MATH1041 Descriptive Statistics – Lecture 2 Lecturer Annotated

Lecture 1 – Numerical & Graphical Summaries for One Variable

Administrative / Context

• Chapter 2 of MATH1041 “Descriptive Statistics” → 4 lectures

• Overarching aim of course: collecting, analysing, interpreting data

• Textbook mapping: Moore et al., 2021, Sections 1.2–1.3

• RStudio is the compulsory software (install + watch MarinStatsLectures videos). Other packages (SAS, SPSS, Excel, …) mentioned but not used.

Learning outcomes (Ch2-L1-O1 … O4)

• Describe a variable numerically & graphically

• Recognise that choice depends on variable type (categorical vs quantitative)

• Comment on graphs (centre, spread, shape, outliers)

• Detect outliers & understand their impact

Variable Types & Strategy Table (single variable perspective)

Categorical → table  + bar chart
Quantitative → mean/SD OR five-number summary + histogram/boxplot

• Decide summary after checking type & number of variables

Statistical Distribution (Def 2.1)

1. Set of possible values (or categories)

2. Corresponding counts / frequencies / proportions (for quantitative: counts per class interval)

• Replacing raw data with distribution loses labels/individual values but yields concise overview

Categorical Variables

Numerical Summary

• Frequency table (Def 2.2): list of categories + counts/percentages

• R: table(x) → optionally wrap with prop.table() ×100 for %

• Examples: Watching Australian Open (Yes/No) ; Mode of transport ; Hemisphere of origin

Graphical Summary

• Bar chart (vertical or horizontal). Order bars logically to aid readability (e.g., descending frequency)

• R base: barplot(table(catVar))

Quantitative Variables

Why not frequency table? (Slide 2.26)

• Potentially infinite distinct values ⇒ table would be huge

Key Features to Capture

• Location (centre)

• Spread (variability)

• Optional: shape (skew, modality)

Measures of Location

• Mean ˉ=n∑i=1nx

• Sensitive to outliers (e.g.

  Gina Rinehart in bar → average wealth skyrockets)

• Median $M =\begin{cases}x<em>{((n+1)/2)} &amp; n \text{ odd}\ \frac{x</em>{(n/2)}+x_{(n/2+1)}}{2}&amp; n \text{ even}\end{cases}$ (Def 2.4)

• Robust to outliers (e.g.

  engineer salary example)

• Guidance: use median when distribution skewed / heavy-tailed

Measures of Spread

• Quartiles (Def 2.5): $Q<em>1$ & $Q</em>3$ are medians of lower/upper halves (median included if n odd, consistent with R)

• Inter-quartile range $IQR = Q<em>3-Q</em>1$ (Def 2.6) – robust

• Standard deviation $s=\sqrt{\frac{\sum<em>{i=1}^{n}(x</em>i-\bar x)^2}{n-1}}$ (Def 2.7) – sensitive to outliers

• Variance $s^2$ (SD squared)

Five-Number Summary

1. Min

2. Q1

3. Median (Q2)

4. Q3

5. Max

• R: summary(x)

• Graphical counterpart: boxplot

Outliers

• Suspected if beyond $Q<em>1-1.5\times IQR$ or $Q</em>3+1.5\times IQR$ (1.5×IQR rule)

• Show as points on “modified boxplot” (default R behaviour)

Graphical Tools

• Histogram: good for shape; choose appropriate bin width (breaks=)

• Boxplot: concise five-number; useful for comparing groups

• Bar plot for discrete quantitative (narrow bars)

• Comment on graph in terms of location, spread, shape, outliers

R Cheat-sheet (one variable)

# numbers
mean(x);  sd(x);  median(x)
quantile(x,c(.25,.5,.75));  IQR(x)
summary(x)
# graphs
hist(x); boxplot(x)

Lecture 2 – Relationships Between Two Variables

Learning outcomes (Ch2-L2-O1 – O3)

1. Pick correct tools when ≥1 variable categorical

2. Describe scatterplot (presence, strength, direction, type)

3. Understand Pearson correlation, its derivation, pros/cons

Summary Table (two variables)

Both categorical → 2-way frequency table + clustered bar chart
Cat + Quant → 5-num (or mean/SD) per group + comparative boxplots / clustered bars
Both quantitative → correlation + scatterplot

Case 1: ≥1 Categorical Variable

• Strategy: split data by categories, then apply single-variable methods within each subgroup, finally compare

• Two-way table: table(cat1,cat2)

• Comparative boxplots: boxplot(y ~ cat)

Case 2: Both Quantitative → Scatterplot

• Axes: X – explanatory (if any), Y – response

• Describe nature using 4 × dichotomies1. Existent

  non-existent

  2. Strong

     weak

  3. Increasing (+)

     decreasing (–)

  4. Linear

     non-linear

• Outliers may signal data error or interesting structure (example: Palm Beach 2000 election)

Pearson Correlation (Def 2.8)

• Standardise each variable $x^\bullet<em>i=\frac{x</em>i-\bar x}{s<em>x},\; y^\bullet</em>i=\frac{y<em>i-\bar y}{s</em>y}$

• Coefficient $r=\frac{\sum<em>{i=1}^{n}x^\bullet</em>i\,y^\bullet_i}{n-1}$

• Properties: $-1\le r\le1$; sign = direction; magnitude = linear strength

• Cautions- Measures linear association only (Anscombe, NBA points vs age)

  • Sensitive to outliers (interactive examples pushing r from 0.1 to 0.99)

• Non-linear dependence? → use alternative measures (e.g.

  Hellinger correlation)

R Cheat-sheet (two vars)

plot(y ~ x)           # scatterplot
cor(x,y)              # Pearson r
by(y,cat,summary)     # num summaries by group
barplot(table(cat1,cat2),beside=TRUE)  # clustered bars

Lecture 3 – Least-Squares Regression (Simple Linear)

Learning outcomes (Ch2-L3-O1 – O4)

• Distinguish explanatory (X) vs response (Y)

• Understand construction & meaning of least-squares line

• Use line for prediction; interpret slope & intercept

• Recognise danger of extrapolation

Model Form

$\hat y = b<em>0 + b</em>1 x$

• $b_1$ (slope): change in $\hat y$ for one-unit increase in x

• $b_0$ (intercept): predicted y at $x=0$ (may be nonsensical)

Least-Squares Criterion

• Choose $b<em>0,b</em>1$ to minimise $\sum<em>{i=1}^{n}(y</em>i-\hat y_i)^2$ (sum of squared residuals)

• Historical: Gauss/Laplace (~1800)

Interpretation Examples

• Ski-jump scores: $\hat y=110.44+1.16x$ ⇒ every extra point in round 1 adds $\approx$1.16 points total

• Snake length–weight example for visual intuition

Prediction & Extrapolation

• Prediction: plug new x within data range → $\hat y$

• Extrapolation: using x outside observed range; may produce impossible results (US farmer population → negative)

R Workflow

fit <- lsfit(x,y)        # returns list with coefficients
coef(fit)                # b0,b1
plot(x,y); abline(coef(fit))
# predict
x.new <- 137
y.pred <- coef(fit)[1] + coef(fit)[2]*x.new

Lecture 4 – Residuals & $r^2$

Learning outcomes (Ch2-L4-O1 – O4)

1. Always visualise data (Anscombe’s quartet)

2. Use residual plots to assess linear model assumptions

3. Compute & interpret $r^2$ (strength of regression)

4. Handle outliers; concepts of leverage & influence

Residuals (Def 2.9)

$e<em>i = y</em>i - \hat y<em>i = y</em>i - (b<em>0+b</em>1x_i)$

• Provide diagnostic info; measured in same units as y

Residual Plot (Def 2.10)

• Scatter of $e<em>i$ vs $x</em>i$ (or vs $\hat y$)

• Ideal pattern: random cloud centred at 0 (no curvature, equal spread)

• Systematic structure (arches, funnels) ⇒ violation of linearity/constant variance

• Example: Janka hardness shows arch → switch to quadratic model

Coefficient of Determination $r^2$

• For simple linear regression with intercept: $r^2 = r^2$ (square of Pearson correlation)

• Interpretation: proportion of variance in y explained by model

  $</p><p>r^2 = \frac{\text{Var}(\hat y)}{\text{Var}(y)}</p><p>$

• Range 0–1 (report as %). Example: temperature explains 81 % of variation in log electricity use

Alternative formula (standardised data)

$</p><p>r^2 = 1 - \frac{\sum (y^\bullet<em>i-\hat y^\bullet</em>i)^2}{n}</p><p>$

⇒ closer points to fitted line in standardised scale → higher r²

Outliers, Leverage, Influence

• Residual large → vertical outlier

• Leverage: extreme x value

• Influential: point whose inclusion markedly changes fit

• Handling rules-of-thumb:- If outlier doesn’t affect results → keep

  • If clear data error / atypical case → justify removal & report both analyses

• Examples: Mitsubishi Lancer ad (high price), Lift capacity dataset

Summary of Regression Diagnostics Workflow

1. Plot data, compute r

2. Fit least-squares line (within appropriate x-range)

3. Plot residuals; look for non-random patterns & equal variance

4. Calculate $r^2$; gauge practical usefulness

5. Investigate outliers/leverage; consider alternative models if needed

Global R Command Reference

# One variable
mean(x); sd(x); median(x); IQR(x)
summary(x); table(cat)
barplot(table(cat)); hist(x); boxplot(x)

# Two variables
plot(y~x)                 # scatter
cor(x,y)                  # correlation r
lsfit(x,y)                # simple linear regression
residuals(lsfit(x,y))     # residuals
abline(lsfit(x,y))
by(y,cat,summary)         # summaries by category

# Misc
quantile(x,c(.25,.75))
IQR(x)

Key Terminology

• Statistical distribution, frequency table

• Mean $\bar x$, Median $M$, Quartiles $Q<em>1,Q</em>3$, IQR, SD $s$, Variance $s^2$

• Five-number summary

• Bar chart vs Histogram vs Boxplot

• Shape: symmetric, left-skew, right-skew; modality (uni/bi)

• Outlier (1.5×IQR rule)

• Scatterplot, Positive/Negative association, Strength, Linearity

• Pearson correlation $r$

• Explanatory vs Response variable

• Least-squares regression line (intercept $b<em>0$, slope $b</em>1$)

• Prediction; Extrapolation

• Residuals, Residual plot

• $r^2$ (coefficient of determination)

• Leverage, Influence

Ethical & Practical Considerations

• Data collection quality dictates validity; flawed collection → flawed study

• “Association ≠ Causation” (observational vs experimental designs)

• Randomisation, comparison, replication principles for experiments

• Always inspect data graphically before trusting numerical summaries

• Report methodology transparently (e.g., treatment of outliers, model assumptions)