Summarising data effectively: graphical methods and numerical summary measures, with the first hands-on Python set-up for scientific computing.

• S1Introduction to the course · Descriptive Statistics

  Course set-up and start of Topic 1 — describing one- and two-variable data.

  • Graphical methods. Stem-and-leaf displays and dotplots preserve raw values; histograms bin them to reveal shape; boxplots compress a batch to its five-number summary; scatterplots and contingency tables/bar plots expose relationships between two variables.
  • Numerical summaries. The mean is the balance point of the data; the median is its midpoint; the sample proportion is a mean of 0/1 indicators; the variance measures average squared spread.

  Sample mean $\bar x=\frac{1}{n}\sum_{i=1}^{n}x_i$  ·  sample variance $s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar x)^2$.

  Key ideaThe mean uses every value and is pulled by outliers; the median only depends on rank order and is robust. Comparing them is the quickest read on skew.

  Devore Ch.1 §1.1–1.4

  Populations & samples, pictorial/tabular displays, measures of location and of variability — the whole descriptive toolkit (S1–S3).

  ↳ demo: descriptive stats & boxplot
• S2Descriptive Statistics (cont.)

  Continues graphical methods and numerical summaries — location, spread and variance.

  • Five-number summary & boxplots. Minimum, $Q_1$, median, $Q_3$, maximum; the box spans the interquartile range and whiskers flag potential outliers (beyond $1.5\times\mathrm{IQR}$).
  • Spread. Variance vs standard deviation, and why $n-1$ (degrees of freedom) makes $s^2$ an unbiased estimator of $\sigma^2$.

  Interquartile range $\mathrm{IQR}=Q_3-Q_1$; an observation is a suspected outlier if $xQ_3+1.5\,\mathrm{IQR}$.

  WorkedFor $\{2,4,4,4,5,5,7,9\}$: $\bar x=5$, $s^2=\tfrac{1}{7}\big[(2{-}5)^2+\dots+(9{-}5)^2\big]=4.57$, so $s\approx2.14$. The median is $4.5$ — slightly below the mean, hinting at a right tail.

  Devore Ch.1 §1.1–1.4

  Same sections — focus shifts to §1.3–1.4 (measures of location and variability).

  ↳ demo: mean vs median
• S3Descriptive Statistics (cont.)

  Wrap-up of Topic 1: shape, skew and the relationship between measures of centre.

  • Shape & skew. Symmetric, right- (positive) and left- (negative) skewed distributions, and how tails drag the mean.
  • Centre under asymmetry. Right-skew typically gives mean $>$ median $>$ mode; the reverse for left-skew.

  Key idea"The mean chases the tail." Reporting median + IQR alongside mean + SD guards against a single summary misrepresenting a skewed batch.

  Devore Ch.1 §1.1–1.4

  Consolidation reading of Ch.1; revisit histogram shape descriptions in §1.2.

  ↳ demo: histogram, dotplot & skew
• S4Python Intro — Jupyter & Google ColabPython

  Python Topic 0/1 set-up: scientific computing environment and notebooks.

  • Notebooks. Cells, kernels and the read–eval–print workflow; running Python in the cloud with Google Colab (no local install).
  • Python basics. Variables, types, lists and functions — the vocabulary the rest of the track builds on.

  Key ideaA notebook interleaves code, output and prose, which makes statistical work reproducible and easy to narrate — the same habit professional data analysts rely on.

  Sundnes — Scientific Programming with Python

  Sundnes Ch.1–2: getting started, variables, loops and functions; the Colab "technical note".

• S5Python Topic 1 — Descriptive Statistics with PythonPython

  Implement descriptive statistics in code using NumPy.

  • NumPy arrays. Vectorised operations replace explicit loops; np.mean, np.median, np.std(ddof=1), np.percentile.
  • From formula to code. Re-deriving Topic 1's summaries on real arrays and checking them against hand calculations.

  Workedx = np.array([2,4,4,4,5,5,7,9]); x.mean() → 5.0; np.var(x, ddof=1) → 4.57 — matching the S2 example, now in one line.

  Sundnes — Scientific Programming with Python

  Sundnes Ch.5 (arrays & plotting with NumPy/Matplotlib) applied to descriptive statistics.

A mathematical framework for randomness and uncertainty — quantifying the likelihoods of outcomes.

• S6Probability — foundations

  Topic 2 begins: building the probabilistic model of an experiment.

  • Sample spaces & events. The sample space $S$ is the set of all outcomes; an event is a subset of $S$, combined with union, intersection and complement.
  • Axioms & properties. $P(A)\ge 0$, $P(S)=1$, countable additivity for disjoint events — yielding the complement and addition rules.

  Addition rule: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$; complement: $P(A')=1-P(A)$.

  Key ideaEverything else in the chapter is bookkeeping on top of three axioms. For equally-likely outcomes, probability collapses to counting: $P(A)=|A|/|S|$.

  Devore Ch.2 — all sections

  Ch.2 §2.1–2.2: sample spaces, events and the axioms/properties of probability.

  ↳ demo: permutations vs combinations
• S7Probability (cont.) — conditional & independence

  Conditional probability, the multiplication rule and independence in depth.

  • Conditioning. $P(A\mid B)$ rescales probability to the world in which $B$ has occurred; the multiplication rule reverses it.
  • Bayes & independence. Total probability partitions $S$; Bayes inverts conditioning; independence means $P(A\cap B)=P(A)P(B)$.

  $P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$  ·  Bayes: $P(A_i\mid B)=\dfrac{P(B\mid A_i)P(A_i)}{\sum_j P(B\mid A_j)P(A_j)}$.

  WorkedA test is 99% sensitive/95% specific for a disease with 1% prevalence. By Bayes, a positive result gives $P(\text{disease}\mid +)=\frac{.99(.01)}{.99(.01)+.05(.99)}\approx 0.17$ — the base rate dominates.

  Devore Ch.2 — all sections

  Ch.2 §2.4–2.5: conditional probability, Bayes' theorem and independence.

  ↳ demo: conditional probability & independence
• S8Probability (cont.) — counting & applications

  Counting techniques applied to probability computations; consolidation of Topic 2.

  • Counting. The product rule, permutations (order matters) and combinations (order ignored) feed equally-likely probability calculations.
  • Putting it together. Mixed problems combining counting, conditioning and independence.

  Permutations $P_{n,k}=\dfrac{n!}{(n-k)!}$; combinations $\binom{n}{k}=\dfrac{n!}{k!\,(n-k)!}$.

  Key ideaAsk "does order matter?" first. Same selection, two answers: $\binom{n}{k}$ if not, $P_{n,k}$ if so.

  Devore Ch.2 — all sections

  Ch.2 §2.3: counting techniques, applied back to §2.1–2.5.

  ↳ demo: counting

The two fundamental kinds of random variable — discrete then continuous — their mass and density functions, expectation, variance, and the canonical distribution families. Includes the midterm and the SciPy Python track.

• S9Discrete Random Variables — Topic 3 begins

  Definition of a random variable, the probability mass function, expectation and variance.

  • Random variable & PMF. A function mapping outcomes to numbers; the PMF $p(x)=P(X=x)$ assigns probability to each value, summing to 1.
  • Expectation & variance. $E[X]$ is the long-run average; $\mathrm{Var}(X)$ measures dispersion around it.

  $E[X]=\sum_x x\,p(x)$  ·  $\mathrm{Var}(X)=E[X^2]-(E[X])^2$  ·  $\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)$.

  Key ideaExpectation is linear ($E[aX+b]=aE[X]+b$) even when variables are dependent; variance is not, and shifts by $b$ leave it unchanged.

  Devore Ch.3 — all sections

  Ch.3 §3.1–3.3: random variables, PMFs/CDFs, expected values and variance.

  ↳ demo: binomial distribution
• S10Discrete RVs (cont.) — Bernoulli & Binomial

  The Bernoulli and Binomial models; mean $np$ and variance $np(1-p)$.

  • Bernoulli. A single success/failure trial with parameter $p$ — the atom of the Binomial.
  • Binomial. Number of successes in $n$ independent identical trials; PMF, mean, variance and mode.

  $X\sim\mathrm{Bin}(n,p)$: $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$, $E[X]=np$, $\mathrm{Var}(X)=np(1-p)$.

  Worked10 fair coin flips, $P(\text{exactly }6\text{ heads})=\binom{10}{6}(.5)^{10}=\tfrac{210}{1024}\approx0.205$; expected heads $=np=5$.

  Devore Ch.3 — all sections

  Ch.3 §3.4: the Binomial probability distribution.

  ↳ demo: binomial distribution
• S11Discrete RVs (cont.) — Hypergeometric & Negative Binomial

  Sampling without replacement and waiting-time counts.

  • Hypergeometric. Successes in a sample of $n$ drawn without replacement from a finite population of $N$ with $M$ successes.
  • Negative Binomial. Number of failures before the $r$-th success — a "waiting-time" count.

  Hypergeometric: $P(X=k)=\dfrac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}}$, mean $n\frac{M}{N}$.

  Key ideaThe Binomial assumes replacement (constant $p$); the Hypergeometric corrects for finite populations and approaches the Binomial when $N\gg n$.

  Devore Ch.3 — all sections

  Ch.3 §3.5: the Hypergeometric and Negative Binomial distributions.

• S12Discrete RVs (cont.) — Poisson

  The Poisson model for rare-event counts; mean $=$ variance $=\lambda$.

  • Poisson process. Counts of events in a fixed interval at constant rate, independent of disjoint intervals.
  • Binomial limit. Arises as $n\to\infty$, $p\to0$ with $np\to\lambda$.

  $X\sim\mathrm{Pois}(\lambda)$: $P(X=k)=\dfrac{e^{-\lambda}\lambda^k}{k!}$, $E[X]=\mathrm{Var}(X)=\lambda$.

  WorkedA call centre averages $\lambda=3$ calls/min. $P(\text{0 calls in a minute})=e^{-3}\approx0.050$; expecting equal mean and variance is the Poisson signature.

  Devore Ch.3 — all sections

  Ch.3 §3.6: the Poisson distribution and the Poisson process.

  ↳ demo: poisson distribution
• S13Discrete RVs (cont.) — consolidation

  Comparing discrete families and choosing the appropriate model.

  • Model selection. Matching a real scenario to the right family by its generating mechanism (fixed $n$? replacement? counts over time?).
  • Moments review. Side-by-side expectation/variance across the discrete families.

  Key ideaIdentify the family from the story, not the numbers: "successes in $n$ fixed trials" → Binomial; "rare counts per interval" → Poisson; "draws without replacement" → Hypergeometric.

  Devore Ch.3 — all sections

  Whole-chapter review of Ch.3 ahead of the midterm.

• S14Review session — Topics 1, 2 & 3Review

  Consolidation ahead of the midterm: descriptive statistics, probability, discrete RVs.

  FocusMixed problems that stitch the three topics together — e.g. summarising simulated data (T1), conditioning (T2), then identifying the generating discrete model (T3).

• S15Midterm ExamExam · 20%

  Closed-book, covering Topics 1–3 (no Python). Laptop with proctoring + handheld calculator; one double-sided A4 formula sheet allowed. Date tentative — may shift to S16/S17.

  Covers Devore Ch.1–3. No GenAI permitted. Bring: proctored laptop, simple calculator, one A4 (two-sided) formula sheet.

• S16Continuous Random Variables — Topic 4 begins

  PDFs and CDFs; expectation and variance for continuous variables.

  • Density & distribution functions. A PDF $f(x)\ge0$ integrates to 1; probabilities are areas under it, and $P(X=x)=0$ for any single point.
  • CDF. $F(x)=P(X\le x)$ is the running area, with $f=F'$.

  $P(a\le X\le b)=\int_a^b f(x)\,dx=F(b)-F(a)$  ·  $E[X]=\int x f(x)\,dx$.

  Key ideaFor continuous variables, only intervals carry probability. The CDF is the bridge between density and probability and is what software returns.

  Devore Ch.4 — all sections

  Ch.4 §4.1–4.2: probability density functions, CDFs, and expected values for continuous variables.

  ↳ demo: continuous PDF & CDF
• S17Continuous RVs (cont.) — Normal distribution

  The normal density, standardisation and the 68–95–99.7 rule.

  • Standardisation. Any normal becomes the standard normal $Z\sim N(0,1)$ via the $z$-score, so one table/function serves all.
  • Empirical rule. ~68% / 95% / 99.7% of mass within 1 / 2 / 3 standard deviations.

  $z=\dfrac{x-\mu}{\sigma}$  ·  $f(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\,e^{-(x-\mu)^2/2\sigma^2}$.

  WorkedIQ $\sim N(100,15^2)$. $P(X>130)=P(Z>2)\approx0.0228$ — the upper ~2% tail, straight from the empirical rule.

  Devore Ch.4 — all sections

  Ch.4 §4.3: the normal distribution, the standard normal and normal-table use.

  ↳ demo: normal & the empirical rule
• S18Continuous RVs (cont.) — Exponential & Gamma

  Continuous families beyond the normal; their densities and moments.

  • Uniform & Exponential. Constant density on an interval; the Exponential models waiting times and is memoryless.
  • Gamma & Lognormal. Sums of exponentials (Gamma); a variable whose log is normal (Lognormal), common for multiplicative/positive data.

  Exponential: $f(x)=\lambda e^{-\lambda x}$, $E[X]=1/\lambda$, $\mathrm{Var}(X)=1/\lambda^2$ for $x\ge0$.

  Key ideaThe Exponential is the continuous partner of the Poisson: if events arrive as $\mathrm{Pois}(\lambda)$, the gaps between them are $\mathrm{Exp}(\lambda)$.

  Devore Ch.4 — all sections

  Ch.4 §4.4: the Exponential and Gamma distributions (plus the Uniform and Lognormal cases).

  ↳ demo: continuous PDF & CDF
• S19Python Topic 2 — Computing probabilities with SciPyPython

  Working with distributions in SciPy.

  • scipy.stats objects. A common interface across distributions: .pmf/.pdf, .cdf, and the inverse CDF .ppf (quantile function).
  • Replacing tables. Computing the S17 IQ probability or a critical value without a printed normal table.

  Workedfrom scipy.stats import norm; 1 - norm.cdf(130, 100, 15) → 0.0228; norm.ppf(0.975) → 1.96.

  Sundnes — Scientific Programming with Python

  Sundnes (NumPy/SciPy material): PMF/PDF, CDF and inverse-CDF evaluation in scipy.stats.

• S20Continuous RVs (cont.)

  Further work with continuous distributions and their applications.

  • Applied practice. Reliability, lifetimes and measurement-error problems across the continuous families.
  • Quantiles. Reading percentiles and critical values off CDFs.

  Key idea"Probability vs quantile" are inverse questions: the CDF turns a value into a probability; its inverse (the ppf) turns a probability back into a value.

  Devore Ch.4

  Ch.4 applied problems; revisit §4.3–4.4.

• S21Continuous RVs (cont.) — consolidation

  Wrap-up of Topic 4 and model selection for continuous data.

  • Model selection. Symmetric/bell-shaped → Normal; positive and right-skewed → Lognormal/Gamma; waiting times → Exponential.
  • Bridge to inference. Why the Normal recurs as a limit (sets up the CLT in Module 4).

  Key ideaMatch the support and shape to the data: a model defined on $[0,\infty)$ can never describe a quantity that goes negative.

  Devore Ch.4

  Whole-chapter review of Ch.4.

Working with several random variables at once, then the distributions of statistics — the foundation for inference. Closes with Monte Carlo, the computer exam and the final exam.

• S22Joint Probability Distributions — Topic 5

  Working simultaneously with two or more random variables.

  • Joint, marginal & conditional. A joint PMF/PDF describes pairs $(X,Y)$; marginals recover one variable; independence means the joint factors.
  • Covariance & correlation. Covariance measures co-movement; correlation $\rho$ rescales it to $[-1,1]$.

  $\mathrm{Cov}(X,Y)=E[XY]-E[X]E[Y]$  ·  $\rho=\dfrac{\mathrm{Cov}(X,Y)}{\sigma_X\sigma_Y}$.

  Key idea$\rho$ captures only linear association: independence implies $\rho=0$, but $\rho=0$ does not imply independence.

  Devore Ch.5 §5.1–5.2

  Ch.5 §5.1 jointly distributed RVs; §5.2 expected values, covariance and correlation.

  ↳ demo: covariance, correlation & least squares
• S23Statistics & their Distributions — Topic 6 begins

  Statistics as random variables and the distribution of the sample mean.

  • A statistic is random. Because it is computed from a random sample, a statistic such as $\bar X$ has its own sampling distribution.
  • Sample mean. Its centre and spread in terms of the population $\mu$ and $\sigma$.

  $E[\bar X]=\mu$  ·  $\mathrm{Var}(\bar X)=\sigma^2/n$  ·  standard error $=\sigma/\sqrt{n}$.

  Key ideaAveraging shrinks variance by a factor of $n$: the standard error falls like $1/\sqrt{n}$, so quadrupling the sample halves the spread.

  Devore Ch.5 §5.3–5.5

  Ch.5 §5.3: statistics and their sampling distributions.

  ↳ demo: Central Limit Theorem
• S24Statistics & their Distributions (cont.) — sample mean

  The sampling distribution of $\bar X$ and the Central Limit Theorem.

  • Central Limit Theorem. For large $n$, $\bar X$ is approximately normal regardless of the population's shape.
  • Rule of thumb. $n\ge30$ is usually enough unless the population is very skewed.

  $\bar X \;\xrightarrow{\;n\to\infty\;}\; N\!\left(\mu,\dfrac{\sigma^2}{n}\right)$.

  WorkedRoll a die ($\mu=3.5,\sigma^2\approx2.92$) $n=50$ times. $\bar X$ is ≈$N(3.5, 0.0583)$, so $P(\bar X>4)\approx P(Z>2.07)\approx0.019$ — even though one roll is uniform, not normal.

  Devore Ch.5 §5.3–5.5

  Ch.5 §5.4: the distribution of the sample mean and the Central Limit Theorem.

  ↳ demo: Central Limit Theorem
• S25Statistics & their Distributions (cont.) — linear combinations

  Distribution of a linear combination of random variables.

  • Mean & variance of $\sum a_i X_i$. Means always add linearly; variances add (with cross terms) and pick up the squared weights.
  • Normal stays normal. A linear combination of independent normals is itself normal.

  $E\big[\sum a_i X_i\big]=\sum a_i E[X_i]$; if independent, $\mathrm{Var}\big[\sum a_i X_i\big]=\sum a_i^2\mathrm{Var}(X_i)$.

  Key ideaThe $a_i^2$ in the variance is why differences are as variable as sums: $\mathrm{Var}(X-Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)$ for independent $X,Y$.

  Devore Ch.5 §5.3–5.5

  Ch.5 §5.5: the distribution of a linear combination.

  ↳ demo: Law of Large Numbers
• S26Statistics & their Distributions (cont.) — consolidation

  Wrap-up of Topic 6 and connections to inference.

  • LLN vs CLT. The LLN says $\bar X\to\mu$ (a point); the CLT describes the shape of $\bar X$'s fluctuations around $\mu$.
  • Why it matters. These results justify confidence intervals and hypothesis tests in later courses.

  Key ideaLLN = convergence of the average; CLT = the distribution of the error of that average. Together they are the engine of statistical inference.

  Devore Ch.5 §5.3–5.5

  Whole-section review of Ch.5 §5.3–5.5.

  ↳ demo: Law of Large Numbers
• S27Python Topic 3 — Monte Carlo simulationPython

  Approximating probabilities by simulation.

  • Random number generation. Sampling from distributions with NumPy's default_rng; seeding for reproducibility.
  • Monte Carlo estimation. Approximating a probability or expectation as the fraction/average over many simulated draws.

  Workedrng = np.random.default_rng(0); (rng.standard_normal(1_000_000) > 2).mean() → ≈0.0228, the same tail as S17 — now by simulation, with error shrinking like $1/\sqrt{N}$ (the LLN at work).

  Sundnes — Scientific Programming with Python

  Sundnes (random numbers chapter): RNG and Monte Carlo approximation of probabilities.

  ↳ demo: Law of Large Numbers
• S28Review session — Topics 1–6Review

  Comprehensive review ahead of the computer and final exams.

  FocusEnd-to-end problems: describe data, model it with the right distribution, reason about a statistic's sampling distribution, and verify by simulation.

• S29Computer ExamExam · 20%

  Open-book exam: solving and discussing exercises on Python Topics 1, 2 & 3 (descriptive stats, SciPy probabilities, Monte Carlo). GenAI tools (e.g. Colab's Gemini) permitted but not required.

  Open-book; GenAI allowed but neither required nor sufficient for full marks. Assessed on correct, well-explained code and sound reasoning.

• S30Final ExamExam · 30%

  Closed-book, comprehensive over Topics 1–6 (no Python). Laptop with proctoring + handheld calculator; two double-sided A4 formula sheets allowed. Minimum 3.5/10 required to pass the course.

  Covers Devore Ch.1–5. No GenAI. Bring: proctored laptop, simple calculator, two A4 (two-sided) formula sheets. Below 3.5/10 fails the course outright.