Mathematics in the Modern World: Data Management, Statistics, and Patterns in Nature

A set of vocabulary flashcards covering fundamental statistical concepts, sampling methods, descriptive measures of variability and position, normal distribution, linear regression, natural patterns, and the Fibonacci sequence.

Statistics

The science of conducting studies to collect, organize, summarize, analyze, and draw conclusions from data.

Descriptive Statistics

A branch of statistics consisting of methods for the collection, organization, summarization, and presentation of data or information, such as charts, tables, averages, measures of variation, and percentiles.

Inferential Statistics

A branch of statistics consisting of methods for drawing and measuring the reliability of conclusions about a population based on information obtained from a sample of that population.

Sir Ronald Aylmer Fisher

A British statistician and geneticist ($1890$-$1962$) considered the Father of Modern Statistics, who introduced the principle of randomization to experimental design and developed the concept of analysis of variance (ANOVA).

Population

The collection of all individuals or items under consideration in a statistical study.

Sample

That part of the population from which information is obtained; it serves as a representative subset of the whole population.

Variable

A characteristic, number, or quantity that can assume different values and be measured or counted; also referred to as a data item.

Continuous Variable

A numeric variable whose observations can take any value between a certain set of real numbers, including values as small as the measurement instrument allows (e.g., height, time, age, and temperature).

Discrete Variable

A numeric variable whose observations take values based on counts from a set of distinct whole numbers and cannot take fractional values between adjacent numbers (e.g., number of registered cars, business locations, or children).

Categorical Variable

A qualitative variable with non-numeric values that describe a quality or characteristic of a data unit, falling into mutually exclusive and exhaustive categories.

Ordinal Variable

A categorical variable whose values can be logically ordered or ranked higher or lower than another, without necessarily establishing a measurable numeric difference between each category (e.g., academic grades, clothing sizes, attitudes).

Nominal Variable

A categorical variable whose observations take values that cannot be organized into a logical sequence (e.g., sex, business type, eye color, religion, and brand).

Nominal Level of Measurement

A measurement scale that classifies data into mutually exclusive categories where no order or ranking can be imposed (e.g., gender, zip codes, political party, religion, nationality).

Ordinal Level of Measurement

A measurement scale that classifies data into distinct categories in which order or ranking is implied, but precise numerical differences between ranks do not exist (e.g., evaluations, class ranks, letter grades).

Interval Level of Measurement

A measurement scale that ranks numerical data with precise, known distances between units of measurement, but lacks a true, non-arbitrary zero point (e.g., IQ level, temperature in $^\circ\text{F}$).

Ratio Level of Measurement

A measurement scale that possesses all characteristics of interval measurement with the addition of a true, non-arbitrary zero point, enabling meaningful ratio comparisons between observations (e.g., height, weight, area, number of phone calls).

Slovin's Formula

A formula introduced in $1960$ used to determine sample size $n$ when population behavior is unknown, estimating a population proportion at a $95\%$ confidence level: $n = \frac{N}{1 + Ne^2}$, where $N$ is population size and $e$ is the margin of error.

Simple Random Sampling

A probability sampling technique where every member of the population has an equal chance of being selected, performed either with replacement or without replacement using methods such as a lottery, fishbowl, or random number table.

Systematic Random Sampling

A probability sampling technique obtained by arranging the population in order, randomly selecting a starting point, and then choosing every $k\text{th}$ member, where $k = \frac{\text{Population size}}{\text{Sample size}}$.

Stratified Random Sampling

A probability sampling technique obtained by dividing the population into subgroups or strata according to relevant characteristics, and then randomly selecting subjects from each stratum using proportional allocation.

Cluster Random Sampling

A probability sampling technique obtained by dividing the population into geographic or district sections (clusters), selecting one or more clusters at random, and surveying all members within the selected clusters.

Multi-Stage Sampling

A large-scale sampling technique frequently used by pollsters and government agencies that combines two or more probability sampling methods (e.g., simple random, systematic, stratified, and cluster sampling).

Convenience Sampling

A nonprobability sampling technique (also called accidental sampling) that uses the easiest or most convenient method to select available individuals for a sample.

Quota Sampling

A nonprobability sampling technique that ensures equal or proportionate representation of subjects based on specific traits such as age, gender, education, race, religion, or year level.

Purposive Sampling

A nonprobability sampling technique (also known as judgmental, selective, or deliberate sampling) where the researcher deliberately selects individuals who meet specific criteria or experience requirements based on prior knowledge.

Snowball Sampling

A nonprobability sampling technique (also known as networking or chain referral sampling) used when subjects are hard to locate, where initial subjects assist by referring or nominating other individuals possessing the same trait.

Expert Sampling

A nonprobability sampling technique where respondents are deliberately chosen based on their established specialized expertise in a field.

Measures of Variability

Descriptive statistics (also called dispersion, spread, or scattering) that indicate how spread out or clustered data values are; larger values denote greater dispersion, while smaller values denote data closer together.

Range (Ungrouped Data)

The simplest measure of variability, calculated as the difference between the highest value and the lowest value: $R = HV - LV$.

Range (Grouped Data)

A measure of variability for frequency distributions calculated as the upper class boundary of the highest class interval minus the lower class boundary of the lowest class interval.

Variance

A descriptive measure of variability representing the average squared deviation of values from the mean, computed as $\sigma^2 = \frac{\sum(x - \mu)^2}{N}$ for a population and $s^2 = \frac{\sum(x - \bar{x})^2}{n - 1}$ for a sample.

Standard Deviation

A measure of dispersion defined as the square root of the variance, expressed in the original units of measurement as $\sigma = \sqrt{\frac{\sum(x - \mu)^2}{N}}$ for a population or $s = \sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}$ for a sample.

Standard Score ($z$-score)

A measure of position indicating how many standard deviations a data value falls above (positive) or below (negative) the mean, given by $z = \frac{x - \bar{x}}{s}$ for samples or $z = \frac{x - \mu}{\sigma}$ for populations.

Quartiles

Measures of position that divide an ordered set of data into four equal parts ($Q_1$, $Q_2$, $Q_3$, $Q_4$), each representing a quarter or $25\%$ of the data, located using the position formula $Q_k = \frac{k}{4}(n + 1)$.

Deciles

Measures of position that divide an ordered data set into ten equal parts ($D_1$ through $D_{10}$), each representing $\frac{1}{10}$ or $10\%$ of the data, located using the position formula $D_k = \frac{k}{10}(n + 1)$.

Percentiles

Measures of position that divide an ordered data set into one hundred equal parts ($P_1$ through $P_{100}$), each representing $\frac{1}{100}$ or $1\%$ of the data, located using the position formula $P_k = \frac{k}{100}(n + 1)$.

Five-Number Summary

A descriptive statistical summary consisting of five values: the least value, first quartile ($Q_1$), second quartile/median ($Q_2$), third quartile ($Q_3$), and greatest value.

Box-and-Whisker Plot

A graphical representation of data based on the five-number summary, featuring a central box extending from $Q_1$ to $Q_3$ with a line at $Q_2$, and horizontal whiskers connecting the box to the least and greatest values.

Normal Distribution

A continuous, perfectly symmetric, bell-shaped distribution where the mean, median, and mode coincide at the center, dividing the area into equal lower and upper $50\%$ halves with asymptotic horizontal tails.

Empirical Rule

A statistical rule for normal distributions stating that approximately $68\%$ of total area lies within $\pm 1.0\sigma$, $95\%$ lies within $\pm 2.0\sigma$, and $99.7\%$ lies within $\pm 3.0\sigma$ of the mean.

Least-Squares Regression Line

The line of best fit $\hat{y} = ax + b$ for bivariate data that minimizes the sum of squared vertical deviations from each point to the line, where slope is $a = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}$ and intercept is $b = \bar{y} - a\bar{x}$.

Linear Correlation Coefficient ($r$)

A statistical metric ranging between $-1$ and $+1$ that quantifies the strength and direction of a linear relationship between two bivariate variables, where $0$ indicates no linear correlation.

Pattern

A regular, repeated, or recurring form, design, or arrangement, which can consist of geometric shapes, figures, letters, colors, or numerical sequences.

Reflective Symmetry

A pattern (also known as bilateral or line symmetry) where one half of an image or organism is the exact mirror image of the other half across an imaginary axis.

Rotational Symmetry

A type of symmetry where an object or image can be turned around a fixed center point and match its original shape a specific number of times.

Angle of Rotation

The smallest angle for which a figure can be rotated around its center to coincide with itself, calculated by the formula $\text{Angle of Rotation} = \frac{360^\circ}{n}$, where $n$ is the order of symmetry.

Fractals

Self-similar mathematical and natural structures identified by Benoit Mandelbrot in $1975$, where every small part is a miniature replica of the whole at different scales of magnification (e.g., ferns, lungs, trees, lightning).

Spirals

Curved biological and natural patterns that wind outward from a center point, observed in millipede defensive coils, nautilus shells, sunflower seed distributions, and succulent plants.

Meanders

Sinuous curves and bends formed in flowing bodies of water or channels, exemplified by the Agusan River in northeastern Mindanao meandering through a valley floor.

Waves

Disturbances that carry energy through a medium such as air, water, or sand, causing the medium to oscillate (e.g., ocean waves, wind-blown grass, sand dune ripples).

Tessellations

Tiling patterns formed by repeating geometric shapes across a flat surface without any gaps or overlaps, seen in natural hexagonal honeycombs and pineapple scales.

Edn3

A specific gene identified by scientists in $2012$ that controls hair color in feline coat patterns, discovered alongside the gene responsible for tabby cat stripes and cheetah spots.

Exponential Population Growth Model

A continuous mathematical model for population increase expressed as $A = P \cdot e^{rt}$, where $A$ is the population size after time $t$, $P$ is the initial population, $r$ is growth rate, and $e \approx 2.718$ is Euler's constant.

Fibonacci Sequence

An integer sequence ($0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$) governed by the recursive rule $x_n = x_{n-1} + x_{n-2}$, where each term is the sum of the two preceding terms.

Leonardo Pisano Bigollo

An Italian mathematician ($1180$-$1250$) from Pisa, commonly known as Fibonacci, who introduced the Hindu-Arabic numeral system and the Fibonacci sequence to Western Europe in his $1202$ book Liber Abaci.

Golden Ratio ($\phi$)

An irrational proportion approximately equal to $1.618034$, approached by the ratio of successive Fibonacci numbers ($\frac{x_n}{x_{n-1}}$), visualized via golden spiral geometry in rectangles and evident in art and architecture like the Parthenon.

Binet's Formula

A closed-form formula enabling direct calculation of any $n\text{th}$ Fibonacci term using the Golden Ratio: $x_n = \frac{\phi^n - (1 - \phi)^n}{\sqrt{5}}$.