Comprehensive Notes: Measurement Scales, Skewness, and Correlations

Class Notes: Measurement Scales, Skewness, and Correlations

  • Context of the session

    • Instructor-led class activity focused on applying a rubric to psychometrics problems (scales, measurement, and interpretation).
    • Students worked in groups (pairs to four people) to discuss and justify answers using a rubric, then compared responses.
    • Emphasis on using tools (rubric) rather than just intuition; outcomes and predictors were framed in a research design context.
    • Names on slips for credit and later posting on Brightspace under Resources.

  • Key terms and concepts

    • Criterion variable vs outcome variable
    • Outcome = the dependent variable of interest in a study; the downstream result you want to explain or predict.
    • In the example, job productivity (or job performance) is the outcome/criterion variable.
    • Other terminology used: criterion (plural: criteria) and outcome interchangeably in practice.
    • Predictor variables
    • Variables used to predict or explain the outcome. In the example, extroversion and conscientiousness are predictors.
    • Conceptual note: predictors can be multiple; different levels of measurement affect analysis choices.
    • Continuum vs. categories (types of variables)
    • Extroversion is best treated as a continuum (an interval-level variable) rather than a binary type (extroverted vs introverted).
    • Taxonomic labels (types) can lead to misapplied analysis; continua align with interval/ratio scales.
    • Measurement scales (classification and implications)
    • Nominal: categories with no inherent order (e.g., types, labels).
    • Dichotomous (a special nominal scale with two categories): e.g., yes/no.
    • Ordinal: ordered categories, but intervals between categories are not guaranteed to be equal.
    • Interval: ordered categories with equal intervals between adjacent values; no true zero.
    • Likert-type scales: commonly 5-point scales (e.g., strongly agree to strongly disagree) used to measure attitudes; debate whether they are truly interval or only ordinal, but often treated as interval in practice.
    • Rubric-driven vs instinct-driven reasoning
    • The rubric asks to determine whether there is more or less of the construct across response options, guiding the level of measurement (dichotomous vs ordinal vs interval).
    • Emphasizes sticking to rubric rules to prevent over-nuanced, non-compliant interpretations.

  • Likert scales and response formats (applied rules)

    • Question 1: GIF vs JIF (pronunciation) – two options imply a dichotomous (nominal, two-category) scale.
    • Question 2: “I love snow” with responses: strongly agree, agree, neither agree nor disagree, disagree, strongly disagree – Likert-type; treated as ordinal, and often treated as interval for analysis if gaps are assumed equal.
    • Rubric-based decision path:
    • If only two options exist, scale is dichotomous (nominal).
    • If more than two options but gaps are assumed equal, scale is interval; if gaps are not guaranteed equal, treat as ordinal.
    • Important caveat discussed: Even with Likert-type items, the teacher emphasizes using the interval assumption only when justified; otherwise, treat as ordinal and use appropriate non-parametric methods if necessary.

  • Outcomes, predictors, and model framing (examples from the session)

    • Context: A company evaluating why some people perform better on the job.
    • Outcome (criterion variable): Job productivity / job performance.
    • Predictors: Levels of extroversion and conscientiousness (two predictors).
    • Discussion point: How to frame relationships properly
    • If extroversion/conscientiousness are predictors, the model implies variability in job productivity is associated with these traits.
    • Emphasis on viewing extroversion as part of a continuum rather than a simple yes/no label.
    • Conceptual guidance for building a model:
    • Start with the outcome variable (job productivity).
    • Then identify predictor variables (e.g., extroversion, conscientiousness).
    • Decide each variable’s measurement level using the rubric (dichotomous, ordinal, interval).
    • Avoid reversing cause and effect; think in terms of predictive relationships and possible causal direction in a theoretical model.

  • Data interpretation: Means, standard deviation, and skewness

    • Mean and standard deviation basics
    • The mean is the central tendency measure used to summarize a distribution.
    • Standard deviation indicates the average amount of dispersion around the mean.
    • A small standard deviation indicates little variability; a large standard deviation indicates more diversity in responses.
    • Skewness concepts
    • Positive skew: tail on the higher end (right tail); mean tends to be higher than the median; distribution has a tail extending to the right.
    • Negative skew: tail on the lower end (left tail); mean tends to be lower than the median.
    • The session emphasized looking at the distribution to determine skewness, rather than relying solely on instinct.
    • Normal distribution intuition
    • In a normal distribution, the peak (mode/median/mean) aligns at the center, with symmetric tails.
    • Skewness assessment focuses on which side has the longer tail and where the mean sits relative to the median.
    • Example interpretation to connect these ideas
    • If a column of means shows the lowest mean (e.g., resilience) and a right tail is present, that item may be positively skewed.
    • If a variable has a high mean (e.g., manipulative) and a skewed tail, interpret in the context of the observed data and the construct being measured.

  • Variability and “least individual differences” (standard deviation focus)

    • Question: Which variable has the least individual differences?
    • Answer concept: The variable with the smallest standard deviation shows the least variation among individuals.
    • In the discussion, dexterousness (DEXTEROUSNESS) had a very small standard deviation (e.g., around 1.01), indicating relatively little variation around the mean for that item.
    • Practical takeaway: When interpreting data, consider both the mean (central tendency) and the standard deviation (variability) to understand distribution shape and reliability of the measure.

  • Correlations: identifying relationships between constructs

    • Core idea: Determine which pairs of constructs tend to move together (positive correlation) or in opposite directions (negative correlation).
    • Positive correlations (high on one trait associated with high on another):
    • Disorganized ↔ Messy
    • Manipulative ↔ Two-faced (trustworthiness/consistency themes)
    • Brave ↔ Confident
    • Negative correlations (one high, the other low):
    • Shy ↔ Chatty (one end of a continuum, opposite end on the same dimension)
    • Loyal ↔ Two-faced (conceptual opposites on behavior/consistency scale)
    • No correlation (independence):
    • Nosy ↔ Dexterousness (no predictable relationship observed in that example)
    • How to interpret correlations
    • Correlation conveys a tendency, not a guarantee; high values on one variable tend to accompany high values on the other, but exceptions exist.
    • The measurement level affects how correlations are interpreted: with interval data, Pearson r is common; with ordinal data, Spearman rho or Kendall’s tau may be more appropriate.
    • Practical exercise: using the rubric to predict which pairs are expected to co-vary and then testing those expectations with the data.

  • Practical data reasoning and the rubric workflow (how to apply in exams or real data analysis)

    • Step 1: Identify the outcome variable (the criterion/outcome you want to explain).
    • Step 2: Identify predictor variables (what might explain or influence the outcome).
    • Step 3: Determine the measurement level of each variable using the rubric (dichotomous, ordinal, interval).
    • Step 4: If comparing two categories, decide if the scale is dichotomous (nominal, two options) or multi-category (ordinal/interval).
    • Step 5: For Likert-type items, decide if you treat as ordinal or interval based on the equal-interval assumption; use the rubric to decide.
    • Step 6: Analyze skewness and variability
    • Examine the column means and the distribution shape to assess skewness.
    • Identify the variable with the least variability (smallest standard deviation) to locate least individual differences.
    • Step 7: Analyze correlations among constructs
    • Propose expected correlations (positive, negative, or none) and then use the data to confirm or revise expectations.
    • Step 8: Be explicit in labeling and labeling rationale
    • Use precise terms: outcome variable, predictor variables, measurement level, correlation interpretation, etc.

  • Important methodological reminders from the session

    • Do not rely solely on intuitive impressions when solving measurement questions; use the rubric to guide decisions.
    • When in doubt, start with the outcome variable and work your way back to predictors and measurement scales.
    • Be mindful of how wording (e.g., “types of people” vs. continua) affects the interpretation and the choice of statistical tools.
    • Use diagrams or simple drawings to reason about skewness and standard deviation, as demonstrated in the session.
    • Real-world relevance: understanding measurement scales and the correct interpretation of statistics is essential for valid data analysis in psychology, organizational behavior, and many social sciences.

  • Quick reference formulas (LaTeX)

    • Mean: xˉ=1notal<em>i=1nx</em>i\bar{x} = \frac{1}{n} otal<em>{i=1}^n x</em>i
    • Standard deviation (sample): s =
      rac{1}{n-1}

ight)

  • Pearson correlation coefficient: r{xy} = rac{ extstyle rac{1}{n} otal{i=1}^n (xi-ar{x})(yi-ar{y})}{ ext{(std)}(x) ext{(std)}(y)} = rac{
    rac{1}{n} ext{S}{xy}}{ ilde{ ext{S}}x ilde{ ext{S}}y} where extS</em>xy=1n−1otal<em>i=1n(x</em>i−xˉ)(yi−yˉ)ext{S}</em>{xy} = \frac{1}{n-1} otal<em>{i=1}^n (x</em>i-\bar{x})(y_i-\bar{y})

  • Skewness (first moment ratio form): extSkew=E[(X−Xˉ)3]extSD(X)3ext(sampleversion:1notal(xi−xˉ)3s3ext)ext{Skew} = \frac{E[(X-\bar{X})^3]}{ ext{SD}(X)^3} ext{(sample version: } \frac{\frac{1}{n} otal (x_i-\bar{x})^3}{s^3} ext{)}

  • Normal distribution intuition: symmetric around the mean; mean = median = mode; if skewness is zero, distribution is symmetric.

    • Final takeaways for exam prep

  • Be able to classify variables by scale (dichotomous, ordinal, interval) using the rubric’s “more or less” logic.

  • Be able to identify the outcome variable and one or more predictor variables in a given research context.

  • Understand when to treat Likert-type data as ordinal vs interval and the implications for analysis.

  • Interpret means, standard deviations, and skewness to describe distributions and identify which items show the most/least variability or skew.

  • Describe hypothesized vs observed correlations, and distinguish between positive, negative, and no correlation.

  • Use the continuum vs categories framing to justify the measurement approach and the appropriate statistical toolbox.

    • Wrap-up and next steps

  • The instructor plans to move into correlations and measurement choices in the next session.

  • Students were encouraged to bring name tags and to review group notes posted under Resources for missed classes.