What you'll learn
Exploring One-Variable Data (~15–23% of AP Statistics) is about describing distributions of a single variable: shape, center, spread, and unusual features.
Types of variables
- Categorical (groups) → summarise with counts/proportions, bar charts.
- Quantitative (numbers) → summarise with center/spread, histograms, dotplots, boxplots.
Describing a distribution: SOCS
- Shape: symmetric, skewed left, or skewed right. A long right tail = skewed right (mean > median).
- Outliers: by the 1.5 × IQR rule, a value below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.
- Center: mean (sensitive to outliers) or median (resistant).
- Spread: range, IQR (Q3 − Q1), and standard deviation.
Measures
- Mean = sum ÷ n. Median = middle value. Mode = most frequent.
- Standard deviation measures typical distance from the mean.
- Adding a constant to all values shifts center but not spread; multiplying scales both.
The normal distribution
For approximately normal data, the empirical (68–95–99.7) rule: ~68% within 1 SD, ~95% within 2 SD, ~99.7% within 3 SD of the mean.
z-scores
A z-score = (value − mean) / SD, giving the number of standard deviations from the mean. z = +2 means two SD above the mean. z-scores let you compare values from different distributions.
Exam tips
- When asked to "describe a distribution," always cover shape, outliers, center, spread in context.
- Choose median/IQR for skewed data; mean/SD for roughly symmetric data.
- Use z-scores to standardise comparisons.
Common mistakes
- Describing distributions without context (units, variable).
- Reporting mean/SD for clearly skewed data.
- Forgetting that adding a constant doesn't change the standard deviation.