Statistics & Research Methods – Core Vocabulary

Descriptive vs. Inferential Statistics

• Descriptive Statistics

  • Purpose: Summarise and present data only; no conclusions beyond the sample.
  • Graphical tools: frequency tables, histograms, bar charts, boxplots.
  • Numerical measures
  • Central tendency: mean, median, mode.
  • Variability: range, variance, standard deviation (SD), inter-quartile range (IQR).
  • Key point: tells you what the sample looks like, not whether it generalises.

• Inferential Statistics

  • Purpose: Use a sample to draw conclusions about a population.
  • Core elements of every test
  • Hypotheses: H0 (no effect) vs. HA (effect exists).
  • Test statistic: t, F, U, \chi^2 etc.
  • Significance level: \alpha (conventionally 0.05).
  • Decision rule: Reject H_0 if p<\alpha (e.g. p<0.05 ⇒ statistically significant).

Parametric vs. Non-parametric Tests

• Parametric Tests

  • Assumptions: normality, equal variances (homoscedasticity), interval/ratio DV.
  • Data scale: continuous.
  • Advantage: greatest power if assumptions met.
  • Examples: one-sample, paired & independent t-tests; one-way & repeated-measures ANOVA.

• Non-parametric Tests

  • Assumptions: far fewer (no normality needed, skew/ordinal allowed).
  • Data scale: ordinal or non-normal.
  • Disadvantage: generally lower power under ideal parametric conditions.
  • Examples (two-condition designs)
  • Sign test (paired): counts direction only.
  • Wilcoxon signed-rank (paired): uses ranks of differences.
  • Mann–Whitney U (independent): compares ranks between groups.

t-Tests

One-Sample t

• Question: Does sample mean \bar{X} differ from known \mu_0?
• Statistic: t = \frac{\bar{X}-\mu_0}{s/\sqrt{N}}
• Degrees of freedom: df = N-1.
• Interpretation: large |t| ⇒ small p ⇒ reject H_0.

Paired / Within-Subjects t

• Design: same participants measured twice (e.g., before vs. after).
• Create difference scores D = X1 - X2.
• Statistic: t = \frac{\bar{D}-0}{s_D/\sqrt{N}}
• df = N-1 (N = pairs).
• Significant t ⇒ mean difference \neq 0.

Independent / Between-Subjects t

• Design: two unrelated groups.
• Equal variances formula: t = \frac{\bar{X}1-\bar{X}2}{\sqrt{\frac{s1^2}{N1}+\frac{s2^2}{N2}}}
• df = N1 + N2 - 2.
• If variances unequal: use Welch’s t.

Non-parametric Alternatives for Two Conditions

• Sign Test (paired): direction only; report p + sign of median change.
• Wilcoxon Signed-Rank (paired): ranks of differences; report Z, p, medians.
• Mann–Whitney U (independent): ranks across groups; report Z (or U), p, medians.

ANOVA Family

One-Way (Between-Subjects) ANOVA

• Use: compare k \ge 3 independent group means.
• Statistic: F = \frac{MS{between}}{MS{within}}.
• df{between}=k-1, df{within}=N-k.
• Large F ⇒ at least one mean differs ⇒ follow with post-hoc tests (Tukey HSD, Bonferroni, etc.).
• Reporting: “F(df{between},df{within}) = value, p = …”.

Repeated-Measures ANOVA

• Design: same subjects in all conditions.
• Additional assumption: sphericity (equal variances of pairwise difference scores).
  • Test with Mauchly’s test; if violated, correct using Greenhouse–Geisser or Huynh–Feldt \varepsilon.
• Post-hoc: paired t-tests with multiple-comparison correction (e.g., Bonferroni).
• Reporting example: “F(df{GG},df{GG}) = value, p = …, \varepsilon_{GG}=…”.

Experimental Designs

• Within-Subjects

  • Same participants experience all IV levels.
  • Pros: higher power, fewer participants.
  • Cons: order, carry-over & practice/fatigue effects.
  • Mitigation: counterbalancing or randomised order.

• Between-Subjects

  • Different participants per condition.
  • Pros: no order effects.
  • Cons: more participants needed for same power; possible group differences ⇒ random assignment essential.

Variables & Data Fundamentals

• Independent Variable (IV): manipulated factor with multiple levels.
• Dependent Variable (DV): measured outcome that “depends” on IV manipulation.
• Parameter vs. Statistic: parameter (population, Greek symbols \mu,\sigma); statistic (sample, Latin \bar{x}, s).

Scales of Measurement

• Nominal: categories, no order (e.g., eye colour).
• Ordinal: ordered ranks, unequal intervals (e.g., Likert scale).
• Interval: equal spacing, no true zero (e.g., ^\circ C).
• Ratio: equal spacing + absolute zero (e.g., reaction time).
  ⇒ Hierarchy: Nominal < Ordinal < Interval < Ratio.

Descriptive Statistics & Variability

• Mean: \bar{x}=\frac{\sum x_i}{N}.
• Median: middle ordered value.
• Mode: most frequent score.
• Range: \text{max}-\text{min}.
• Variance: s^2 = \frac{\sum (x_i-\bar{x})^2}{N-1}.
• Standard deviation: s = \sqrt{s^2}.
• Importance: central tendency + variability provide a complete picture; small SD ⇒ clustered data, large SD ⇒ dispersed.

Hypothesis Testing Concepts

• Null hypothesis H_0: no effect or difference.
• Alternative hypothesis H_A: predicted effect/difference.
• Directional (one-tailed) vs. non-directional (two-tailed).
• Type I error: false positive (reject true H_0).
• Type II error: false negative (fail to reject false H_0).
• Control Type I by setting \alpha (commonly 0.05). Power ≈ 1-\beta (prob. of avoiding Type II).

p-Values & Significance

• p-value: probability of data (or more extreme) given H_0.
• Rule: if p<\alpha ⇒ statistically significant ⇒ reject H_0.
• Report exact p (e.g., p=0.032) unless very small (e.g., p<0.001).

Normal Distribution & z-Scores

• Normal curve: symmetric, unimodal; mean = median = mode.
• 68-95-99.7 rule: \pm1,2,3 SD cover 68%, 95%, 99.7% of data.
• Standard normal: \mu=0,\sigma=1.
• Single point z: z = \frac{X-\mu}{\sigma}; sampling distribution z = \frac{\bar{X}-\mu}{\sigma/\sqrt{N}}.
• High |z| ⇒ low tail probability ⇒ evidence against H_0 (rarely used when \sigma unknown).

Sampling & Randomisation

• Population → (random sampling) → sample → descriptive stats → infer population parameters.
• Random assignment within experiments: equate groups on extraneous variables.
• Counterbalancing in within-subjects: distribute order effects evenly.

Reporting Guidelines (APA-style)

• Parametric: report mean ± SD (e.g., M=12.4\pm2.1).
• Non-parametric: report median [IQR/range].
• Always state: test statistic, degrees of freedom, p-value, and conclusion.
  • Example: “t(9)=4.40, p=0.002, one-tailed; significant.”

Quick Reference: Core Formulas & df

• One-sample t: t=\frac{\bar{X}-\mu_0}{s/\sqrt{N}}, df=N-1.
• Paired t: t=\frac{\bar{D}}{s_D/\sqrt{N}}, df=N-1.
• Independent t: t=\frac{\bar{X}1-\bar{X}2}{\sqrt{s1^2/N1+s2^2/N2}}, df=N1+N2-2 (pooled).
• One-way ANOVA: F = \frac{MS{between}}{MS{within}}, df{between}=k-1, df{within}=N-k.

Worked Example (Paired t)

• Given: \bar{D}=2.1, s_D=1.51, N=10.
• Compute: t = \frac{2.1}{1.51/\sqrt{10}} \approx 4.40, df=9.
• Compare to critical t{\alpha=0.05,df=9}\approx1.83 (one-tailed) ⇒ reject H0.

Ethical & Practical Implications

• Correct choice of test guards against invalid conclusions (over-claiming or missing real effects).
• Reporting full descriptive and inferential stats promotes transparency & reproducibility.
• Awareness of Type I/II trade-off vital when designing studies (e.g., set \alpha lower for high-stakes research, raise power with larger N).