Introductory Statistics: Variables, Distributions, Measures of Center and Variability, and Linear Regression

Vocabulary practice flashcards covering fundamental statistical concepts, variable scales, frequency tables, distributions, measures of central tendency and variability, normal curves, z-scores, correlation, and least-squares regression.

Nominal qualitative variable

A qualitative variable for which observations represent categories that cannot be meaningfully ordered (such as political affiliation, fruit types, religion, or college major).

Ordinal qualitative variable

A qualitative variable where categories have a logical and broadly agreed-upon ordering (such as sauce hotness rated mild, medium, or hot).

Ranked variable

A dataset where each single observation is an integer indicating relative standing or value; observations within the same category receive different ranks, and values cannot be considered as existing on a real number line.

Quantitative variable

A set of observations where each observation is a number that represents a count (integers) or an amount (real numbers), which can be treated on a real number line.

Independent variable ($\text{IV}$)

A variable in an experimental study that is manipulated or controlled by the investigator, often assigning subjects to levels using a random assignment rule.

Dependent variable ($\text{DV}$)

A variable in an experiment that is observed rather than manipulated, whose values are assumed to be affected by the independent variable.

Observational study

A study that focuses solely on determining the relationship between two observed variables without manipulation, making the distinction between independent and dependent variables inapplicable.

Noise factors (random variants)

Random factors that affect the outcome of a score, study, or research result.

Inferential statistics

Statistical methods used to determine whether differences or relationships observed could be due to chance variation and noise factors rather than a true effect.

Ungrouped frequency distribution

A descriptive frequency table useful for small integer datasets where each individual possible value of the variable constitutes its own class.

Grouped Frequency Distribution

A statistical table that organizes raw quantitative data into grouped intervals (classes) and displays the exact count of data points (absolute frequency) in each interval.

Relative Frequency Distribution

A statistical table that scales raw counts relative to the total number of observations, displaying the fraction, decimal, or percentage of the dataset in each class interval.

Cumulative frequency distribution

A data summary showing the frequency (or count) of items less than or equal to the upper class limit of each class, applicable to quantitative and ordinal categorical data.

Histogram

A graphical representation reserved strictly for quantitative data where adjacent bars touch and bar heights correspond to class frequencies or relative frequencies.

Bimodal distribution

A frequency distribution featuring two distinct peaks or modes.

Mean

A measure of central tendency calculated as the sum of all observations divided by the total number of observations ($\bar{X} = \frac{\sum X}{n}$ for samples, or $\mu = \frac{\sum X}{N}$ for populations).

Median

The middle score in an ordered dataset (or the average of the two middle scores if the count is even) that splits the dataset so that exactly half of the values lie below and half lie above it.

Mode

The most frequently occurring value or values in a dataset, representing the peak of a distribution.

Range

A measure of variability computed as the difference between the largest and smallest observation in a dataset ($\text{Range} = X_{\text{max}} - X_{\text{min}}$).

Interquartile range ($\text{IQR}$)

The distance covering the middle $50\%$ of data, calculated as the third quartile ($75^{\text{th}}$ percentile) minus the first quartile ($25^{\text{th}}$ percentile), which remains resistant to extreme outliers.

Mean absolute deviation ($\text{MAD}$)

The average absolute numerical distance between each observation and the mean of the dataset, defined for a population as $\text{MAD} = \frac{\sum |X - \mu|}{N}$.

Sum of squares ($SS$)

The sum of squared deviations of scores from the mean, defined as $SS = \sum (X - \mu)^2$ or calculated computationally as $\sum X^2 - \frac{(\sum X)^2}{N}$.

Population variance ($\sigma^2$)

A measure of variability for a population defined as the sum of squared deviations from the population mean divided by the population size ($\sigma^2 = \frac{SS}{N}$).

Sample variance ($s^2$)

A measure of variability for sample data calculated as the sample sum of squares divided by the sample size minus one ($s^2 = \frac{ss}{n - 1}$).

Population standard deviation ($\sigma$)

The square root of the population variance ($\sigma = \sqrt{\sigma^2}$), measuring data spread and distance on the number line relative to the mean.

$Z$-score

A standardized value indicating the exact number of standard deviations a raw observation ($X$) lies above or below the mean, calculated as $Z = \frac{X - \mu}{\sigma}$.

Standard normal distribution

A specific normal distribution obtained after transforming raw normal scores into $Z$-scores, characterized by a mean of $\mu = 0$ and a standard deviation of $\sigma = 1$.

Probability density ($f(x)$)

The height of a theoretical continuous probability curve at a specific value of $x$, describing relative likelihood rather than absolute probability, where the entire area under the curve equals $1.0$.

Correlation ($r$ or $\rho$)

A standardized coefficient ranging between $-1$ and $1$ that measures the strength and direction of the linear relationship between two variables.

Least Squares Regression

A linear prediction method that fits a straight line ($Y' = bX + a$) to paired quantitative data by minimizing the sum of squared differences between observed values and predicted values ($\sum (Y - Y')^2$).

Regression residuals

The vertical differences between the actual observed values and predicted values of a dependent variable ($Y - Y'$).

Standard error of the estimate ($s_{y|x}$)

The typical amount by which observed values of $Y$ deviate from predicted values ($Y'$), representing the variability of $Y$ not explained by the linear regression on $X$.