Standardizing Distributions

z-Score (Standard Score)

A standardized metric that expresses an individual raw score's distance and direction from the distribution center in units of standard deviations, setting the distribution mean to 0 and standard deviation to 1 (calculated as z = (x - mu) / sigma for populations or z = (x - x_bar) / s for samples). In lecture, the professor illustrated this using four job candidates (Han with score 60, mean 80, SD 20 -> z = -1.0; Xave with score 90, mean 80, SD 20 -> z = +0.5; Beverly with score 90, mean 95, SD 2 -> z = -2.5; Crystal with score 140, mean 120, SD 10 -> z = +2.0), demonstrating how z-scores allow direct comparisons across different classes and scales. (Slide 5-8; Transcript 00:00:23, 00:01:49, 00:09:56; Textbook pp. 81, 265-267)

Shape Preservation Property of z-Transformation

Transforming raw scores into z-scores shifts the center to 0 and rescales the standard deviation to 1, but does NOT alter the shape, skewness, or kurtosis of the underlying distribution. In lecture, the professor demonstrated using jamovi pain severity data (N = 300) that raw severity (mean = 5.67, SD = 0.888, skewness = -0.718, kurtosis = -0.0399) and standardized Z-severity (mean = -3.98e-17 ~ 0, SD = 1.00, skewness = -0.718, kurtosis = -0.0399) yield identical density curves and shape statistics. (Slide 9-10; Transcript 00:26:26, 00:27:30; Textbook pp. 81, 265-267)

Standardized Mean Difference (Cohen's d)

An effect size metric that quantifies the difference between two group means in standard deviation units rather than raw score units, calculated by dividing the raw mean difference by the average pooled standard deviation. In lecture, the professor illustrated this by comparing male (mean = 5.83, SD = 0.900, n = 118) and female (mean = 5.55, SD = 0.865, n = 182) pain severity scores, transforming a raw difference of 0.28 points into a standardized effect size of Cohen's d = 0.318, indicating that the two gender groups differ by 0.318 standard deviation units. (Slide 11-12; Transcript 00:31:58, 00:32:24, 00:42:28; Textbook pp. 265-267)

Population Parameter

A fixed, theoretical numerical value (e.g., population mean mu, population standard deviation sigma) that describes a characteristic of the entire target population; it is usually unknown because researchers rarely have access to every individual in a population. (Slide 13; Transcript 00:53:47, 00:53:51; Textbook p. 151)

Sample Estimate (Sample Statistic)

A numerical summary value (e.g., sample mean x-bar, sample standard deviation s) calculated from an observed sample subset, used by researchers to approximate the unknown population parameter. (Slide 13; Transcript 00:54:06, 00:54:19; Textbook p. 151)

Sampling Error

The random numerical difference between a sample estimate and the true population parameter (calculated as Sample Mean minus Population Mean, x-bar - mu), which varies across different random samples and can be positive (sample mean above population mean) or negative. (Slide 13; Transcript 00:56:23, 00:56:34; Textbook p. 151)

Minnesota Multiphasic Personality Inventory (MMPI) T-Scores

A standardized clinical scoring scale used in personality assessment that transforms raw questionnaire scores into standardized T-scores with a set mean of 50 and a standard deviation of 10 to allow uniform diagnostic comparison across clinical subscales. (Textbook pp. 265-266)

Weighted Pooled Variance Formula

The mathematical formula for combining variance estimates from two independent samples of unequal sizes by weighting each sample variance by its degrees of freedom (N1 - 1 and N2 - 1) before dividing by total degrees of freedom (N1 + N2 - 2). (Textbook pp. 252, 267)

Jamovi Computed Variable Formula for z-Scores

The exact software command entered into the jamovi formula box = Z(variable_name) or = (variable - VMEAN(variable)) / VSTDEV(variable) to automatically create a new standardized variable with mean 0 and standard deviation 1. (Slide 9; Transcript 01:16:03, 01:21:00)

Average (Pooled) Standard Deviation Formula in Cohen's d

The explicit square-root formula used in lecture to compute the denominator for Cohen's d across two groups: sqrt(((n1 - 1)s1^2 + (n2 - 1)s2^2) / (n1 + n2 - 2)), which accounts for unequal sample sizes between groups. (Transcript 00:38:00, 00:41:48)

Why do researchers standardize variables into z-scores when comparing individuals from different groups or classes?

Because raw scores from different distributions or measurement scales have different centers and spreads, making raw comparisons like apples to oranges; converting to z-scores places all individuals on a single universal scale with mean 0 and standard deviation 1. (Slide 5-8; Transcript 00:01:23, 00:01:58, 00:16:32)

What happens to the mean, standard deviation, skewness, and kurtosis of a variable after z-score standardization?

The mean becomes 0 (or computationally close to 0, e.g., -3.98e-17) and the standard deviation becomes 1, while skewness and kurtosis remain completely unchanged because standardizing rescales units without altering distribution shape. (Slide 9-10; Transcript 00:27:30, 00:28:05)

How do you perform an Independent Samples t-Test in jamovi to obtain Cohen's d for comparing two group means?

Go to Analyses -> T-Tests -> Independent Samples T-Test, move the continuous outcome into Dependent Variables and the categorical variable into Grouping Variable, and check the Effect Size checkbox to display Cohen's d in the results table. (Transcript 01:18:14, 01:18:21)

Why is the raw mean difference between two groups often insufficient for evaluating effect size in psychological research?

Because raw mean differences depend entirely on arbitrary measurement units (e.g., points on a Likert scale), whereas standardized effect sizes (Cohen's d) express the difference in standard deviation units, allowing researchers to evaluate whether a difference is meaningfully large regardless of scale. (Slide 11; Transcript 00:32:24, 00:32:49, 01:15:20)