Introductory Statistics: Data Collection, Summarization, Regression, and Probability

Comprehensive vocabulary flashcards covering introductory statistics concepts including study design, sampling bias, graphical and numerical data summaries, linear regression, and basic probability rules.

Individuals

The entities that are measured in a study.

Population

The entire group that is studied.

Census

The process of collecting data for every individual in a population.

Sample

A subset of a population for which data is actually collected.

Parameter

A numerical summary of a population (for example, the actual average time that all US college students spend on social media each week).

Statistic

A numerical summary of a sample (for example, the computed average from survey data of $2000$ students).

Descriptive Statistics

The practice of summarizing collected data through tables, graphs, and numerical summaries such as averages or percentages to provide an overview of the data.

Inferential Statistics

A method that takes results from a sample, extends them to the population, and measures the reliability of the result.

Variable

Any characteristic of the individuals within the population.

Quantitative Variable

A variable whose observations are numerical.

Qualitative Variable

A variable whose observation belongs to a set of categories.

Nominal Variable

A qualitative variable used to represent names or categories that have no inherent order.

Ordinal Variable

A qualitative variable with categories that indicate an order or ranking, but where intervals between values are not clearly defined.

Binary Variable

A qualitative variable with only two possible levels, such as yes or no.

Selection Bias

Bias resulting from the non-random selection of the study population.

Self-Selection

A condition where participants choose whether or not to be included in a study.

Attrition Bias

The systematic drop-out of participants from a study over time.

Experiment

A study in which researchers assign individuals to certain experimental conditions and observe outcomes on the response variable of interest.

Observational Study

A study in which researchers observe and record behaviors without imposing any conditions or treatments.

Simple Random Sampling

A sampling method in which every member of the population has an equal and independent chance of being selected.

Stratified Sampling

A sampling method in which the population is divided into distinct subgroups and a random sample is taken from every single subgroup in proportion to its size.

Cluster Sampling

A sampling method in which the population is divided into natural groups or clusters, entire clusters are randomly selected, and all (or a random sample of) members within those chosen clusters are studied.

Systematic Sampling

A sampling method in which population members are placed in order and individuals are selected at regular fixed intervals after starting at a random point.

Convenience Sample

A non-random sample composed of individuals who are easily obtained.

Voluntary Response Sample

A sample where individuals are self-selected into participating.

Sampling Bias

A bias occurring when the sampling technique favors one part of the population over another.

Nonresponse Bias

A bias that occurs when individuals selected to be in the sample do not respond.

Response Bias

A bias that occurs when collected answers do not reflect the true opinions of the respondent, often due to lying or confusing wording.

Control Group

A baseline group in an experiment used to compare against other treatments.

Placebo

A treatment with no active ingredient given to help subjects respond better than those receiving nothing, due to psychological reasons.

Single-Blind

An experimental setup in which subjects do not know which treatment they are receiving.

Double-Blind

An experimental setup in which neither the subjects nor the researchers know which treatment the subjects are receiving.

Replication

Applying a treatment to more than one subject or repeating an experiment to confirm characteristics are not one-off events.

Frequency Distribution

A table that lists each category of data along with the number of observations in each category.

Relative Frequency Distribution

A table that lists each category of data together with the proportion of observations in each category.

Bar Graph

A graph that features a bar for each category, where the height of each bar corresponds to its frequency or relative frequency.

Pareto Chart

A bar graph with categories ordered by their frequency from tallest to shortest.

Pie Chart

A circular statistical graphic where the area of each sector is proportional to the relative frequency of that category, used to compare parts to a whole.

Histogram

A graphical display showing the distribution of continuous numerical data across intervals, indicating the frequency or relative frequency of each class.

Class Width

An approximate value used for interval sizing in histograms, calculated as: $\text{Class width} \approx \frac{\text{largest data value} - \text{smallest data value}}{\text{number of classes}}$.

Dot Plot

A simple display where dots are placed along a number line to show where each data value lies.

Stem and Leaf Plot

A display that organizes ordered data by separating each number into a stem (leading digit or digits) on the left and a leaf (final digit) on the right.

Symmetric Distribution

A distribution shape in which the right and left sides of the curve are similar reflections of each other.

Skewed Left

A distribution shape where the left tail extends further than the right tail, and the mean is typically less than the median ($\text{mean} < \text{median}$).

Skewed Right

A distribution shape where the right tail extends further than the left tail, and the mean is typically greater than the median ($\text{mean} > \text{median}$).

Unimodal

A distribution that exhibits a single peak or mode.

Bimodal

A distribution that exhibits two distinct peaks or modes.

Uniform Distribution

A distribution where the frequency across each class is relatively the same.

Mean

The numerical average of a set of observations, represented by $\bar{x}$ for a sample mean and $\mu$ for a population mean.

Median

The middle observation when all data values are arranged in ascending numerical order; if $n$ is even, it is the average of the two middle observations.

Mode

The specific observation that occurs most frequently in a dataset.

Resistant

A term describing a numerical summary (such as the median) that is not easily influenced by extreme observations or outliers.

Range

A basic measure of spread calculated as the maximum data value minus the minimum data value ($\text{Max} - \text{Min}$).

Deviation

The directed distance of an individual value from the mean, calculated as: $\text{Deviation} = x_i - \bar{x}$.

Standard Deviation

A measure of how spread out numbers in a dataset are from their mean, denoted as $s$ for a sample and $\sigma$ for a population.

Variance

The square of the standard deviation, representing an average of the squared deviations from the mean; denoted as $s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$ for a sample and $\sigma^2$ for a population.

Outlier

An observation that is unusually large or small relative to the other values in a dataset.

Percentile

For any set of ordered observations, a specific value positioned such that $p\%$ of observations fall below it and $(100 - p)\%$ fall above it.

Interquartile Range (IQR)

The distance between the first and third quartiles, defined by $\text{IQR} = Q_3 - Q_1$, which represents the range of the middle $50\%$ of observations.

1.5x IQR Rule

A method to spot outliers using cutoff fences: $\text{Lower fence} = Q_1 - 1.5 \times \text{IQR}$ and $\text{Upper fence} = Q_3 + 1.5 \times \text{IQR}$, where any value falling outside these fences is an outlier.

Univariate Analysis

The statistical analysis of a single variable to understand its distribution or characteristics.

Bivariate Analysis

The statistical analysis of the relationship between two variables.

Response Variable

The variable (denoted $y$) that measures the outcome of a study.

Explanatory Variable

The variable (denoted $x$) that explains or influences changes in the response variable.

Scatterplot

A graphical display showing the relationship between two quantitative variables measured on the same individual, plotting the explanatory variable on the x-axis and the response variable on the y-axis.

Linear Correlation Coefficient

A unitless measure denoted by $r$ (where $-1 \le r \le 1$) that describes the direction, form, and strength of the linear relationship between two quantitative variables.

Lurking Variable

An unobserved or unmeasured third variable that influences both the explanatory variable and the response variable, creating a false association or masking a true one.

Least-Squares Regression Line

The mathematical line $\hat{y} = b_0 + b_1 x$ that minimizes the sum of squared residuals to predict $y$ based on $x$, where $b_0$ is the y-intercept and $b_1$ is the slope.

Residual

The prediction error for an observation, calculated as: $\text{Residual} = \text{Observed } y - \text{Predicted } y$.

Coefficient of Determination ($R^2$)

The square of the correlation coefficient, which indicates the percentage of variability in the response variable that is explained by the linear model.

Probability Experiment

An act or process of observation with uncertain results that can be repeated, where the outcome of any single trial cannot be known ahead of time.

Law of Large Numbers

A principle stating that as an experiment is repeated more times, the relative frequency or average of the results gets closer to the true expected value.

Sample Space

The set (denoted $S$) of all possible outcomes of a probability experiment.

Event

A collection of one or more outcomes from a probability experiment.

Unusual Event

An event whose probability of occurrence is less than $0.05$ ($5\%$).

Complement of an Event

All outcomes in the sample space that are not in event $A$, denoted $A^C$, with probability given by: $P(A^C) = 1 - P(A)$.

Mutually Exclusive Events

Events that have no outcomes in common (also termed disjoint), meaning $P(A \text{ and } B) = 0$.

Conditional Probability

The probability of event $A$ occurring given that event $B$ has already occurred, defined as: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$.

Independent Events

Two events for which knowing whether one event occurs does not affect the probability of the other occurring, satisfying $P(A|B) = P(A)$ and $P(A \text{ and } B) = P(A) \times P(B)$.