{9}{5} = 1.80

3. Conclusion: Since $1.80 > 1.25$, the ACT score is relatively better because it is more standard deviations above the mean than the SAT score.

Common Mistakes & Pitfalls

1. Missing Context in Comparisons: Never write generic statements like "The mean is higher." Always write, "The mean weight of the elephants is higher than the mean weight of the hippos."
2. Confusing Skewness and Mean/Median Relationship:
   • Skewed Right: Mean > Median (The tail drags the mean up).
   • Skewed Left: Mean < Median (The tail drags the mean down).
3. Using the Empirical Rule on Non-Normal Data: You cannot use the 68-95-99.7 rule if the distribution is not stated to be Normal or is clearly skewed.
4. Misinterpreting z-scores as Percentages: A z-score of 2.0 does not mean 2%. It corresponds (approximately) to the 97.7th percentile (area to the left is 0.9772).
5. Area vs. Value: Remember that table values (or calculator output normalcdf) give you the area (probability), not the x-value. Conversely, invNorm takes an area and gives you a z-score or x-value.