Quantitative Data Analysis: Levels of Measurement, Descriptive Statistics, and Inferential Testing

A comprehensive vocabulary review deck covering quantitative data analysis concepts including measurement scales, descriptive statistics, sampling distributions, confidence intervals, survey weighting, t-tests, ANOVA, and Chi-Square analysis.

Measurement

The assignment of numerals to attributes of objects or events according to specified methodological rules.

Nominal Measurement

A level of measurement where numerals act solely as labels or identifiers to designate categories or subgroups, carrying no numerical meaning or relative position.

Ordinal Measurement

A level of measurement where numerals indicate rank order or relative standing, but the distances between intervals are unknown and not necessarily equal.

Equal Interval Scale Measurement

A level of measurement where numerically equal distances on the scale represent equal distances in the property being measured, permitting all mathematical operations.

Composite Ordinal Scale

A scale created by statistically combining multiple ordinal items, producing an emergent composite measure that approximates equal intervals.

Interval Scale

A subdivision of equal interval measurement where the distances between intervals are equivalent, but the scale lacks a meaningful absolute zero (e.g., degrees Celsius).

Ratio Scale

A subdivision of equal interval measurement where intervals are equivalent and the scale possesses a true, meaningful zero value (e.g., income, height).

Valid Percentage

The percentage calculated as a proportion of the total sample minus any missing values.

Incidence

The number of new or newly diagnosed cases of a condition occurring within a specified population over a given period.

Prevalence

The actual total number of cases alive with a condition during a specified period of time, typically one year.

Lifetime Prevalence

The total number or proportion of people in a population who have experienced a particular problem or condition at any point during their lives.

Outlier

An extreme, isolated observation located far from the central cluster of values that can compromise the representativeness of the mean.

Skewed Distribution Types

Asymmetrical distributions categorized into negative skew (elongated tail at lower values, mean $<$ median $<$ mode), no skew (perfectly symmetrical bell curve where mode $=$ median $=$ mean), and positive skew (elongated tail at higher values, mode $<$ median $<$ mean).

Skewness Rule of Thumb

An empirical guideline stating that a distribution can be considered symmetrical if $|\text{skewness statistic}| < 2 \times |\text{standard error}|$.

Normal Distribution

An ideal symmetrical, bell-shaped distribution in which the mean, median, and mode are all identical and located at the center.

Inter-Quartile Range (IQR)

A range-based measure of dispersion defined as the distance between the $25^{\text{th}}$ and $75^{\text{th}}$ percentiles, capturing the central $50\%$ of observations.

Sum of Squares (SS)

The sum of squared distances of each observation from the sample mean, calculated as $\sum (M - X)^2$.

Variance

A measure of dispersion representing the average squared distance of observations from the mean, calculated as $\frac{\sum (M - X)^2}{N}$.

Standard Deviation (SD)

The square root of the variance, measuring the average distance of scores from the mean expressed in the original units of measurement.

Bessel Correction

An adjustment that divides the sum of squared deviations by $n - 1$ instead of $N$ to eliminate bias when estimating population variance and standard deviation from a sample.

Population Parameter

A true numerical characteristic describing an entire population, equivalent to a sample statistic.

Central Limit Theorem

The principle stating that the sampling distribution of a sample statistic (such as the mean) approaches a normal distribution as sample size increases, regardless of the population distribution shape.

Sampling Distribution

The theoretical probability distribution formed by calculating a statistic from repeated independent random samples of identical size drawn from the same population.

Sampling Error

The extent to which an estimate obtained from a sample differs from the true population value purely due to random chance.

Standard Error (SE)

The standard deviation of a sampling distribution, calculated for a sample mean as $SE = \frac{s}{\sqrt{N}}$.

Confidence Interval (CI)

A range of plausible values around a sample point estimate expected to contain the true population parameter at a specified probability, such as $\bar{X} \pm 1.96 \times SE$ for a $95\%$ confidence level.

Standard Normal Distribution Curve

A standard normal distribution curve $N(0, 1)$ enclosing $68.27\%$ of values within $\mu \pm 1\sigma$, $95\%$ of area within $\pm 1.96\sigma$, $95.45\%$ within $\mu \pm 2\sigma$, $99\%$ of area within $\pm 2.576\sigma$, and $99.73\%$ within $\mu \pm 3\sigma$.

Z-Score

A standardized score indicating how many standard deviations an observation falls from the mean, calculated as $Z = \frac{X - M}{SD}$.

t-Distribution

A family of symmetrical, bell-shaped distributions with heavier tails than the standard normal distribution, used to calculate confidence intervals and significance when sample sizes are small or population variance is estimated.

Non-Coverage Error

A survey error occurring when members of the target population are systematically omitted from the sampling frame.

Non-Response Error

A survey error that arises when selected individuals in a sample refuse or fail to participate, potentially biasing sample estimates.

Sample Weight

A numerical factor assigned to each individual case to correct for sampling design and non-response, calculated as $(\text{share in the population}) \div (\text{share in the sample})$.

Calibration

A survey weighting procedure that adjusts sample cases so that the weighted sample totals match known population census totals across socio-demographic variables.

Gross Weights

Survey weights scaled so that the sum of the weights equals the total population count rather than the sample size.

Null Hypothesis ($H_0$)

A formal statistical hypothesis postulating no difference, effect, or relationship in the population (e.g., mean difference $= 0$).

Alternative Hypothesis ($H_1$)

A statistical hypothesis positing that a genuine non-zero difference, effect, or relationship exists in the population.

Alpha Level ($\alpha$)

The predetermined significance threshold (commonly $0.05$ or $0.01$) representing the probability of rejecting the null hypothesis when it is true.

Independent Samples t-Test

A parametric inferential test used to determine whether the difference between the means of two independent groups on a scale variable is statistically significant.

SPSS Levene's Test Decision Rule

The protocol in an SPSS independent samples test output where a Levene's test significance value of $p < 0.05$ rejects variance equality, directing the researcher to read the 'Equal variances not assumed' row.

Welch's t-Test

An independent samples t-test variant calculated without assuming equal group variances, employing Welch-Satterthwaite adjusted degrees of freedom which are typically non-integers.

Eta Squared ($\eta^2$)

An effect size measure representing the proportion of variance in a scale variable explained by group membership, calculated from a t-test as $\eta^2 = \frac{t^2}{t^2 + df}$.

Cohen's d

A standardized effect size quantifying the difference between two sample means in pooled standard deviation units, calculated as $d = \frac{M_1 - M_2}{s_p}$.

Dual Effect

A phenomenon where two groups have similar means despite possessing very different distributions because one group exhibits more extreme scores at both distribution tails.

Bivariate Analysis Decision Map

A flowchart directing bivariate test selection based on group count ($2$ groups for t-test, $3$ or more for ANOVA), Levene's test for equality of variances, statistical significance, post-hoc procedures, and effect sizes.

One-Way Analysis of Variance (ANOVA)

An inferential statistical test used to evaluate mean differences across three or more groups by calculating an $F$-ratio comparing between-group variance to within-group variance ($F = \frac{MS_{\text{between}}}{MS_{\text{within}}}$).

Familywise Error Rate

The cumulative probability of committing at least one Type I error across $c$ pairwise comparisons, calculated as $\text{Familywise error} = 1 - (1 - \alpha)^c$.

Tukey's Honestly Significant Difference (HSD) Test

A post-hoc pairwise comparison procedure conducted after a significant ANOVA that controls the overall familywise error rate at $0.05$ under the assumption of equal variances.

Games-Howell Test

A post-hoc pairwise comparison procedure executed following a significant Welch's ANOVA that controls familywise error when group variances are unequal.

Expected Frequency ($E$)

The theoretical frequency calculated for a crosstabulation cell under the null hypothesis of no relationship, calculated by multiplying the group total by the overall percentage (e.g., $621 \times 0.711 = 441.53$).

Chi-Square Test of Independence ($\chi^2$)

A non-parametric test assessing the relationship between two categorical variables by evaluating departures of observed cell counts from expected cell counts using $\chi^2 = \sum \frac{(O - E)^2}{E}$.