Business Statistics: Summary Measures

Vocabulary flashcards covering key statistical summary measures including central location, dispersion, relative position, linear association, and multiperiod returns.

Central Location

Refers to how numerical data tend to cluster around some middle or central value.

Arithmetic Mean

The primary measure of central location, calculated by adding up all observations and dividing by the total number of observations.

Sample Mean

A sample statistic denoted by $\bar{x}$, calculated as $\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}$.

Population Mean

A population parameter denoted by $\mu$, calculated as $\mu = \frac{x_1 + x_2 + \dots + x_N}{N}$.

Median

The middle value of a variable that divides arranged data in half, with an equal number of observations lying above and below it.

Mode

The observation that occurs most frequently in a dataset.

Unimodal

Describing a dataset or distribution that has exactly one mode.

Bimodal

Describing a dataset or distribution that has two modes.

Weighted Mean

A measure of central location used when observations contribute differently to the average, computed as $\bar{x}_w = \sum w_i x_i$.

Symmetric Distribution

A distribution where one side of the histogram is a mirror image of the other, resulting in equal mean, median, and mode values.

Skewness Coefficient

A numerical measure of skewness where zero indicates symmetry, a positive value indicates extreme observations in the right tail, and a negative value indicates extreme observations in the left tail.

Percentile

A measure of location dividing a variable into two parts, where approximately $p$ percent of observations are less than the $p^{\text{th}}$ percentile and $(100 - p)$ percent are greater.

Five-Number Summary

A set of summary statistics consisting of the minimum, first quartile ($Q_1$), median ($Q_2$), third quartile ($Q_3$), and maximum.

Boxplot

A visual display (also called a box-and-whisker plot) used to graphically depict a five-number summary and identify potential outliers.

Interquartile Range

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

Range

The simplest measure of dispersion, calculated as the difference between the maximum and minimum observations ($\text{Range} = \text{Max} - \text{Min}$).

Mean Absolute Deviation

The average of the absolute differences between the observations and their mean.

Variance

A measure of dispersion defined as the average of the squared differences between the observations and their mean.

Standard Deviation

The positive square root of the variance, expressing dispersion in the original units of the variable.

Coefficient of Variation

A unitless measure of relative dispersion that adjusts for differences in the magnitudes of the means, calculated as $CV = \frac{s}{\bar{x}}$ for a sample or $CV = \frac{\sigma}{\mu}$ for a population.

Mean-Variance Analysis

An investment performance analysis framework evaluating an asset's rate of return in terms of reward (mean) and risk (variance or standard deviation).

Sharpe Ratio

A reward-to-variability ratio calculated as $\frac{\bar{x} - R_f}{s}$, measuring the excess return per unit of risk.

Chebyshev's Theorem

A theorem stating that the proportion of observations within $k$ standard deviations from the mean is at least $1 - \frac{1}{k^2}$ for any $k > 1$, applicable to any distribution shape.

Empirical Rule

A rule for symmetric, bell-shaped distributions stating that approximately 68% of observations fall within $\bar{x} \pm s$, 95% within $\bar{x} \pm 2s$, and almost 100% within $\bar{x} \pm 3s$.

z-score

A unitless measure of relative position calculated as $z = \frac{x - \bar{x}}{s}$, expressing the distance of an observation from the mean in standard deviations.

Covariance

A measure of association that quantifies the direction of a linear relationship between two variables.

Correlation Coefficient

A unit-free measure bounded between $-1$ and $1$ describing both the direction and strength of the linear relationship between two variables.

Geometric Mean Return

A multiplicative average used to evaluate multiperiod investment returns over several years, given by $G_R = \left( (1 + R_1) \times (1 + R_2) \times \dots \times (1 + R_n) \right)^{\frac{1}{n}} - 1$.

Compound Growth Rate

The average geometric growth rate across $n$ multiperiod observations calculated as $G_g = \left( \frac{x_n}{x_1} \right)^{\frac{1}{n-1}} - 1$.