Unit 1 One-Variable Data: Comparing Distributions and Using the Normal Model

Distribution

A description of what values a quantitative variable takes and how often it takes them (overall pattern, not just one summary number).

SOCS comparison (Center, Spread, Shape, Outliers)

A common AP Statistics structure for comparing quantitative distributions by describing differences/similarities in center, spread, shape, and outliers/unusual features, in context.

Center

Where the typical value of a distribution is located (often summarized by the mean or median).

Spread

How variable the data values are; how much the distribution extends or how consistent values are (e.g., IQR or standard deviation).

Shape

The overall form of a distribution (e.g., symmetric vs skewed; unimodal vs bimodal; presence of gaps/clusters).

Outlier

An unusually large or small value compared with the rest of the data; can distort the mean/standard deviation and may indicate an error or a rare case.

Histogram

A graph for quantitative data (especially large data sets) that groups values into bins and shows frequency; good for visualizing shape.

Boxplot

A display that summarizes a distribution using the five-number summary and highlights center (median), spread (IQR), and potential outliers.

Side-by-side boxplots

Multiple boxplots on the same scale used to compare groups easily, especially for differences in medians, IQRs, and outliers.

Mean

The arithmetic average of a quantitative variable; sensitive to outliers and skew.

Median

The middle value in an ordered data set; resistant to outliers and skew and often preferred for skewed distributions.

Interquartile range (IQR)

A resistant measure of spread: IQR = Q3 − Q1; represents the spread of the middle 50% of the data.

Standard deviation

A measure of spread describing the typical distance of values from the mean; sensitive to outliers and extreme values.

Right-skewed distribution

A distribution with a long tail to the right (toward larger values); often has a few unusually high values.

Left-skewed distribution

A distribution with a long tail to the left (toward smaller values); often has a few unusually low values.

Unimodal distribution

A distribution with one clear peak (one mode).

Bimodal distribution

A distribution with two distinct peaks (two modes), often suggesting a mixture of two subgroups.

1.5×IQR rule

A common rule to flag potential outliers: values below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR) are considered outliers.

Normal distribution

A unimodal, symmetric, bell-shaped density curve with tails extending indefinitely; fully described by mean μ and standard deviation σ.

Normal parameters (μ and σ)

The two numbers that determine a Normal model: μ is the mean (center) and σ is the standard deviation (spread). Written X ~ N(μ, σ).

Density curve

A smooth curve where area represents probability; total area under the curve equals 1, and probabilities correspond to areas (not heights).

Empirical Rule (68–95–99.7)

For a Normal distribution: about 68% of observations fall within 1σ of μ, 95% within 2σ, and 99.7% within 3σ.

Standard Normal distribution

The Normal distribution with mean 0 and standard deviation 1: Z ~ N(0, 1); used as a reference after standardizing to z-scores.

z-score

The number of standard deviations a value x is from the mean: z = (x − μ)/σ (or (x − x̄)/s when standardizing using sample statistics).

Percentile

A measure of relative standing: the percentile of a value is the percent of observations at or below that value (e.g., 90th percentile means ~90% are ≤ that value).