Quantitative Analysis I — Statistics

Learning Objectives

By the end of this session, you will be able to:

• Compute and interpret descriptive statistics — measures of central tendency, dispersion, and distribution shape — and select the appropriate summary statistics for different variable types and distributions
• Explain the logic of null hypothesis significance testing — including p-values, Type I and Type II error, statistical power, effect size, and confidence intervals — and apply this logic to draw appropriate conclusions from statistical output
• Select, run, and interpret the correct statistical test — independent and paired t-tests, one-way ANOVA with post-hoc comparisons, and Pearson/Spearman correlation — based on research question, variable types, and data structure
• Check and report the assumptions underlying parametric tests — normality, homogeneity of variance, independence — and choose appropriate non-parametric alternatives when assumptions are violated
• Report statistical results in APA 7th edition format, including test statistics, degrees of freedom, p-values, effect sizes, and confidence intervals, with interpretations that connect back to the research questions

Session Planner

Suggested breakdown of the 4-hour contact session.

Time	Segment	Activity	Mode
0:00–0:08	Opening	Recap Unit 2; transition: "You've designed your methodology. Now: when the data comes in, what do you actually DO with it?" The analysis plan from your methodology chapter becomes real this week.	Whole class
0:08–0:30	Lecture 1	Descriptive statistics — types of variables, central tendency, dispersion, shape; data screening and cleaning; the logic of inference — populations, samples, sampling distributions, the Central Limit Theorem	Lecture
0:30–0:55	Lecture 2	Hypothesis testing framework — null and alternative hypotheses, p-values, Type I/II error, power, effect size, confidence intervals. t-tests: one-sample, independent, paired — when to use each, assumptions, interpretation	Lecture
0:55–1:15	Activity	Test selection exercise: given 10 research scenarios with variable types and RQs, select the appropriate statistical test. Justify the choice. Identify what the null hypothesis would be.	Pairs
1:15–1:25	Discussion	Debrief test selection; address common mismatches (e.g., using correlation when a t-test is appropriate, ignoring the independence assumption)	Whole class
1:25–1:40	Break	—	—
1:40–2:05	Lecture 3	One-way ANOVA — between-groups vs within-groups, the F-ratio, post-hoc tests (Tukey, Bonferroni), effect size (eta-squared). Correlation — Pearson vs Spearman, interpreting r and r², correlation matrix reporting. Assumption testing for all parametric tests.	Lecture
2:05–2:20	Software Demo	Live demonstration: run a complete analysis in SPSS/R/Python — import data, screen for errors, check assumptions, run t-test and ANOVA, interpret output, report in APA format. Students follow along on their own machines.	Demo
2:20–3:50	Lab Work	Part A: Descriptive statistics and data screening; Part B: t-tests and ANOVA on provided datasets; Part C: Apply analyses to your own pilot or secondary data	Individual
3:50–4:00	Exit Ticket	Submit analysis output and interpretation; identify the test you are least confident about applying correctly	Individual

1. Descriptive Statistics — Knowing Your Data Before You Test It

Descriptive statistics are the foundation of all quantitative analysis. Before you test hypotheses, before you run regressions, before you draw any inferential conclusions, you must know your data — its central tendencies, its spread, its shape, its outliers, its missing values. Skipping descriptive statistics is like driving without looking at the dashboard: you may be moving, but you have no idea what is happening under the hood.

1.1 Types of Variables — Your Choice Determines Your Analysis

Variable Type	Definition	Examples	Appropriate Descriptive Statistics	Appropriate Graphs
Nominal (Categorical)	Categories with no inherent order — labels, not quantities	Gender (Male/Female/Other), Industry (IT/Manufacturing/Finance), City, Operating System	Frequency, mode, percentage; no mean or median (meaningless)	Bar chart, pie chart (limited use)
Ordinal	Categories with a meaningful order but unknown/unequal distances between categories	Education level, Likert-scale items (individual items, not scale totals), satisfaction ratings, socioeconomic status	Median, mode, frequency, interquartile range; mean is debatable (common in practice but technically assumes equal intervals)	Bar chart, stacked bar
Interval / Ratio (Continuous)	Numerical values with equal intervals; ratio variables additionally have a meaningful zero	Age, income, temperature (interval), height, weight, response time, test score, number of purchases (ratio)	Mean, median, mode, standard deviation, variance, range, skewness, kurtosis	Histogram, box plot, density plot, scatter plot

1.2 The Descriptive Statistics Checklist — What to Report

1.3 Data Screening — The Most Important Step Students Skip

Before running a single inferential test, you must screen your data. Data screening catches errors that, if undetected, produce invalid results. The screening process is not glamorous, but it is what separates competent quantitative researchers from students who get numbers from software and assume they are correct.

1. Check for impossible values: Age = 250. Likert response = 7 on a 1–5 scale. Negative reaction time. These are data entry errors. Scan minimum and maximum values for every variable.
2. Check for missing data: How much missing data is there? Is it random (MCAR), related to observed variables (MAR), or related to unobserved variables (MNAR)? For capstone purposes: report the percentage of missing data for each variable. If >5% missing, justify your handling approach (listwise deletion, pairwise deletion, mean imputation, or multiple imputation). Never silently delete rows with missing data.
3. Check for outliers: Univariate outliers: values more than 3 SD from the mean, or outside 1.5 × IQR on a box plot. Multivariate outliers: unusual combinations of values across multiple variables. For each outlier, decide: is it an error (fix or remove), a genuine extreme value (retain but note), or an influential case (run analysis with and without it to check)?
4. Check distributions: Are continuous variables approximately normal? Check histograms, Q-Q plots, skewness/kurtosis values, and Shapiro-Wilk test. This determines whether you can use parametric tests (Section 2) or need non-parametric alternatives.
5. Check for duplicates: Are there duplicate rows? (e.g., the same participant submitted the survey twice). Check for duplicate IDs, identical response patterns, or suspiciously fast completion times.

2. Inferential Statistics — The Logic of Drawing Conclusions

Inferential statistics enable you to draw conclusions about a population based on data from a sample. The logic is counterfactual and probabilistic: we ask "if nothing were really going on (the null hypothesis), how likely is it that we would observe data like ours purely by chance?" If that probability is very low, we infer that something probably IS going on. This is the logic of null hypothesis significance testing (NHST).

2.1 The NHST Framework

2.2 Type I and Type II Error — The Two Ways to Be Wrong

	H0 is TRUE (no effect)	H0 is FALSE (real effect)
Reject H0 (significant)	Type I Error (α) False positive — concluding there is an effect when there isn't one. Probability = α (typically .05)	Correct Decision (Power = 1−β) True positive — correctly detecting a real effect. Probability depends on effect size, sample size, and α
Fail to Reject H0 (non-significant)	Correct Decision (1−α) True negative — correctly not claiming an effect when none exists. Probability = .95 when α = .05	Type II Error (β) False negative — failing to detect a real effect. Most commonly caused by insufficient sample size (low power)

3. t-Tests — Comparing Means

The t-test is the most fundamental inferential test: it compares means to determine whether an observed difference is larger than would be expected by sampling error alone. Despite its simplicity, the t-test is often used incorrectly — applied to the wrong data structure, with violated assumptions, or without the appropriate variant for the research design.

3.1 The Three t-Tests — When to Use Each

Test	What It Compares	Data Structure	Example RQ	H0
Independent Samples t-test	The means of a continuous DV between TWO independent groups	One categorical IV (2 groups) + one continuous DV. Each participant is in ONE group only.	"Do male and female employees differ in job satisfaction?" (IV: gender, 2 groups; DV: satisfaction score)	H0: μmale = μfemale
Paired Samples t-test	The means of a continuous DV at TWO time points (or under two conditions) within the SAME participants	One continuous DV measured twice on the same participants. Each participant provides two data points.	"Does employee satisfaction change from before (pre-test) to after (post-test) a training intervention?"	H0: μdifference = 0 (the mean of the pre-post differences is zero)
One-Sample t-test	The mean of a continuous variable against a KNOWN or HYPOTHESISED population value	One continuous variable; a known or hypothesised population mean to compare against	"Is the mean customer satisfaction score in this sample significantly different from the industry benchmark of 3.5?"	H0: μsample = 3.5 (the population mean equals the benchmark)

3.2 Assumptions of the t-Test

Assumption	What It Means	How to Check	What to Do If Violated
Independence	Observations are independent of each other — one participant's score does not influence another's. THE most important assumption.	Check the study design. Are participants nested in groups (teams, classes, organisations)? If so, independence may be violated (use multilevel modelling or clustered standard errors).	Use paired test for dependent observations; use multilevel modelling for nested data. If independence is severely violated, standard errors will be wrong and results invalid.
Normality	The DV is approximately normally distributed in each group (for independent t-test) or the differences are approximately normal (for paired t-test)	Visual: histogram, Q-Q plot. Statistical: Shapiro-Wilk test. Practical: skewness and kurtosis values. With n ≥ 30 per group, the CLT makes the t-test robust to moderate non-normality.	If severe non-normality (skewness > |2|) and n < 30: use Mann-Whitney U test (independent) or Wilcoxon signed-rank test (paired) — non-parametric alternatives that don't assume normality.
Homogeneity of Variance	The variance of the DV is approximately equal across groups (for independent t-test only)	Levene's test. If p > .05, variances are not significantly different — assumption met. If p ≤ .05, variances differ — assumption violated.	Use Welch's t-test (does not assume equal variances). SPSS reports both; the "equal variances not assumed" row is Welch's. This is the safer default when variances differ.

3.3 Reporting a t-Test in APA Format

4. One-Way ANOVA — Comparing Three or More Means

When you need to compare means across three or more groups, running multiple t-tests inflates the Type I error rate. Three groups require three pairwise comparisons; at α = .05 per test, the familywise error rate is approximately 1 − (0.95)³ = .14 — nearly three times the acceptable rate. ANOVA solves this by testing all group means simultaneously: "are there ANY differences among these groups?"

4.1 The Logic of ANOVA

ANOVA partitions the total variance in the DV into two components: between-groups variance (differences among group means — the effect you are testing) and within-groups variance (differences among individuals within the same group — error). The F-ratio is: F = (between-groups variance) / (within-groups variance). If the group means differ substantially (large between-groups variance) relative to the natural variation within groups (small within-groups variance), F will be large, and the result will be significant.

4.2 One-Way ANOVA — Between-Groups

Element	Description
When to Use	One categorical IV with 3+ independent groups; one continuous DV. Example: comparing employee satisfaction across four departments (IT, Marketing, Finance, HR).
Assumptions	Same as independent t-test: independence, normality (within each group), homogeneity of variance (Levene's test).
Key Output	F(dfbetween, dfwithin) = F-value, p-value, η² (eta-squared: proportion of variance in DV explained by the IV).
Post-Hoc Tests	Tukey's HSD: Most common; controls familywise error; compares all pairwise means. Use when group sizes are equal or approximately equal. Bonferroni: More conservative (lower power); divides α by number of comparisons. Use when you have specific planned comparisons. Games-Howell: Use when the homogeneity of variance assumption is violated (does not assume equal variances).
APA Report	"A one-way between-groups ANOVA was conducted to compare the effect of [IV] on [DV]. There was a significant effect of [IV] on [DV] at the p < .05 level: F([dfbetween], [dfwithin]) = [F], p = [p], η² = [value]. Post-hoc comparisons using Tukey's HSD test indicated that the mean score for [Group A] (M = [mean], SD = [SD]) was significantly different from [Group B] (M = [mean], SD = [SD]), p = [p]. No other comparisons were significant."

4.3 Non-Parametric Alternatives

Parametric Test	Non-Parametric Alternative	When to Use the Alternative
Independent t-test	Mann-Whitney U test (Wilcoxon rank-sum)	Severe non-normality with small samples (n < 30 per group); ordinal DV
Paired t-test	Wilcoxon Signed-Rank test	Severely non-normal differences with small samples
One-way between-groups ANOVA	Kruskal-Wallis H test	Severe non-normality with small n per group; ordinal DV; heterogeneity of variance that Games-Howell cannot address

5. Correlation — Measuring the Strength of Relationships

Correlation quantifies the strength and direction of the linear relationship between two continuous variables. It is the foundation of regression (Week 14), the basis for reliability analysis (Cronbach's alpha is a form of correlation), and one of the most commonly reported statistics in quantitative capstone research. It is also one of the most commonly misinterpreted: correlation does not imply causation, a point that cannot be stated too frequently.

5.1 Pearson and Spearman Correlation

Dimension	Pearson Correlation (r)	Spearman Correlation (rs or ρ)
What It Measures	The strength and direction of the LINEAR relationship between two continuous variables	The strength and direction of the MONOTONIC relationship between two variables (can be used with ordinal data or non-normal continuous data)
Range	−1 to +1. −1 = perfect negative linear relationship; 0 = no linear relationship; +1 = perfect positive linear relationship	−1 to +1. Interpreted similarly to Pearson, but based on ranks, not raw values
Assumptions	Both variables are continuous, approximately normally distributed, linearly related, with no extreme outliers	Variables are at least ordinal; relationship is monotonic (as one increases, the other consistently increases or decreases, but not necessarily in a straight line)
r² — Coefficient of Determination	The proportion of variance in one variable that is shared with (explained by) the other. r = .50 → r² = .25 → 25% shared variance. An apparently "moderate" correlation of .50 actually shares only 25% of variance.	Not directly interpretable as variance explained; Spearman's r² approximates shared variance but is less intuitive than Pearson's

5.2 Interpreting Correlation Coefficients — Cohen's Guidelines

|r|	Interpretation	r²	What This Means in Plain Language
.10	Small / Weak	1%	The variables share 1% of their variance — knowing one tells you almost nothing about the other. With a large sample, even r = .10 can be statistically significant. It is still a small effect.
.30	Medium / Moderate	9%	The variables share 9% of their variance. This is a meaningful effect in many social science contexts. Many published correlations in management and psychology are in the .20–.40 range.
.50	Large / Strong	25%	The variables share 25% of their variance. This is a substantial effect. Three-quarters of the variance is still unexplained — correlation, even "large" ones, leave much unknown.

5.3 Correlation Matrix — How to Report Multiple Correlations

Key reporting conventions: (1) Report means and SDs in the first columns. (2) The diagonal is dashes (a variable correlates perfectly with itself). (3) Report only the lower or upper triangle — the matrix is symmetric. (4) Use asterisks for significance levels and define them in a note. (5) Always include the sample size (N). (6) If you have 6+ variables, a correlation matrix in a table is more appropriate than listing all pairs in the text.

Think Deeper — Cross Questions

Discuss in pairs before sharing with the class.

Quick Check — Test Selection

For each research scenario, select the most appropriate statistical test.

Knowledge Check — Interactive Quiz

Test your understanding of quantitative statistics.

Lab Activity — Hands-On Statistical Analysis

Part A: Descriptive Statistics and Data Screening (45 min)

1. Import your dataset (or use the provided practice dataset) into SPSS, R, or Python.
2. Run descriptive statistics for all variables: N, mean, median, SD, min, max, skewness, kurtosis. For categorical variables: frequencies and percentages.
3. Screen for problems: Check min/max for impossible values. Identify missing data patterns. Run box plots to identify outliers. Check distributions with histograms and Shapiro-Wilk tests.
4. Document every screening decision: For each outlier found, note: (a) is it an error or a genuine value? (b) what did you do with it? For missing data: how much is missing, for which variables, and how will you handle it?

Part B: t-Tests and ANOVA on Practice Data (60 min)

Part C: Apply to Your Own Data (45 min)

If you have pilot data, secondary data, or a practice dataset that mirrors your expected capstone data: run the analyses specified in your methodology chapter. If you do not yet have data, use the provided practice datasets. Complete the Analysis Report Table:

Exit Ticket

Submit with your analysis output.

Key Takeaways — Week 13