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May 2026 Statistics I — Weeks 1–4 study guide
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May 2026 Statistics I — Weeks 1–4 study guide
1930 words
10 min read
2026-07-18
Reading compass
Now · Week 1 — data and descriptive questions
Statistics I, Weeks 1–4: data, uncertainty, and evidence
Statistics turns variable observations into defensible claims. The central habit is to name the population, sample, variable, and measurement scale before calculating anything.
Week 1 — data and descriptive questions
A population is the full group of interest; a sample is the observed subset. A variable may be categorical (labels), ordinal (ordered labels), discrete numerical (countable values), or continuous numerical (measurable values). The type determines which summaries are meaningful: averaging category labels is not an interpretation.
Descriptive statistics summarise observed data; inferential statistics reason beyond the observed data. Do not confuse a sample’s pattern with a guaranteed population truth.
Sampling checkpoint: watch an estimate move
An estimate is allowed to differ from the population parameter; that variation is the reason we describe uncertainty. The more dangerous failure is bias: a systematically unrepresentative selection can produce a stable-looking answer that targets the wrong group. Use the toy population below to compare ordinary random variation with a deliberately front-loaded selection.
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Retrieval prompt: a sample mean changes after a redraw. Is that evidence of a mistake, or an expected feature of sampling? State what would make you worry about the sampling method instead.
Week 2 — centre and spread
The mean uses every numerical value but moves strongly under outliers. The median is the middle ordered observation and is more resistant to extremes. The mode is the most frequent value and can be useful for categories. Range, interquartile range, variance, and standard deviation describe spread; no single measure is “the best” without knowing the distribution and question.
Retrieval prompt: create a five-value dataset where mean and median differ. Explain which summary you would report for incomes and why.
Week 3 — probability language
A sample space contains possible outcomes; an event is a subset of that space. Probability quantifies uncertainty on a scale from 0 to 1 under a model. Complement, union, intersection, conditional probability, independence, and mutual exclusivity are separate ideas. Mutually exclusive events cannot happen together; independent events do not change each other’s probabilities. Neither implies the other in general.
Use a table, tree, or set diagram to make the event relationship visible before applying a formula.
Week 4 — distributions and interpretation
A distribution describes how values are arranged: its shape, centre, spread, and unusual values. Histograms and box plots answer different questions; choose the representation that preserves the relevant structure. A skewed distribution has a long tail, and that tail explains why mean and median can tell different stories.
Always state units and the comparison group. A statistically correct calculation without context can still be a misleading conclusion.
The statistical language ladder
| Level | Terms | What the level lets you do |
|---|---|---|
| Source | population, sample, census, sampling frame | Say who the claim is about and who was actually observed. |
| Measurement | variable, observation, unit, category, scale | Decide whether a calculation is meaningful at all. |
| Description | frequency, proportion, centre, spread, shape | Summarise what the observed data look like. |
| Chance | outcome, sample space, event, complement | Model uncertainty explicitly rather than by intuition. |
| Interpretation | bias, association, causation, context | Keep a numerical result from becoming an overclaim. |
Measurement scales, precisely
- Nominal: categories have names but no natural order, such as course section. Frequencies and modes make sense; an average does not.
- Ordinal: categories are ordered, such as a satisfaction rating. Order is meaningful; equal spacing between labels is not automatically guaranteed.
- Interval: numerical differences are meaningful but zero is conventional, as with Celsius temperature.
- Ratio: numerical differences and ratios are meaningful because zero represents absence of the measured quantity, as with time or count.
This taxonomy is not academic decoration. It tells you whether “twice as much,” subtraction, or an average has an interpretable meaning.
Original worked examples — explain the number, not only the arithmetic
Example A: centre resists different kinds of distortion
Take the invented waiting-time data
4, 5, 5, 6, 30 minutes. The mean is (4 + 5 + 5 + 6 + 30) / 5 = 10; the median is 5. Neither number is inherently “correcter.” The mean answers a total-time-per-person question; the median better describes a typical ordinary visitor when one unusually long wait matters but should not dominate the story.Now replace
30 with 7. The mean becomes 5.4, close to the median 5. The change shows why distribution shape, rather than a memorised rule, should choose the summary.Example B: probability relationships before formulas
Suppose a bag holds cards labelled
A, A, B, C. Let event E be “the card is A” and F be “the card has a vowel label.” Here E is a subset of F; the events overlap. Draw the outcomes first, then count. A diagram prevents the frequent error of treating overlap as if the events were mutually exclusive.Example C: conditional probability changes the reference group
“What fraction of all students are in group B?” and “among students who submitted, what fraction are in group B?” have different denominators. The word given changes the population under discussion. Before writing
P(B | submitted), write the denominator in a sentence: only submitted students are now in view.Interactive event checkpoint
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Start at Independent. Its events overlap, so overlap alone cannot mean dependence. Switch to Mutually exclusive and notice that the intersection becomes empty. Finally choose Dependent overlap and compare
P(B) with P(B | A): the second value uses a reduced sample space because A is now known.Original retrieval task: Make an event A with two outcomes and an event B with one different outcome. Explain why they are mutually exclusive. Then add one shared outcome. Are they necessarily independent now? Use the displayed probabilities to justify your answer rather than the appearance of the tiles.
Distribution literacy: a visual reading protocol
When you see a histogram, dot plot, or box plot, read it in this order:
- Axis and units: What is measured? Are bins equal width? Is zero meaningful?
- Shape: Is there one mound, several clusters, symmetry, or a long tail?
- Centre: Where do typical observations sit? Does mean or median suit the shape?
- Spread: How widely do observations vary? Compare IQR for resistant spread and standard deviation for squared-distance spread.
- Exceptions: Are there outliers, data-entry mistakes, or an important second population?
A box plot reports quartiles compactly, but it does not reveal every cluster. A histogram reveals shape, but bin choices can change its apparent story. Choose the display to answer a question, not because a chart looks familiar.
Error taxonomy — how statistical claims go wrong
- Selection bias: the observed sample systematically differs from the population of interest.
- Measurement error: a variable is recorded inaccurately or with inconsistent definitions.
- Confounding: another variable explains part or all of an apparent association.
- Base-rate neglect: a vivid conditional statistic is treated as if it represented the overall population.
- Causal overreach: an association is described as a cause without a design that supports causation.
- Precision theatre: too many decimal places make a rough measurement appear more certain than it is.
Before calculating: identify the statistical object
A statistic is not just a formula result. First name the object being described.
- A population is the full group a question is about; a sample is the observed subset. A sample mean estimates a population mean but is not automatically equal to it.
- An observation is one measured unit; a variable is the characteristic measured on each unit. “Student” is not a variable; “weekly study hours” is.
- A parameter describes a population; a statistic describes a sample. This difference matters whenever you generalise beyond the records in front of you.
- A distribution records how values, categories, or probabilities are arranged. A centre alone cannot tell you whether observations are tightly clustered, skewed, or split into two groups.
Three useful non-examples
- The median is not “the middle value before sorting.” Order is part of the definition.
- A probability of
0.2is not a promise that exactly two of the next ten outcomes will occur. It is a modelled long-run proportion or degree of belief under stated assumptions. - Two variables changing together does not establish that one caused the other. A common cause or selection effect can produce the association.
Distribution reading protocol
Read a plot in this order: shape → centre → spread → unusual values → context. Shape asks whether the distribution is symmetric, skewed, uniform, or multimodal. Centre gives a typical location; spread tells you how representative that centre is. Only after those steps should you compare groups.
For example, two classes can both have a mean score of 70. If one class ranges from 68 to 72 while the other ranges from 20 to 100, “the typical score is the same” is technically true but educationally incomplete. The variation changes the interpretation.
Interactive distribution checkpoint
Interactive study checkpoint
Distribution reading lab
Change one property at a time. Read the curve as a model of a variable, not as a picture to memorise.
Model result
84.1%
of modelled observations are at or below x = 1.0.
value of xrelative densityμ 0.0x 1.0The shaded probability is P(X ≤ 1.0). The cut-off sits past the centre into the right side. This is a model probability—not a guarantee about the next individual observation.
Use the lab as a three-minute experiment, not as a formula machine.
- Start with Balanced, then increase the spread while leaving the centre and cut-off unchanged. Describe both changes: the curve becomes wider and its highest point falls. The area is still one whole model.
- Use Shifted group. The same cut-off now means something different because the reference distribution moved. Say the new reference group aloud before reading the percentage.
- Set the cut-off at the centre. For a symmetric normal model, the cumulative probability is about one half. That statement depends on the model's symmetry; it is not a rule for every real-world histogram.
Interpretation check: “84% are at or below 1” is incomplete. Finish it as: “Under this normal model, about 84% of observations of [named variable, with units] are at or below 1 [unit].” Name the variable and units before trusting a decimal.
Original retrieval lab
Can you make a sample statistic useful without copying the chapter?
Write from memory first. This saves only on this device and never declares an answer correct; it makes the gap visible so you know exactly what to reread.
Explain the difference between a population parameter and a sample statistic, why an estimate can vary, and what selection bias would change.
Your recall draft0/3 parts drafted
- Classify each as categorical or quantitative, then state a plausible measurement scale: course track, number of attempts, temperature in Celsius, and rank in a queue.
- Give a dataset where the mean is more informative than the median, and one where the median is safer. Name the feature that changes your choice.
- A study reports that 60% of commuters use the bus. Write two questions you must ask before treating that value as a population claim.
- Explain the reference group in: “Among students who submitted, 80% passed.” Why is it not automatically the pass rate for every enrolled student?
Your solution should always state the denominator and unit. “80%” is incomplete until you can finish the sentence “80% of what, measured how, and over which period?”
Four retrieval checks
- Classify the scale of a student’s rank, number of attempts, and temperature in Kelvin. Explain each choice.
- Invent a six-value dataset in which the median is more representative than the mean, and say why.
- Give one pair of mutually exclusive events and one pair of independent events. Explain why the labels are not interchangeable.
- Read a skewed distribution in one complete sentence that includes units, centre, spread, and the tail direction.