Quantitative Data Analysis and Visualisation — Comprehensive Study Notes Summary & Study Notes

📊 Descriptive statistics — key concepts

Descriptive statistics summarise numerical data using measures of location and dispersion. Common summaries: mean, median, mode, variance, standard deviation, minimum, maximum, range, quartiles. Visual summaries include histograms and boxplots.

➗ Measures of central tendency and dispersion

• Sample mean: $\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i$. This is the arithmetic average and is sensitive to outliers.
• Median: the middle value when data are ordered. For even $n$, average the two middle values.
• Mode: the most frequent value(s).
• Sample variance: $S^2=\frac{1}{n-1}\sum_{i=1}^n (x_i-\bar{x})^2$. Use $n-1$ for an unbiased estimator of population variance.
• Sample standard deviation: $S=\sqrt{S^2}$.
• Range: $,\text{range}=\text{max}-\text{min}$. Gives overall spread but is sensitive to extremes.
• Quartiles (Q1, Q2, Q3) split the ordered data into four equal parts; Q2 is the median. These are used in boxplots.

📈 Boxplots and histograms — interpretation

• Histogram: shows the distribution shape and preserves the original data bins. Useful for seeing modality and approximate distribution form.
• Boxplot: highlights median, quartiles, whiskers (approximate spread), and outliers. Good for comparisons across groups.
• Skewness from plots: A longer left whisker or long left tail indicates left-skew, a longer right whisker/right tail indicates right-skew, and symmetric whiskers indicate approximate symmetry.

🧪 Hypothesis testing for difference of two means (large samples)

When comparing two independent groups with large sample sizes ($n\ge 30$), a z-test for difference of means can be used (using sample s.d. as estimates of population s.d.).

• Null hypothesis: $H_0:;\mu_1-\mu_2=0$ (no difference).
• Alternative (one-sided or two-sided) depending on the research question (e.g. $H_1:;\mu_1-\mu_2>0$ if expecting group 1 larger).
• Test statistic: $z=\displaystyle\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1-\mu_2)}{\sqrt{S_1^2/n_1+S_2^2/n_2}}$. Under $H_0$ take $\mu_1-\mu_2=0$.
• Decision: compare $z$ to critical value $z_{\alpha}$ (one-sided) or use p-value from standard normal.

Example (numbers from a practice problem): $n_1=n_2=40$, $\bar{x}_1=85$, $\bar{x}_2=78$, $S_1=12$, $S_2=10$. Compute $z=\frac{85-78}{\sqrt{12^2/40+10^2/40}}\approx 2.834$. For a one-sided test at $\alpha=0.05$, critical $z\approx 1.645$, so reject $H_0$ and conclude group 1 has larger mean.

⚖️ Interpreting test results

• If $|z|>z_{\alpha/2}$ (two-sided) or $z>z_{\alpha}$ (one-sided), reject $H_0$ at level $\alpha$.
• Report conclusion in context (e.g. "statistically significant evidence that young professionals spend more on groceries than retired individuals at the 5% level").

🔢 Chi-squared test for independence (contingency tables)

Use this to test association between two categorical variables (e.g. packaging type and spoilage).

• Observed table: cells $O_{ij}$, row totals and column totals.
• Expected counts under independence: $E_{ij}=\frac{(\text{row total}_i)(\text{col total}_j)}{n}$.
• Chi-squared statistic: $\chi^2=\sum_{i}\sum_{j} \frac{(O_{ij}-E_{ij})^2}{E_{ij}}$.
• Degrees of freedom: $(r-1)(c-1)$ for an $r\times c$ table.
• Decision: compare $\chi^2$ to critical value from $\chi^2_{\nu}$ or use p-value.

Example (numbers from the exam): Observed: Carboard spoiled 100, not spoiled 119 (row total 219); Plastic spoiled 200, not spoiled 153 (row total 353); column totals: spoiled 300, not spoiled 272; total $n=572$. Expected example: $E(\text{Carboard, spoiled})=\frac{219\times300}{572}\approx114.86$. Calculated $\chi^2\approx 6.55$ with $df=1$. Critical at $\alpha=0.05$ is $\approx3.841$, so reject independence — conclude an association between packaging and spoilage.

📉 Multiple linear regression — interpretation of R output

• Model form: $E(Y)=\beta_0+\beta_1 X_1+\beta_2 X_2+\dots$.
• Coefficient interpretation: each coefficient is the expected change in $Y$ for a one-unit increase in that predictor, holding others constant.
• From example R output: estimated model $E(\text{sales})=2.938889+0.045765(\text{TV})+0.188530(\text{radio})-0.001037(\text{newspaper})$.
• Significance: use t-statistics and p-values for each coefficient. Very small p-values (e.g. < 0.001) indicate strong evidence the coefficient differs from zero.
• Prediction: plug in predictor values into the fitted equation for a point prediction. Example: for TV=10, radio=10, newspaper=10, $E(\text{sales})=2.938889+0.045765\times10+0.188530\times10-0.001037\times10\approx 5.278889$ (thousand units).
• Residual: observed minus predicted. If observed sales = 10 (thousand), residual $=10-5.278889\approx 4.721111$ (thousand).
• R-squared: proportion of variance explained by the model. Higher $R^2$ indicates better fit but beware overfitting. Prefer simpler models if they explain similar variance (parsimony).

Model selection example: $R^2$ for TV+radio+newspaper = 0.8972 and for TV+radio = 0.8972. The additional predictor (newspaper) gives no improvement in $R^2$, so prefer the simpler TV+radio model.

🧮 Degrees of freedom and sample size from regression output

• In the linear model output, residual degrees of freedom = $n - k$ where $k$ is the number of parameters (including intercept). So $n=\text{residual df}+k$.
• Example: residual df = 196, $k=4$ (intercept + 3 predictors), so $n=196+4=200$ regions.

🧾 Useful formulas (formula-sheet style)

• Standard error of the sample mean: $\text{SE}(\bar{X})=\sqrt{S^2/n}$.
• Standard error for difference of two means: $\sqrt{S_1^2/n_1+S_2^2/n_2}$.
• Large-sample $100(1-\alpha)$ CI for mean: $\left(\bar{X}-z_{\alpha/2}\frac{S}{\sqrt{n}},;\bar{X}+z_{\alpha/2}\frac{S}{\sqrt{n}}\right)$.
• Hypothesis test for proportion uses standard error $\sqrt{\pi_0(1-\pi_0)/n}$ under $H_0$.
• Chi-squared statistic: $\chi^2=\sum\frac{(O_k-E_k)^2}{E_k}$.
• Logistic regression form: $\log\left(\frac{P(Y=1)}{P(Y=0)}\right)=\beta_0+\beta_1X_1+\dots+\beta_pX_p$.
• Poisson regression (with offset $t$): $\log(Y)=\log(t)+\beta_0+\beta_1X_1+\dots+\beta_pX_p$.

🔎 Reference z / t values (common quantiles)

• One-sided $\alpha=0.05$ critical: $z\approx 1.645$.
• Two-sided $\alpha=0.05$ critical: $z_{0.975}\approx 1.959964$.
• Useful R-derived values: $\text{qnorm}(0.95)\approx 1.644854$, $\text{qnorm}(0.975)\approx 1.959964$.

✅ Practical tips for exam-style problems

• Always state hypotheses in context and specify whether the test is one-sided or two-sided.
• Show formula, substitute numbers, compute the test statistic, state critical value or p-value, then give a contextual conclusion.
• For contingency tables, check expected counts (all should be reasonably large for the Chi-squared approximation to hold).
• For regression interpretation, comment on significance (p-values), sign and magnitude of coefficients, goodness-of-fit ($R^2$), and practical uncertainty (residuals, prediction intervals not just point predictions).

These notes summarise the core tools used in the sample exam: descriptive summaries and plots, z-tests for mean differences, Chi-squared tests for independence, and interpretation of multiple linear regression output.