Exploring One-Variable Data

What this unit covers

The topics below follow the published Statistics course framework for Unit 1. This unit is worth 15–23% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Describing Distributions13 min · 3 objectivesClassify variables as categorical or quantitative and choose an appropriate graph · Describe a distribution by its shape, center, spread, and unusual features in context · Read shape from a graph, including skew direction and modality
• Summary Statistics: Center & Spread14 min · 3 objectivesCompute and interpret the mean, median, standard deviation, and IQR · Explain which measures are resistant to outliers and choose accordingly · Predict how the mean and median compare under different skew
• The Normal Model & z-Scores14 min · 3 objectivesStandardize a value into a z-score and interpret it in context · Apply the 68–95–99.7 (empirical) rule to a Normal distribution · Use z-scores to compare values from different distributions
• Outliers & Boxplots12 min · 3 objectivesApply the 1.5 × IQR rule to identify outliers · Construct and read a boxplot using the five-number summary · Explain how outliers affect resistant versus non-resistant statistics

Formulas in Unit 1

Every term in Unit 1

All 21 terms we publish for Exploring One-Variable Data, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

1.5 × IQR outlier rule

A value is an outlier below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

z-score

z = (x − μ)/σ, the number of standard deviations from the mean. Allows comparison across different distributions and units.

Categorical vs quantitative variables

Categorical places individuals in groups; quantitative takes numerical values you can average. Zip codes are numbers but categorical.

Describing a distribution

Shape, Outliers, Center, Spread — and always in context. Omitting context is the commonest lost point in the whole course.

Skewness direction

Skewed right has a long right tail and mean above median; skewed left is the reverse. The tail names the skew, not the bulk of the data.

Mean vs median resistance

The median resists outliers; the mean does not. Report the median and IQR for skewed data, mean and standard deviation for roughly symmetric data.

Standard deviation

The typical distance of a value from the mean. It is zero only when every value is identical, and is never negative.

Interquartile range

IQR = Q3 − Q1, the spread of the middle half. Resistant to outliers, unlike the range.

Boxplot limitations

Shows the five-number summary but hides multimodality — two very different distributions can produce identical boxplots.

Effect of transformations

Adding a constant shifts center but not spread; multiplying scales both center and spread by that constant.

Normal distribution and the empirical rule

About 68%, 95% and 99.7% of values lie within one, two and three standard deviations of the mean.

Percentile

The percentage of observations at or below a value. The 90th percentile is not the same as a score of 90.

Density curve

A smooth model with total area 1 under it. Area corresponds to proportion, which is why probability is an area.

Comparing distributions

Compare shape, center and spread explicitly with comparative language — "the median for group A is higher than for group B" — not two separate descriptions.

Why context is scored

Every description must name the variable and its units. "The distribution is skewed right" is incomplete; "the distribution of commute times is skewed right" is not.

Choosing a graph type

Bar charts and pie charts for categorical data; dotplots, stemplots, histograms and boxplots for quantitative. Using a bar chart for quantitative data loses the point.

Histogram vs bar chart

Histogram bars touch because the variable is continuous; bar chart bars are separated because the categories are distinct.

Effect of an outlier on mean and median

The mean is pulled toward the outlier; the median barely moves. This is what "resistant" means.

Standardizing and comparing

Converting to z-scores lets you compare values from different distributions, such as an SAT score against an ACT score.

Normal probability calculations

Sketch, shade, standardize, then find the area. The sketch is worth doing because it catches the reversed-tail error.

Cumulative relative frequency graph

Read percentiles directly: the height at a value is the proportion at or below it, so the median is where the curve crosses 0.5.

What examiners penalize here

• When a prompt says "skewed" or mentions outliers, default to **median and IQR**. When it says "roughly symmetric" or "bell-shaped," the **mean and standard deviation** are appropriate. Matching the summary to the shape is a routine AP decision point.
• Show the fence arithmetic explicitly on free response: state IQR = Q3 − Q1, multiply by 1.5, then add to Q3 and subtract from Q1. Graders want to see the boundary values, not just the word "outlier."

Practice Statistics

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 9 units of AP Statistics

1. Unit 1 · Exploring One-Variable Data
2. Unit 2 · Exploring Two-Variable Data
3. Unit 3 · Collecting Data
4. Unit 4 · Probability & Random Variables
5. Unit 5 · Sampling Distributions
6. Unit 6 · Inference for Proportions
7. Unit 7 · Inference for Means
8. Unit 8 · Inference for Categorical Data: Chi-Square
9. Unit 9 · Inference for Quantitative Data: Slopes