NSE 212 Week 7: Analyzing Quantitative Data — Descriptive and Bivariate Statistics

Vocabulary flashcards covering descriptive and bivariate statistics, including measures of central tendency, measures of variability, graphical displays, distribution shapes, correlation coefficients, and linear regression models.

Descriptive statistics

Statistical procedures that organize, summarize, and describe observed data to identify typical values, spread, distributions, and patterns before inference is attempted, without by themselves establishing causation or generalizability.

Mean ($\bar{x}$)

A measure of central tendency calculated as the sum of values divided by the number of valid observations ($\bar{x} = \frac{\sum x_i}{n}$); it incorporates every score and is sensitive to extreme outliers.

Median

The middle score in an ordered distribution locating the $50\text{th}$ percentile (calculated by averaging the two middle values when $n$ is even); it is resistant to extreme scores.

Mode

The most frequently occurring value or category in a distribution; it can be used with nominal data where arithmetic averaging is meaningless and may have multiple values or ties.

Percentile

A relative positional score indicating where a value falls within an ordered distribution (for example, a $75\text{th}$-percentile score does not mean $75\%$ of questions were answered correctly).

Variability

A statistical concept describing how different observations are from one another, reflecting the degree of dispersion or spread around the center of a distribution.

Range

A measure of spread calculated as the maximum value minus the minimum value; it relies exclusively on extreme values and is highly sensitive to them.

Inclusive range

A variation of the range calculated as $h - l + 1$, intended specifically for discrete counted values rather than continuous measurements.

Sample variance ($s^2$)

A measure of variability computed as $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$, where squaring deviations prevents positive and negative differences from canceling out, yielding squared measurement units.

Sample standard deviation ($s$)

A measure of spread calculated as the square root of the sample variance ($s = \sqrt{s^2}$), which returns the dispersion metric to the original units of measurement.

Degrees of freedom divisor ($n - 1$)

The denominator used when calculating sample variance that corrects for bias, making the sample variance an unbiased estimator of the population variance under standard sampling assumptions.

Interquartile Range (IQR)

A resistant summary of variability that measures the spread of the middle $50\%$ of an ordered distribution and requires quartile cutoff values to calculate.

Frequency distribution

A method of tallying how often each score, category, or non-overlapping class interval occurs, which reconciles to the total valid sample size $n$.

Valid Percent

A percentage value computed in statistical software that excludes missing data cases from the denominator, distinct from "Percent" which includes missing cases.

Cumulative Percent

The sequential sum of ordered valid percentages, which is meaningful for ordered ordinal or quantitative data but not for unordered nominal categories.

Histogram

A chart displaying the distribution of quantitative data across adjacent numerical intervals, where continuous spacing and bin width affect the visual representation.

Frequency polygon

A graph that connects data frequencies plotted at the midpoints of ordered class intervals, useful for comparing distribution shapes across compatible scales.

Scatterplot

A bivariate plot displaying paired quantitative observations with one variable on each axis, used to visually assess association direction, form, strength, and outliers.

Normal distribution

A symmetric, bell-shaped distribution with asymptotic tails approaching the horizontal axis, where the mean, median, and mode coincide in the idealized unimodal case.

Positive skew (Right skew)

An asymmetric distribution characterized by a long tail extending toward higher values, which pulls the mean upward relative to the median.

Negative skew (Left skew)

An asymmetric distribution characterized by a long tail extending toward lower values, which pulls the mean downward relative to the median.

Bivariate statistics

Statistical techniques that examine two variables simultaneously to describe or evaluate their relationship or association.

Pearson correlation coefficient ($r$)

A statistic ranging from $-1$ to $+1$ that summarizes the direction and strength of a linear association between two quantitative variables.

Direct association (Positive association)

A relationship where two variables change in the same direction, meaning higher values on one variable correspond to higher values on the other variable.

Inverse association (Negative association)

A relationship where two variables move in opposite directions, meaning higher values on one variable correspond to lower values on the other variable.

Correlation matrix

A rectangular display of pairwise correlations across multiple variables, with diagonal cells equal to $1$ and mirrored off-diagonal cells representing identical pairs.

Coefficient of determination ($r^2$)

The square of the Pearson correlation coefficient ($r^2$), representing the proportion of variance in one variable accounted for by the linear relationship with another variable.

Coefficient of nondetermination (Alienation, $1 - r^2$)

The proportion of variance in a bivariate relationship that remains unaccounted for or unexplained by the linear model, computed as $1 - r^2$.

Spearman correlation

A non-parametric correlation coefficient that assesses monotonic associations by evaluating ranked or ordered values.

Phi coefficient

A correlation coefficient used specifically to measure the strength and direction of association between two binary categorical variables.

Point-biserial correlation

A correlation coefficient used to examine the relationship between one binary categorical variable and one continuous quantitative variable.

Rank-biserial correlation

A correlation method used to evaluate the association between a binary categorical grouping variable and rank or ordinal data.

Contingency table (Crosstab)

A grid summarizing the joint distribution of paired categorical variables using cell counts along with row, column, or overall percentages.

Simple linear regression

A prediction model expressed as $\hat{Y} = a + bX$, where intercept $a$ and slope $b$ are used to estimate the average outcome value ($\hat{Y}$) from a predictor value ($X$).

Residual

The difference between an observed data value and the corresponding value predicted by a regression line ($Y - \hat{Y}$).