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Statistics

The science of collecting, organizing, summarizing, and analyzing information to draw conclusions or answer questions, while providing a measure of confidence in those conclusions.

Parameter

A numerical summary of an entire population.

Statistic

A numerical summary calculated from a subset or sample of a population.

Descriptive statistics

Methods that organize and summarize data through numerical measures, tables, and graphs.

Inferential statistics

Methods that take results from a sample, extend them to the population, and report a measure of reliability for the conclusion.

Qualitative (categorical) variable

A variable that classifies individuals based on an attribute or characteristic into non-numeric categories.

Quantitative variable

A variable that provides numerical measures of individuals, where arithmetic operations such as addition and subtraction are meaningful.

Discrete variable

A quantitative variable with either a finite number of possible values or a countable number of values ($0, 1, 2, 3, \dots$).

Continuous variable

A quantitative variable with an infinite, uncountable number of possible values that can take on any real value between two given points.

Nominal level of measurement

A measurement level where values consist of names, labels, or categories with no natural or meaningful ordering.

Ordinal level of measurement

A measurement level possessing nominal properties, but where values can be arranged in a meaningful order or rank.

Interval level of measurement

A measurement level where differences between values are meaningful, but a value of zero does not represent the absence of the quantity.

Ratio level of measurement

A measurement level where differences and ratios between values are meaningful, and a value of zero represents the complete absence of the quantity.

Explanatory variable

The independent variable in a study that is manipulated or observed to determine its effect on an outcome.

Response variable

The dependent variable in a study that represents the outcome being measured.

Observational study

A study that measures the response variable without attempting to influence or manipulate the explanatory variables or subjects.

Designed experiment

A controlled study where a researcher intentionally assigns subjects to treatment groups and manipulates explanatory variables to record the response.

Lurking variable

An explanatory variable not considered or known by the researcher that influences the response variable.

Confounding variable

An explanatory variable considered in a study whose effect cannot be distinguished from the effect of another explanatory variable.

Cross-sectional study

An observational study that collects data from individuals at a single specific point in time or over a very short duration.

Retrospective (case-control) study

An observational study that looks back in time using existing records to compare individuals with a specific characteristic to those without.

Prospective (cohort) study

An observational study that identifies a group of subjects and follows them forward over time to collect data on outcomes.

Simple random sampling

A sampling technique where every possible sample of size $n$ from a population of size $N$ has an equal chance of being selected.

Frame

A complete list of all individuals within a population used to select a sample.

Systematic sample

A sample obtained by selecting every $k^{\text{th}}$ individual from a population after randomly choosing a starting point between 1 and $k$.

Stratified sample

A sample produced by dividing the population into non-overlapping, homogeneous subgroups (strata) and taking a simple random sample from each subgroup.

Cluster sample

A sample produced by dividing the population into sections (clusters), selecting a random sample of clusters, and surveying all individuals within the chosen clusters.

Convenience sample

A non-random sampling method where subjects are selected based on easy availability or accessibility, often leading to suspect results.

Sampling bias

Bias that occurs when the selection technique favors one segment of the population over another, resulting in undercoverage.

Nonresponse bias

Bias that arises when selected individuals who decline to participate hold systematically different opinions or traits than those who do participate.

Response bias

Bias that occurs when survey responses do not reflect the true feelings of respondents due to interviewer error, misleading answers, or question wording.

Nonsampling error

An error arising from undercoverage, nonresponse, response bias, or data-entry mistakes, which can occur even during a census.

Sampling error

An error that occurs because a sample gives incomplete information about a population due to natural variability.

Placebo

An innocuous or inactive treatment designed to look, feel, or taste like the experimental treatment, given to a control group.

Double-blind experiment

An experimental setup in which neither the subject nor the researcher interacting with the subject knows which treatment is being administered.

Statistics

The science of collecting, organizing, summarizing, and analyzing information to draw conclusions or answer questions, while providing a measure of confidence in those conclusions.

Parameter

A numerical summary of an entire population.

Statistic

A numerical summary calculated from a subset or sample of a population.

Descriptive statistics

Methods that organize and summarize data through numerical measures, tables, and graphs.

Inferential statistics

Methods that take results from a sample, extend them to the population, and report a measure of reliability for the conclusion.

Qualitative (categorical) variable

A variable that classifies individuals based on an attribute or characteristic into non-numeric categories.

Quantitative variable

A variable that provides numerical measures of individuals, where arithmetic operations such as addition and subtraction are meaningful.

Discrete variable

A quantitative variable with either a finite number of possible values or a countable number of values ($0, 1, 2, 3, \dots$).

Continuous variable

A quantitative variable with an infinite, uncountable number of possible values that can take on any real value between two given points.

Nominal level of measurement

A measurement level where values consist of names, labels, or categories with no natural or meaningful ordering.

Ordinal level of measurement

A measurement level possessing nominal properties, but where values can be arranged in a meaningful order or rank.

Interval level of measurement

A measurement level where differences between values are meaningful, but a value of zero ($0$) does not represent the absence of the quantity.

Ratio level of measurement

A measurement level where differences and ratios between values are meaningful, and a value of zero ($0$) represents the complete absence of the quantity.

Explanatory variable

The independent variable in a study that is manipulated or observed to determine its effect on an outcome.

Response variable

The dependent variable in a study that represents the outcome being measured.

Observational study

A study that measures the response variable without attempting to influence or manipulate the explanatory variables or subjects.

Designed experiment

A controlled study where a researcher intentionally assigns subjects to treatment groups and manipulates explanatory variables to record the response.

Lurking variable

An explanatory variable not considered or known by the researcher that influences the response variable.

Confounding variable

An explanatory variable considered in a study whose effect cannot be distinguished from the effect of another explanatory variable.

Cross-sectional study

An observational study that collects data from individuals at a single specific point in time or over a very short duration.

Retrospective (case-control) study

An observational study that looks back in time using existing records to compare individuals with a specific characteristic to those without.

Prospective (cohort) study

An observational study that identifies a group of subjects and follows them forward over time to collect data on outcomes.

Simple random sampling

A sampling technique where every possible sample of size $n$ from a population of size $N$ has an equal chance of being selected.

Frame

A complete list of all individuals within a population used to select a sample.

Systematic sample

A sample obtained by selecting every $k^{\text{th}}$ individual from a population after randomly choosing a starting point between $1$ and $k$.

Stratified sample

A sample produced by dividing the population into non-overlapping, homogeneous subgroups (strata) and taking a simple random sample from each subgroup.

Cluster sample

A sample produced by dividing the population into sections (clusters), selecting a random sample of clusters, and surveying all individuals within the chosen clusters.

Convenience sample

A non-random sampling method where subjects are selected based on easy availability or accessibility, often leading to suspect results.

Sampling bias

Bias that occurs when the selection technique favors one segment of the population over another, resulting in undercoverage.

Nonresponse bias

Bias that arises when selected individuals who decline to participate hold systematically different opinions or traits than those who do participate.

Response bias

Bias that occurs when survey responses do not reflect the true feelings of respondents due to interviewer error, misleading answers, or question wording.

Nonsampling error

An error arising from undercoverage, nonresponse, response bias, or data-entry mistakes, which can occur even during a census.

Sampling error

An error that occurs because a sample gives incomplete information about a population due to natural variability.

Placebo

An innocuous or inactive treatment designed to look, feel, or taste like the experimental treatment, given to a control group.

Double-blind experiment

An experimental setup in which neither the subject nor the researcher interacting with the subject knows which treatment is being administered.

Raw data

Data that has been collected but has not yet been organized or summarized.

Frequency distribution

A tabular summary listing each data category alongside the count of occurrences for each category.

Relative frequency distribution

A tabular summary listing each category alongside its proportion or percentage of total observations, computed as $\text{Relative Frequency} = \frac{\text{category frequency}}{\text{sum of all frequencies}}$.

Bar graph

A display for qualitative data using non-touching, equal-width rectangles whose heights correspond to category counts or proportions.

Pareto chart

A bar graph with visual bars arranged in decreasing order of frequency or relative frequency.

Side-by-side bar graph

A chart displaying two relative frequency data sets next to each other to enable direct comparison across categories.

Pie chart

A circular statistical graphic divided into sectors proportional to category relative frequencies, where sector angles equal $\text{relative frequency} \times 360^\circ$.

Histogram

A display for quantitative data using contiguous, touching rectangles whose heights represent frequency or relative frequency and widths represent class intervals.

Lower class limit

The smallest numerical value that can belong to a specific class interval.

Upper class limit

The largest numerical value that can belong to a specific class interval.

Class width

The difference between consecutive lower class limits, estimated as $\text{Class width} \approx \frac{\text{largest value} - \text{smallest value}}{\text{number of classes}}$ rounded up to a convenient integer.

Dot plot

A quantitative graphic where observations are arranged horizontally in increasing order, placing a dot above a value each time it appears.

Uniform distribution

A distribution shape where frequencies are distributed evenly across all values or classes.

Bell-shaped distribution

A symmetric distribution shape with the highest frequency in the center, tapering off equally toward both tails.

Skewed right

A distribution shape where the tail extends significantly further to the right side of the peak.

Skewed left

A distribution shape where the tail extends significantly further to the left side of the peak.

Stem-and-leaf plot

A quantitative graph that splits each observation into leading digits (the stem) and the final digit (the leaf), retaining the original raw data values.

Class midpoint

The value calculated by adding consecutive lower class limits and dividing the sum by $2$.

Frequency polygon

A quantitative line graph plotting points above class midpoints at heights matching class counts, connected by straight line segments.

Cumulative frequency distribution

A tabular summary displaying the total count of observations less than or equal to the upper limit of a given class.

Ogive

A line graph plotting cumulative frequency or cumulative relative frequency against upper class limits.

Time-series plot

A chart with time on the horizontal axis and variable observations on the vertical axis, connecting sequential data points with line segments.