Categorical and Quantitative Variables, Density Curves, and Numerical Summaries

Vocabulary flashcards covering key statistical concepts, definitions, and graphical displays from Chapter 1 on looking at data distributions.

Cases

Collection of objects in a data set.

Variable

A characteristic of a person or a thing that can be assigned a number or a category.

Quantitative Variable

A variable that records the amount of something (numerical information).

Categorical Variable

A variable that records which of several categories a person or thing is in.

Pie Chart

A graph in which a circle is divided into sectors, where each sector shows the relative frequency of each category.

Bar Graph

A graph drawn using vertical bars where the height of each bar represents the frequency or relative frequency of each category, with vertical bars kept separate.

Stemplot

A table in which each possible value is split into a stem (all but the final digit) and a leaf (the final digit).

Histogram

A graph drawn using vertical bars where the height of each bar represents the frequency or relative frequency of each possible value or each range of values in the data set.

Unimodal Distribution

A distribution with one peak.

Bimodal Distribution

A distribution with two peaks.

Multimodal Distribution

A distribution with two or more peaks.

Symmetric Distribution

A distribution whose graph can be divided into two parts that are mirror images of each other.

Left-skewed Distribution

A distribution that has a longer left tail.

Right-skewed Distribution

A distribution that has a longer right tail.

Density Histogram

A histogram in which bar heights are rescaled so that the total area equals $1$, with height calculated as $\text{Density} = \frac{\text{Proportion in interval}}{\text{Interval width}}$.

Density Curve

A smooth curve that provides an idealized model of an overall distribution, where the total area under the curve equals $1$ and the area under the curve between two values represents the proportion of observations in that interval.

Normal Distribution

A distribution with a density curve that is symmetric, unimodal, and bell-shaped, completely determined by its mean $\mu$ and standard deviation $\sigma$.

68-95-99.7 Rule

A rule stating that for any Normal distribution, approximately $68\%$ of observations lie within $1$ standard deviation of $\mu$, $95\%$ lie within $2$ standard deviations of $\mu$, and $99.7\%$ lie within $3$ standard deviations of $\mu$.

z-score

The standardized value of an observation $x$, calculated as $z = \frac{x - \mu}{\sigma}$, which tells how many standard deviations $x$ lies from the mean $\mu$.

Standard Normal Distribution

The distribution obtained after standardizing a Normal variable, having mean $\mu = 0$ and standard deviation $\sigma = 1$.

Notation $z_\alpha$

Notation that denotes the $z$-score having area $\alpha$ to its right under the standard Normal curve.

Normal Quantile

The $z$-score that corresponds to a given cumulative proportion.

Normal Quantile Plot

A plot comparing observed values with values expected from a Normal distribution, where an approximately straight-line pattern indicates that a Normal model may be reasonable.

Sample Mean ($\bar{x}$)

The arithmetic average of observations $x_1, x_2, \dots, x_n$, calculated as $\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}$.

Median

The midpoint of ordered observations, dividing the data set such that $50\%$ of observations lie below it and $50\%$ lie above it.

Resistant Measure

A statistical measure (such as the median) that is not strongly affected by a small number of extreme observations.

Range

A measure of spread calculated as $\text{Range} = \text{Maximum} - \text{Minimum}$.

Sample Variance ($s^2$)

A measure of spread around the mean, calculated as $s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}$.

Sample Standard Deviation ($s$)

The square root of the sample variance, calculated as $s = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}}$, which measures variation around the mean in original units.

Quartiles

Values ($Q_1$, $Q_2$, $Q_3$) that divide ordered data into four equal parts, each containing approximately $25\%$ of the observations.

Interquartile Range ($\text{IQR}$)

The measure of spread for the middle $50\%$ of observations, calculated as $\text{IQR} = Q_3 - Q_1$.

Five-Number Summary

A numerical summary of a quantitative distribution consisting of the Minimum, $Q_1$, Median, $Q_3$, and Maximum.

1.5 \times \text{IQR} Rule

A rule that identifies potential outliers as observations falling below the lower fence $Q_1 - 1.5(\text{IQR})$ or above the upper fence $Q_3 + 1.5(\text{IQR})$.

Boxplot

A visual display that summarizes a quantitative distribution using the five-number summary and separately plots potential outliers in a modified version.