Statistics foundations: A Practical Course with Ines Ben Youssef

Start with the purpose

Place Statistics foundations inside a worked problem where the setup matters as much as the answer. Name the result a learner is trying to produce and the constraints that make the skill useful.

How Statistics foundations actually works

These are the load-bearing ideas. Everything later in the course is an application of one of them, so it is worth reading slowly and returning to when something stops making sense.

• Mean and median answer different questions. The mean uses every value and is dragged by outliers, while the median is the middle value and resists them. For skewed data such as income or response times, the median describes a typical case far better than the mean does.
• Sample variance divides by n - 1. Dividing squared deviations by n underestimates the population variance, because the deviations were measured from the sample mean rather than the unknown true mean. Dividing by n - 1 instead removes that bias.
• A p-value is not the probability a hypothesis is true. A p-value gives the probability of observing data at least as extreme as yours assuming the null hypothesis holds. It says nothing about the size or importance of an effect, which is why an effect size and confidence interval should always sit beside it.

Misconceptions worth clearing early

Each of these is common, understandable, and expensive to leave in place. Recognising them now saves rework later.

• Reading correlation as causation. A strong correlation coefficient feels like an explanation of the mechanism. Fix: Correlation is equally consistent with reverse causation or a shared confounder. Only a controlled experiment or a carefully designed natural experiment supports a causal claim.
• Comparing means without checking spread. Two averages are trivially easy to place side by side. Fix: Report the standard deviation or an interval alongside them. A difference of 2 carries no weight when the spread within each group is 30.
• Sign errors when rearranging. Moving several terms in one written step hides the operation being applied. Fix: Apply one operation to both sides at a time and write the step down, even when it feels obvious.

Build the mental model

Connect the key terms as a process rather than a word list. Use this sequence: represent the information, choose a method, calculate carefully, and verify the result.

Catch the common miss

Compare a surface-level attempt with one that shows a correct setup, visible steps, checked units or assumptions, and a clear interpretation. Explain the single difference that matters most.

Watch the whole process

Trace a model from the initial prompt to a worked solution set with a reasoning note. Mark each point where the learner must observe, choose, or verify rather than act automatically.

Worked example: Positive predictive value of a highly accurate test

Follow each step and predict the next before you read it. Predicting first is what turns a demonstration into practice.

1. Assume a condition affects 1% of people, the test detects 99% of true cases, and it wrongly flags 5% of healthy people.
2. Among 10,000 people, 100 have the condition and 99 of those test positive.
3. Of the 9,900 without it, 5% test positive, which is 495 false positives.
4. Total positives are 99 + 495 = 594, so the chance that a positive result is genuine is 99/594, about 16.7%.

When the base rate is low, most positive results are false no matter how impressive the accuracy figure sounds on its own.

Where this usually goes wrong

Watch for these while you work through the demonstration rather than afterwards.

• The gambler's fallacy. A long run of one outcome feels like it is owed a correction. Fix: Independent trials have no memory. A fair coin that has landed heads nine times still lands heads next with probability one half.
• Confusing the two directions of a conditional. The two read almost identically when stated in English. Fix: Write both out with explicit populations. The chance a positive test means disease is very different from the chance a diseased person tests positive.
• Sign errors when rearranging. Moving several terms in one written step hides the operation being applied. Fix: Apply one operation to both sides at a time and write the step down, even when it feels obvious.

Practice with scaffolding

Repeat the model with one detail changed. Keep the prompts visible and say or write the reason for each choice before continuing.

Check before feedback

Use a correct setup, visible steps, checked units or assumptions, and a clear interpretation as the quality test. Make one self-correction before asking the tutor to review the result.

Read the situation

Translate the scenario into a clear task. Separate facts, assumptions, constraints, and information that is interesting but not relevant to Hypothesis testing.

Choosing well under real constraints

Applied work is mostly judgement under limits: less time, less information, and more competing goals than a textbook example allows. These are the decision rules that hold up in practice.

• Several methods could solve the problem: Choose the one whose setup you can state clearly; a slower method you understand beats a faster one you half-recall.
• You are stuck at the setup: Solve a smaller version with easy numbers first, then generalise the structure you used.
• The answer looks wrong but the arithmetic checks out: Recheck the translation from words to symbols; the error is usually upstream of the calculation.

Reading the situation before acting

Before choosing an approach, state three things explicitly: what result the situation actually requires, which constraints are fixed rather than preferences, and what evidence would tell you the approach is working. Skipping this step is the most common reason competent work solves the wrong problem.

• A thesis must be contestable. A working thesis is a sentence an informed reader could reasonably dispute; without a possible counter-position it is a summary. Social media affects teenagers is a topic, whereas platform design rather than screen time drives the observed harm is a thesis.

Practitioner notes

Small pieces of working knowledge that rarely appear in introductory material.

• Keep a single thesis line at the top of the file and rewrite it at the start of every session. When the sentence stops changing, the argument has settled.
• If the thesis needs a semicolon and three clauses, it is probably two arguments. Choose the one you actually have evidence for.

Compare real options

Generate two workable approaches and test both against the purpose. Do not hide the tradeoff; name what each option improves and what it gives up.

Make the reasoning visible

Produce a worked solution set with a reasoning note and attach a short decision note. The note should make the result auditable, not merely confident.

Use the rubric, not a feeling

Review the work for a correct setup, visible steps, checked units or assumptions, and a clear interpretation. Record evidence for each judgment so feedback points to something observable.

Diagnostic checklist

Run this before you revise anything. Diagnosing first prevents the common failure of polishing the parts that were already fine.

• Check: Reading correlation as causation — is this present in your work?
• Check: Comparing means without checking spread — is this present in your work?
• Check: Sign errors when rearranging — is this present in your work?
• Check: Dropping or mixing units — is this present in your work?

The quality bar

This is what finished work looks like in this field. Use it as the standard for your revision rather than a general sense of improvement.

• The setup states what each symbol means, including units
• Steps are visible enough for a reader to find the exact point of any disagreement
• The result is checked against the original problem and interpreted in context

Diagnose before editing

Name the symptom, then ask what decision or missing step produced it. Choose the cause you can address rather than changing everything at once.

Revise and compare

Make one purposeful revision and compare the two versions. Keep the change only if it improves the intended result without creating a larger problem.

Define a finishable brief

Choose a specific audience, result, and boundary for the Quantitative literacy capstone. Reduce scope until the project can be finished and reviewed in one focused cycle.

Bringing the parts together

A capstone is judged on coherence, not on the number of techniques it includes. Return to the core ideas and make sure the work demonstrates them rather than decorating them.

• Mean and median answer different questions. The mean uses every value and is dragged by outliers, while the median is the middle value and resists them. For skewed data such as income or response times, the median describes a typical case far better than the mean does.
• Sample variance divides by n - 1. Dividing squared deviations by n underestimates the population variance, because the deviations were measured from the sample mean rather than the unknown true mean. Dividing by n - 1 instead removes that bias.
• A p-value is not the probability a hypothesis is true. A p-value gives the probability of observing data at least as extreme as yours assuming the null hypothesis holds. It says nothing about the size or importance of an effect, which is why an effect size and confidence interval should always sit beside it.

Standards that make the work credible

These are the marks of work that would be taken seriously by someone who does this professionally.

• The setup states what each symbol means, including units
• Steps are visible enough for a reader to find the exact point of any disagreement
• The result is checked against the original problem and interpreted in context

Practitioner notes

Small pieces of working knowledge that rarely appear in introductory material.

• Plot the raw data before computing anything. Anscombe's quartet is four datasets with matching means, variances and regression lines but four completely different shapes.
• State the sample size next to every percentage. 60% of 5 respondents and 60% of 5000 respondents carry entirely different weight.

Build with checkpoints

Plan foundation, first draft, verification, and revision milestones. At each checkpoint, save evidence instead of relying on memory.

Present and continue

Present a worked solution set with a reasoning note with a concise rationale. Use the final rubric to choose one strength to retain and one next practice target.