Statistical & Mathematical Methods for Data Science · Chapter 3

Statistics — A Gentle Introduction, taught visually

Everything in PPT 3, rebuilt so you can see it: what statistics actually is, the statistician's three hats, the scientific-method loop, how to spot shady science, the new vocabulary (IV, DV, confounders), the four levels of measurement — and the chapter's real calculation core: rounding, Σx, x̄, and Σx² vs (Σx)². Every definition is paired with a worked example, and the case scenarios are long and multi-part, the way a real exam case is.

Definitions — each with an example Case scenarios — descriptive, multi-part Calculations — Σ, x̄, rounding drills

What is statistics? One sentence, two jobs

The chapter's definition is short enough to memorise word-for-word — and it hides two jobs: organising information, and then communicating what it means. Exam definitions are usually marked against this exact wording.

The definition — learn it verbatim

“Statistics is a way to organise information to make it easier to understand what the information might mean.”

And its second job: statistics provides a conceptual understanding so results can be communicated to others in a clear and accurate way. Organising without communicating is only half the subject.

Example: A teacher has 120 quiz scores in a spreadsheet. As a raw list they mean nothing. The moment she sorts them, counts how many fall in each band, graphs them and finds the average, the list has been organised — and when she tells the class “most of you scored between 60 and 80, the average was 71,” she has communicated it. Both halves are statistics.

🧮 Organise

Sort, count, tabulate, graph and summarise raw numbers until a pattern becomes visible.

Example: Turning 10,000 hospital admission records into a table of admissions per month.

🔍 Understand

Ask what the organised information might mean — note the word “might.” Statistics deals in evidence, not certainty.

Example: Admissions spike every winter — the pattern suggests a seasonal cause worth investigating.

📣 Communicate

Present the finding clearly and accurately enough that someone else can act on it — or challenge it.

Example: A one-page report with a chart a hospital manager can understand in two minutes.

Exam trap: If a scenario says a researcher “ran the numbers and kept the results in a drawer,” that is not complete statistics by this chapter's definition — the communication half is missing. And if an answer choice says statistics “proves” something with certainty, it is wrong: the definition says what information might mean.

What is a statistician? Three hats, one person

The chapter describes the statistician through three characters. Tap each hat — examiners love asking which role a scenario describes, and the unscrupulous attorney is the favourite trap.

The Curious Detective — examines the crime scene

Examines the crime scene. The crime scene is the experiment — the detective-statistician starts by studying how the data was produced.

Looks for clues. The data from the experiment are the clues. No clues, no case.

Develops suspicions about the culprit. Questions (hypotheses) formed at the scene determine how the questions will be answered.

Remains sceptical. Relies on sound clues (good statistics) and information from the scene — not the “fad” of the day.

Example: A café owner notices sales dip every Tuesday. Like a detective she first inspects the “scene” (the till records), gathers clues (footfall, weather, a rival's Tuesday discount), forms a suspicion (“the rival's discount is the culprit”), and stays sceptical instead of blaming the first trendy explanation she reads online.

The Honest Attorney — builds an argument from the facts

Examines the facts of the case. Examines the data: Is the data sound? What might the data mean?

Creates a legal argument using the facts. Tries to come up with a reasonable explanation for what happened.

Cross-examines his own argument. Is there another possible explanation? Do the data actually support the hypothesis?

The dark twin — the unscrupulous or naive attorney: either by choice or by lack of experience, the data are manipulated or forced to support the hypothesis. Worst case: ignoring disconfirming data, or making the data up. If a scenario mentions cherry-picked results or deleted inconvenient rows, this is the character being described.

Example: A researcher argues a new study app works. The honest version reports all results — including the group where it didn't help — and asks whether motivation, not the app, explains the gain. The unscrupulous version quietly drops that group from the report.

The Good Storyteller — makes it publishable

For findings to be published, they must be put together in a clear, coherent manner that relates four things:

What happened? — the study and its design.

What was found? — the results, honestly reported.

Why is it important? — the significance of the finding.

What does it mean for the future? — implications and next steps.

Example: “We surveyed 800 students (what happened). Late-night phone use was linked to lower quiz scores (what was found). Universities could use this in student wellbeing programmes (why it matters). A follow-up experiment should test whether limiting phone use actually improves scores (what it means for the future).”

All statistics are not alike

Two separate distinctions live under this heading — people (conservative vs liberal statisticians) and jobs (descriptive vs inferential statistics). Keep them apart; the exam mixes them deliberately.

Conservative statistician

Uses the tried and true methods; prefers conventional rules and common practices.

Advantage: more accepted by peers and journal editors
Advantage: guards against chance influencing the findings
Disadvantage: new statistical methods are avoided

Example: A reviewer insists on the standard, decades-old test for a dataset “because that is what the field trusts” — safe, respected, but blind to anything the old test can't see.

Liberal statistician

More likely to use new statistical methods; willing to question convention.

Advantage: may discover previously undetected changes, causes and relationships
Disadvantage: more difficulty getting findings accepted by publishers and peers

Example: A researcher applies a brand-new technique to old climate data and spots a pattern the standard methods missed — then spends a year convincing sceptical editors it is real.

Descriptive statistics

Describing the information you actually have: how many (frequency), what it looks like (graphing), what types (tables) — its parameters.

Example: “In our class of 40, the average quiz score was 71, scores ranged from 38 to 97, and this bar chart shows the distribution.” Nothing is claimed about any other class.

Inferential statistics

Making educated guesses (inferences) about a large group (population) based on what we know about a smaller group (sample).

Example: Polling 1,200 randomly chosen voters to estimate how all 2 million voters in a city will vote. The claim reaches beyond the people actually measured — that leap is inference.

Exam trap — the one-word test: if the statement only talks about the group that was measured, it is descriptive. The moment it generalises to people who were never measured, it is inferential. “Sample” and “population” in the same sentence is the loudest trigger phrase in the chapter.

Definition flip-cards — each with an example

Front: the chapter's own wording — try to say the example out loud before you flip. Back: a worked example plus the trap that costs marks. Tap a card to flip it.

The science of science — the scientific method loop

Science builds explanations that are consistent and predictive, rather than changing, reactive and biased. The method is a repetitive cycle, not a straight line. Tap each stage to see what happens there.

Theory

An idea or set of ideas that attempt to explain an important phenomenon. Theories are generated from observations — we ask “why might this happen?” and “is there something that consistently happens given a set of initial conditions?”

Examples from the slides: theories of behaviour; the theory of relativity. How do we know a theory is any good? It produces hypotheses — and if the hypotheses are supported through experiments, we put more belief in the theory.

Hypothesis

Usually generated by a theory. It states what is predicted to happen as a result of an experiment or event, in the form: “I think X will happen as a result of Y” — or “If Y occurs, then X will result.”

Example: Theory: sleep consolidates memory. Hypothesis: “If students sleep 8 hours before a test (Y), then they will recall more words than students who sleep 4 hours (X).”

Research

Provides the investigator an opportunity to examine an area of interest and/or manipulate circumstances to observe the outcome — its job is to test a theory or hypothesis.

Example: Actually running the sleep study: two groups, different sleep durations, same word-recall test the next morning.

Observations (the results)

The results of an experiment. Observations can do three things: support or detract from a theory, suggest revision of a theory, or generate a new theory — which is why the loop never ends.

Example: The 8-hour group recalls 18 words, the 4-hour group 11 — supporting the theory. If both recalled 15, the theory would need revision.

Exam trap — order matters: observations → theories → hypotheses → research → new observations. A hypothesis comes from a theory and is tested by research; a lone guess with no theory behind it, or a “hypothesis” written after seeing the results, breaks the cycle the chapter describes.

Why do we need statistics? To be better consumers of information

Statistics helps us understand what was observed, communicate what was found, make an argument, answer a question — and above all, judge other people's claims. The chapter gives an eight-question checklist for any study you are quoted. Each question below carries the failure it is designed to catch.

1 · Who was surveyed or studied?

Are the participants like you or your interest group — or were they all men, all one ethnic group, all twenty-somethings? If not, the result may not transfer to you.

2 · Why did they participate?

Just for the money? Paid a lot — which can bend performance and ratings? Desperate for a cure? Did they have something to prove? Motive contaminates answers.

3 · Was there a control group — and a placebo?

If not, how do you know the treatment worked? Did participants know they received it — or was the change just the placebo effect, the belief in the treatment?

4 · How many people participated?

Too few → a real difference may be missed. Too many → almost any tiny difference gets “detected,” even one that isn't meaningful. Both extremes mislead.

5 · How were the questions worded?

Does the wording point at the “expected” answer? Does it accurately reflect what is being studied? Is it at the right level for the participant?

6 · Was causation assumed from correlation?

Most media-reported studies are correlational (relationships only) but reported as if they were experiments (causation). A link is not a cause.

7 · Who paid for the study?

Does the funder have a reason to want a particular result — a pharmaceutical company, a political party, a specific interest group?

8 · Was it peer-reviewed?

Peer-reviewed journals examine submissions more rigorously. A claim in the New England Journal of Medicine and the same claim in the National Enquirer do not carry equal weight.

How the exam uses this: a scenario describes a study and asks “what should a critical consumer ask?” — the answer is never “nothing, it was published.” Pick the checklist item the scenario is quietly violating: the funder, the missing control group, the loaded wording, or correlation dressed up as causation.

Good science vs shady science

Good science is a toolkit of safeguards. Shady science is what happens when the safeguards are skipped — and then dressed up as fact.

🎲 Random sample

Participants are randomly drawn from the population — everyone has an equal chance of being selected. Keeps the sample representative.

Example: Selecting voters from the full electoral roll by lottery, not by asking whoever walks past your office.

🔀 Random assignment

Participants are randomly placed into groups, so the groups start out equivalent and differences can be blamed on the treatment — not on who ended up where.

Example: Flipping a coin to decide which patients get the new drug and which get the standard one.

💊 Placebo + control group

A control group gets a placebo (an inactive look-alike), so improvement from believing you were treated can be separated from improvement caused by the treatment itself.

Example: Half the patients receive a sugar pill identical to the real drug; neither group knows which.

🙈 Double-blind

Neither the participants nor the experimenters know who is in which group — so expectations on both sides can't steer the results.

Example: The nurse handing out pills and the patient taking them both only see a code number.

📏 Large, representative sample

The sample should be relatively large — able to detect differences — and representative of the population it claims to describe.

Example: Surveying 2,000 citizens across all provinces to describe a country, not 30 students from one classroom.

🧹 Minimising confounders

Good design actively minimises confounding variables — the extra factors that could explain the result instead of the variable you care about.

Example: Testing a reading programme while holding teacher, class time and materials constant across groups.

The shady hall of fame — the chapter's own examples

🧠 “We use only 10% of our brains”

A famous claim repeated for decades with no sound study behind it. Shady because it survives on repetition, not evidence — ask checklist questions 7 and 8: who profits from it, and where was it peer-reviewed?

📺 News “surveys”

A TV channel asks viewers to text in their opinion. The respondents are self-selected — angry or enthusiastic people answer, everyone else doesn't. Not a random sample, so it cannot describe the public.

🎤 Does American Idol pick America's favourite?

Fans can vote many times; non-viewers never vote; the “sample” is whoever felt strongly enough to text. The winner is the favourite of motivated voters, not of America. Same flaw as the news survey, bigger stage.

Random sample ≠ random assignment: sample = who gets into the study (drawn from the population); assignment = which group they land in once inside. A study can have one without the other — a random sample split non-randomly by choice is still broken, and volunteers randomly assigned are still volunteers.

Learning a new language — the words change meaning

“The words sound the same, but it is a whole new game.” In statistics, significance is not everyday importance, a variable is something surprisingly stable, and error is not a mistake. These are the terms the exam defines.

Independent variable (IV)

Manipulated by the experimenter, related to the topic of curiosity, and expected to influence the dependent variable. The IV is the cause you control.

Example: In a sleep study, hours of sleep (4 vs 8) is the IV — the researcher sets it.

Dependent variable (DV)

Measured in the study; the topic of curiosity; it changes as a result of exposure to the IV. The DV is the effect you record.

Example: In the same study, words recalled on the test is the DV — it depends on the sleep the participant got.

Confounding variable — “the fly in the ointment”

An uncontrolled extra variable that changes along with the IV and offers its own explanation for the DV. Who is in control — experimental control (designing it away) or statistical control (accounting for it in analysis)?

Example: The 8-hour group also drank coffee before the test. If they recall more, was it the sleep or the caffeine? Caffeine is the confounder — the fly in the ointment.

Operational definition

Defining a concept by exactly how it will be measured in this study — “what are you talking about?” answered precisely enough that anyone could repeat the measurement.

Example: “Memory” is vague. Operationally: “the number of words correctly recalled from a 20-word list after 10 minutes.” Now it can be measured, checked and replicated.

Error — not a mistake

In statistics, error is the recognition of measurement imperfection. Every measurement wobbles a little; error names that wobble honestly instead of pretending the number is perfect. Its sources include the participant and the study conditions.

Example: A student scores 72 today and 75 tomorrow on the same test — she didn't get smarter overnight; mood, sleep and room noise are measurement error.

Control — two kinds

Experimental control: holding conditions constant by design. Statistical control: measuring the nuisance variable and removing its effect mathematically. Design first, statistics second.

Example: Testing all students in the same quiet room = experimental control. Recording each student's caffeine intake and adjusting for it = statistical control.

The IV/DV memory hook: the experimenter manipulates the Independent variable; the Dependent variable depends on it. If a scenario has no manipulation at all (just observing and measuring), there is no true IV — it is a correlational study, and causation cannot be claimed.

Types of variables — how it can be measured matters

Two sorting systems run in parallel: the kind of data (quantitative vs qualitative) and the shape of the values (discrete vs continuous). A variable gets one label from each system.

Quantitative data

Data values that are numeric — amounts you can add, average and compare in size.

Example from the slides: a math anxiety score. Also: height, income, marks, temperature. Arithmetic on these numbers means something.

Qualitative data

Data values that can be placed into distinct categories according to some characteristic — typically either/or. The values name kinds, not amounts.

Examples from the slides: eye colour, hair colour, gender, types of foods and drinks. You can count how many fall in each category — you cannot average “blue” and “brown.”

Discrete variable

What is measured belongs to unique and separate categories — there are gaps between possible values; nothing exists in between.

Example from the slides: pets — dog, cat, goldfish, rat. Also: number of children (2 or 3, never 2.5). If there are only two categories, the variable is called dichotomous: open/closed, male/female, pass/fail.

Continuous variable

What is measured varies along a line scale and can take all values between any two given values — no gaps; finer measurement is always possible.

Examples from the slides: length, temperature, age, distance, time. Between 20°C and 21°C live 20.5, 20.55, 20.555… forever.

Exam trap — money and counts: “number of students in class” is discrete (you can't have 24.6 students) even though it is numeric and quantitative. And a qualitative variable written with numbers (jersey numbers, ID codes) is still qualitative — the numbers are just labels. Ask: does halfway between two values make sense? If not, it isn't continuous.

Levels of measurement — the NOIR staircase

Four levels, each adding one new power: Nominal names, Ordinal adds rank, Interval adds equal gaps, Ratio adds a true zero. Tap each step — every higher level keeps everything below it.

🎮 Sorter game — which level is it?

Ten variables, four levels. Choose the level for each one; the explanation tells you why — that reasoning is what the exam marks.

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Calculation lab — rounding, Σ, x̄, and the great square trap

“Getting our toes wet” is the chapter's calculation core, and it is pure exam marks: four symbols, four routines. Learn the order of operations for each — the last one is designed to catch you.

Rounding — look one digit past

To round, inspect the digit immediately after the place you keep: less than 5 → go down (keep the digit as it is); greater than 5 → go up (add 1). A bare 5 is conventionally rounded up.

Worked: 51.356 → nearest whole: the next digit is 3 (< 5) → 51. 15.73 → next digit 7 (> 5) → 16. 7.667 → one decimal: next digit 6 → 7.7.

Σx — sigma means “add them all”

Σ (sigma) indicates that the group of numbers will be added together. Nothing more exotic than that — but everything else in this section is built on it.

Worked (from the slides): x = 3, 78, 32, 15 → Σx = 3 + 78 + 32 + 15 = 128.
Your turn: x = 7, 33, 10, 19 → Σx = 69 · x = 62, 21, 73, 4 → Σx = 160.

x̄ (x-bar) — the mean

Add all the data points together (Σx), then divide by the number of data points (N): x̄ = Σx ÷ N.

Worked (from the slides): x = 3, 12, 6, 5, 11, 15, 1, 7 → Σx = 60, N = 8 → x̄ = 60 ÷ 8 = 7.5.
Your turn: x = 3, 7, 1, 4, 4, 2 → Σx = 21, N = 6 → x̄ = 3.5 · x = 28, 36, 22, 40, 34, 29 → Σx = 189, N = 6 → x̄ = 31.5.

Σx² vs (Σx)² — order is everything

Σx²: square each number first, then add. (Σx)²: add all the numbers first, then square the sum. Same ingredients, different order, very different answers — they are not the same.

Worked (from the slides): x = 2, 4, 6, 8 → Σx² = 4 + 16 + 36 + 64 = 120. x = 5, 7, 2, 3 → (Σx)² = (17)² = 289.
Your turn: x = 1, 3, 5, 7 → Σx² = 84 · x = 4, 3, 9, 1 → Σx² = 107 · x = 7, 7, 3, 2, 5 → (Σx)² = (24)² = 576 · x = 3, 8, 1, 2 → (Σx)² = (14)² = 196.

⚠️ The head-to-head the chapter ends on: x = 4, 3, 2, 1

Σx² (square first, then add):
16 + 9 + 4 + 1 = 30

(Σx)² (add first, then square):
(4 + 3 + 2 + 1)² = (10)² = 100

Same four numbers — 30 vs 100. In the exam, read the brackets before you touch the calculator: the brackets in (Σx)² are the instruction. Squaring early keeps numbers small; squaring the sum lets every number multiply with every other.

🧮 Sigma lab — type any numbers, watch all four statistics build

Your data (comma-separated)

Σx — sum

–

x̄ — mean (Σx ÷ N)

–

Σx² — square each, then add

–

(Σx)² — add, then square the sum

–

Σx²

(Σx)²

🎯 Rounding drill — the slide's own numbers

Apply the rule: digit after the cut is < 5 → down, > 5 → up, 5 → up.