Descriptive Statistics: Numerical Measures

Vocabulary practice flashcards covering measures of location, variability, distribution shape, empirical rules, and summary statistics from Chapter 3.

Population Parameters

Numerical measures computed for data from an entire population.

Sample Statistics

Numerical measures computed for data drawn from a sample of a population.

Point Estimator

A sample statistic used as a point estimator or approximation of the corresponding population parameter.

Mean

A measure of central location defined as the average of all data values in a data set, where the sample mean ($\bar{x}$) serves as the point estimator of the population mean ($\mu$).

Median

The middle value in a data set when items are arranged in ascending order; preferred as a measure of central location when extreme values are present.

Mode

The value in a data set that occurs with the greatest frequency.

Bimodal Data

A data set that contains exactly two modes.

Multimodal Data

A data set that contains more than two modes.

Percentile

A value such that at least $p$ percent of items take on this value or less and at least $(100 - p)$ percent take on this value or more.

Quartiles

Specific percentiles that divide a data set into quarters: First Quartile ($Q_1 = 25^{\text{th}}\text{ percentile}$), Second Quartile ($Q_2 = 50^{\text{th}}\text{ percentile} = \text{Median}$), and Third Quartile ($Q_3 = 75^{\text{th}}\text{ percentile}$).

Skewness

An important numerical measure of the shape of a distribution, which equals zero for symmetric distributions, is negative when skewed left, and positive when skewed right.

Range

The simplest measure of variability, calculated as the difference between the largest and smallest values in a data set.

Interquartile Range (IQR)

The difference between the third quartile and the first quartile ($IQR = Q_3 - Q_1$), representing the range for the middle 50% of the data.

Variance

A measure of variability equal to the average of the squared deviations between each observation ($x_i$) and the mean ($\bar{x}$ for a sample, $\mu$ for a population).

Standard Deviation

A measure of variability defined as the positive square root of the variance ($s$ for a sample, $\sigma$ for a population), expressed in the same units as the original data.

Coefficient of Variation (CV)

A relative measure of variability indicating how large the standard deviation is relative to the mean, calculated as $\left(\frac{\text{std. dev.}}{\text{mean}}\right) \times 100\%$.

z-Score

Also called the standardized value, it denotes the number of standard deviations an observation ($x_i$) is from the mean.

Empirical Rule

A rule used for bell-shaped distributions stating that approximately 68% of data values lie within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.

Chebyshev's Theorem

A theorem used in place of the Empirical rule for any distribution shape to determine the minimum proportion of data values within a specified number of standard deviations of the mean.

Outlier

An unusually small or large data value, commonly identified as having a z-score less than $-3$ or greater than $+3$, or lying outside limits set $1.5 \times IQR$ beyond the quartiles.

Five-Number Summary

A five-statistic summary of a data set comprising the Smallest Value, First Quartile ($Q_1$), Median ($Q_2$), Third Quartile ($Q_3$), and Largest Value.

Box Plot

A graphical display based on a five-number summary where a box is drawn between $Q_1$ and $Q_3$, a line represents the median, and limits positioned using $1.5 \times IQR$ help identify outliers.