Statistics Lecture 1: Research Methods, Frequency Distributions, and Central Tendency

Vocabulary flashcards covering foundational concepts in statistics including research design, variables, measurement scales, frequency distributions, stem-and-leaf displays, and measures of central tendency.

Population

The set of all the individuals of interest in a particular study.

Sample

A set of individuals selected from a population, usually intended to represent the population in a research study.

Variable

A characteristic or condition that changes or has different values for different individuals.

Datum

A single measurement or observation, also called a score or a raw score.

Parameter

A numerical value that describes a population, usually derived from measurements of the individuals in the population.

Statistic

A numerical value that describes a sample, usually derived from measurements of the individuals in the sample.

Descriptive Statistics

Statistical procedures used to summarize, organize, and simplify data.

Inferential Statistics

Techniques that allow researchers to study samples and then make generalizations about the population from which they were selected.

Sampling Error

The naturally occurring discrepancy, or error, that exists between a sample statistic and the corresponding population parameter.

Correlational Method

A research method that observes two different variables to determine whether there is a relationship between them, without providing a cause-and-effect explanation.

Chi-Square Test

A statistical test used in correlational studies when the measurement process simply classifies individuals into categories that do not correspond to numerical values.

Experimental Method

A research method that examines the relationship between variables by manipulating one variable to define groups and measuring a second variable, allowing for a cause-and-effect explanation.

Participant Variables

Characteristics such as age, gender, or intelligence that vary from individual to individual and must be controlled in experiments.

Environmental Variables

Characteristics of the environment such as lighting, time of day, or weather that researchers must control in experiments.

Independent Variable

The variable manipulated by the researcher in an experiment, usually consisting of two or more treatment conditions to which subjects are exposed.

Dependent Variable

The variable observed to assess the effect of the treatment in an experiment.

Control Group

Individuals in an experiment who do not receive an experimental treatment (often receiving a placebo) to provide a baseline for comparison.

Nonequivalent Groups Study

A nonexperimental research design comparing preexisting groups where the researcher cannot control participant assignment or ensure equivalent groups.

Prepost Study

A research design where two groups of scores are obtained by measuring the same variable twice for each participant: once before treatment and again after.

Quasi-Independent Variable

In a nonexperimental study, the independent variable used to create the different groups of scores.

Constructs

Internal attributions or characteristics that cannot be directly observed but are useful for describing and explaining behavior.

Discrete Variable

A variable consisting of separate, indivisible categories with no values existing between two neighboring categories.

Continuous Variable

A variable containing an infinite number of possible values falling between two observed values, divisible into an infinite number of fractional parts.

Real Limits

The boundaries of intervals for scores represented on a continuous number line, consisting of an upper real limit at the top and a lower real limit at the bottom.

Nominal Scale

A measurement scale consisting of categories with different names, used to label and categorize observations without quantitative distinctions.

Ordinal Scale

A measurement scale consisting of categories organized in an ordered sequence, ranking observations in terms of size or magnitude.

Interval Scale

A measurement scale with ordered categories of equal size where equal numerical differences reflect equal differences in magnitude, but with an arbitrary zero point.

Ratio Scale

An interval scale with an absolute zero point, where ratios of numbers reflect ratios of magnitude.

Sigma ($\Sigma$)

The Greek letter standing for the mathematical operation 'sum of' in statistical notation.

Frequency Distribution

An organized tabulation showing exactly how many individuals are located in each category on the scale of measurement.

Relative Frequency ($p$)

The proportion of the distribution corresponding to each category, calculated as $p = \frac{f}{N}$, where the sum of all proportions equals $1.00$.

Grouped Frequency Distribution Table

A frequency distribution table used when scores cover a wide range, organizing scores into class intervals (typically about 10 intervals of widths such as 2, 5, 10, or 20).

Histogram

A frequency distribution graph for interval or ratio scales where adjacent bars touch because the numbers are continuous.

Polygon

A frequency distribution graph for interval or ratio scales where dots represent category frequencies and are connected by line segments.

Bar Graph

A frequency distribution graph for nominal or ordinal scales where spaces or gaps are left between adjacent bars to emphasize distinct categories.

Central Tendency

A statistical measure that identifies a single score as representative of the center of a distribution.

Variability

A statistical measure of the degree to which scores are spread over a wide range or clustered together.

Symmetrical Distribution

A distribution shape in which the left side of the graph is roughly a mirror image of the right side.

Positively Skewed Distribution

A distribution in which scores pile up on the left side and the tail tapers off to the right.

Negatively Skewed Distribution

A distribution in which scores pile up on the right side and the tail points to the left.

Percentile Rank

The percentage of individuals with scores equal to or less than a particular $X$ value.

Interpolation

A mathematical process used to estimate intermediate values assuming that scores and percentages change in a regular, linear fashion through an interval.

Stem-and-Leaf Display

An efficient display method where each score is divided into a stem (first digit/s) and a leaf (final digit).

Population Mean ($\mu$)

The average of an entire population, calculated as $\mu = \frac{\Sigma X}{N}$.

Sample Mean ($M$)

The average of a sample, calculated as $M = \frac{\Sigma X}{n}$.

Weighted Mean (Combined Sample Mean)

The overall mean of combined groups calculated by dividing the total sum of scores ($\Sigma X$) by the total number of scores ($n$), where larger samples carry more weight.

Sum of Scores from Frequency Table ($\Sigma fX$)

The total sum of scores in a frequency table obtained by multiplying each score $X$ by its frequency $f$ and summing all products.

Linear Transformation of the Mean

Multiplying or dividing every score in a distribution by a constant value changes the mean by the exact same constant factor.

Median

The measure of central tendency that identifies the midpoint of a distribution when scores are listed in order from smallest to largest.

Mode

The score or category that has the greatest frequency in a distribution.

Bimodal Distribution

A distribution that contains two distinct modes.

Multimodal Distribution

A distribution that contains more than two distinct modes.

Open-Ended Distribution

A distribution with no upper or lower limit for one of its categories, making it impossible to compute $\Sigma X$ or the mean.

Central Tendency in Skewed Distributions

In a positively skewed distribution, the order of central tendency measures from smallest to largest is Mode, Median, Mean; in a negatively skewed distribution, the order is Mean, Median, Mode.