Unit 4 · Topic 4: Statistics and Probability
The Complete IB Mathematics: Analysis and Approaches Guide · IB Mathematics: Analysis and ApproachesUnit 4 · Topic 4: Statistics and Probability
- The Complete IB Mathematics: Analysis and Approaches Guide
- IB Mathematics: Analysis and Approaches
- 5 sections
International Baccalaureate does not publish a per-unit weighting for this exam. Exam structure (official): SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%); HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%). Be ready to carry the same unit skills and representations into a complete answer on every paper. Descriptive statistics summarize a dataset but do not automatically explain causation. Mean and standard deviation respond to scale and outliers; median and interquartile range answer different questions about center and spread.
- How IB Mathematics: Analysis and Approaches assesses this International Baccalaureate publishes no per-unit percentage for IB Mathematics: Analysis and Approaches; Paper 1 and Paper 2 each account for 40% of the SL total and 30% of the HL total. · SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%) | HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%) · show the model with a data table, histogram, cumulative-frequency graph, or box plot, a probability table or distribution curve, a conditional-probability table with the reference group marked
- Key skills Sampling, Histograms, Cumulative Frequency, and Box Plots, Measures of Central Tendency and Dispersion, Linear Correlation and Regression Lines, Probability: Combined Events, Conditional Probability, and Independence, Discrete Random Variables and the Binomial Distribution, The Normal Distribution, Standardization, and Inverse Normal, Bayes' Theorem (HL), Continuous Random Variables, Probability Density Functions, and Variance (HL)
- How to study for Unit 4 This page turns a data table, histogram, cumulative-frequency graph, or box plot, a probability table or distribution curve, a conditional-probability table with the reference group marked into one route: Identify the reference group, support, distribution conditions, and the statistic or probability the question requests..
- The organizing decision Define the population or random variable and verify the model conditions before calculating or interpreting a probability.
What IB Mathematics: Analysis and Approaches Unit 4 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Sampling, Histograms, Cumulative Frequency, and Box Plots
sampling; cumulative frequency; box plot02Measures of Central Tendency and Dispersion
mean; population standard deviation; interquartile range03Linear Correlation and Regression Lines
correlation; regression line; causation04Probability: Combined Events, Conditional Probability, and Independence
conditional probability; intersection; independence05Discrete Random Variables and the Binomial Distribution
random variable; binomial distribution; constant probability06The Normal Distribution, Standardization, and Inverse Normal
normal distribution; standardization; z-score07Bayes' Theorem (HL)
Bayes' theorem; conditioned group; prior probability08Continuous Random Variables, Probability Density Functions, and Variance (HL)
probability density; support; expectationInternational Baccalaureate does not publish a per-unit weighting for this exam.
Official unit name and weighting: IB subject guide / subject brief.
Data, chance, and model conditions
Connect the published share to the unit model
Descriptive statistics summarize a dataset but do not automatically explain causation. Mean and standard deviation respond to scale and outliers; median and interquartile range answer different questions about center and spread.
A probability model begins with a random variable and support. Binomial work checks fixed trials, two outcomes, constant probability, and independence; normal work standardizes or uses technology with stated bounds.
Conditional probability changes the reference group. Independence is a testable relation, while a continuous density must remain nonnegative and integrate to one before its expectation or interval probability is interpreted.
The decision that organizes this unit
Define the system and choose the route before calculating
Define the population or random variable and verify the model conditions before calculating or interpreting a probability.
Identify the reference group, support, distribution conditions, and the statistic or probability the question requests.
Mechanism route and repair branches
Relationships to preserve
- Center and spread describe different features of a dataset.
- Conditional probability changes the reference group used in the denominator.
- A continuous density must be nonnegative and have total area one before expectation is calculated.
Representations to read
- a data table, histogram, cumulative-frequency graph, or box plot
- a probability table or distribution curve
- a conditional-probability table with the reference group marked
Branches to reject
- claiming equal means imply equal distributions
- using a distribution command without checking its conditions
- conditioning on the entire sample instead of the named group
| Key concept | Why it's hard | What scores |
|---|---|---|
| Describe data | A single center can hide skew and outliers | Pairs center, spread, shape, and context |
| Choose a distribution | Conditions are often assumed rather than stated | Defines the variable and checks model conditions |
| Condition or integrate | The denominator or support may silently change | Shows the conditioned group or normalized density |
How IB Mathematics: Analysis and Approaches assesses Topic 4: Statistics and Probability
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Sampling, Histograms, Cumulative Frequency, and Box Plots | sampling; cumulative frequency; box plot | claiming equal means imply equal distributions |
| Measures of Central Tendency and Dispersion | mean; population standard deviation; interquartile range | using a distribution command without checking its conditions |
| Linear Correlation and Regression Lines | correlation; regression line; causation | conditioning on the entire sample instead of the named group |
| Probability: Combined Events, Conditional Probability, and Independence | conditional probability; intersection; independence | claiming equal means imply equal distributions |
| Discrete Random Variables and the Binomial Distribution | random variable; binomial distribution; constant probability | using a distribution command without checking its conditions |
| The Normal Distribution, Standardization, and Inverse Normal | normal distribution; standardization; z-score | conditioning on the entire sample instead of the named group |
| Bayes' Theorem (HL) | Bayes' theorem; conditioned group; prior probability | claiming equal means imply equal distributions |
| Continuous Random Variables, Probability Density Functions, and Variance (HL) | probability density; support; expectation | using a distribution command without checking its conditions |
Compare two datasets with center and spread
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Compare center and population spread
The listed scores are the complete populations for two groups.
Group A scores are 6, 7, 8, 9, 10. Group B scores are 4, 6, 8, 10, 12. Compare their means and population standard deviations, and state which group is more consistent.
Show your working.
Give standard deviations to three significant figures.
A. Both means are 8; σA=√2≈1.41 and σB=√8≈2.83, so Group A is more consistent.
✅ Correct! 😊 Correct. The groups have equal centers, but A's smaller population standard deviation shows that its scores are clustered more closely around 8.
B. Both means are 8, so the groups are equally consistent.
❌ Not quite 😭 Equal means compare center only. The visibly different squared deviations produce different spreads, so consistency cannot be inferred from the means alone.
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C. The population standard deviations are 2 and 8, so Group A is more consistent.
❌ Not quite 😭 The values 2 and 8 are population variances. Standard deviation requires taking their square roots, so the reported statistics are mislabeled.
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D. σA≈1.58 and σB≈3.16, so Group A is more consistent.
❌ Not quite 😭 These values divide by n−1 and use a sample convention. The prompt identifies the data as complete populations, so division by n is required.
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Full explanation
- Step 1Each group totals 40 across five scores, so both population means are 8.
- Step 2For A, squared deviations total 10; dividing by 5 gives variance 2 and σA=√2≈1.41.
- Step 3For B, squared deviations total 40; dividing by 5 gives variance 8 and σB=√8≈2.83.
- Step 4Group A is more consistent because its population standard deviation is smaller.
Practice 2: Condition on the correct reference group
In a survey of 100 students, 60 study art, 40 study music and 24 study both subjects.
Find P(art given music) and determine whether studying art and studying music are independent for this survey.
Show the probability equality used to test independence.
A. P(A|M)=24/40=0.6, and P(A∩M)=0.24=P(A)P(M)=0.6·0.4, so the events are independent.
✅ Correct! 😊 Correct. Conditioning restricts the denominator to music students, and the equality between the intersection and product verifies independence.
B. P(A|M)=24/100=0.24, and the events are dependent because some students study both.
❌ Not quite 😭 This uses the whole survey instead of the music group as denominator. An overlap can occur for independent events, so its existence does not prove dependence.
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C. P(A|M)=40/24, and the events are independent because 40 is less than 60.
❌ Not quite 😭 The conditional fraction is reversed and exceeds 1, so it cannot be a probability. Comparing group sizes is not an independence test.
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D. P(A|M)=24/40=0.6, but the events are dependent because P(A∩M)≠P(A)+P(M).
❌ Not quite 😭 The conditional probability is right, but independence uses a product, not a sum. Here 0.24 equals 0.6·0.4 exactly.
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Full explanation
- Step 1Conditioning on music makes the 40 music students the reference group.
- Step 2Therefore P(A|M)=24/40=0.6.
- Step 3Compute P(A∩M)=24/100=0.24 and P(A)P(M)=(60/100)(40/100)=0.24.
- Step 4Because these values are equal, studying art and music are independent in this survey.
Key terms for Unit 4: Topic 4: Statistics and Probability
Models, uses, and boundaries
- Mean
- The mean is the sum of the observations divided by the number of observations. Use it with a separate measure of spread when comparing datasets.
- Population spread
- Population standard deviation is the square root of the mean squared distance from the population mean. Use it when the example explicitly treats the listed values as the population.
- Binomial probability
- The probability of exactly six successes is 8 choose 6 times 0.7 to the sixth power times 0.3 squared. Use it after checking fixed independent trials, two outcomes, and constant success probability.
- Conditional probability
- The probability of French given music is the overlap probability divided by the probability of music. Use it after restricting the denominator to the music group.
- Continuous expectation
- Expected value is found by integrating x times the density over its support. Use it only after the density has been normalized to total area one.
IB Mathematics: Analysis and Approaches Unit 4 FAQ
How much of IB Mathematics: Analysis and Approaches does Unit 4 carry?
International Baccalaureate does not publish a per-unit weighting for this exam.
What is the first move on a Topic 4: Statistics and Probability problem?
Identify the reference group, support, distribution conditions, and the statistic or probability the question requests.
Which relationships should I preserve?
Center and spread describe different features of a dataset. Conditional probability changes the reference group used in the denominator. A continuous density must be nonnegative and have total area one before expectation is calculated.
Which representations should I practice?
Practice moving among a data table, histogram, cumulative-frequency graph, or box plot, a probability table or distribution curve, a conditional-probability table with the reference group marked.
What error should I check before submitting an answer?
Check for claiming equal means imply equal distributions; using a distribution command without checking its conditions; conditioning on the entire sample instead of the named group.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Compare two datasets with center and spread
- Use binomial and normal models with conditions
- Condition on the correct reference group
- Normalize a continuous density and find expectation
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
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International Baccalaureate does not publish a per-unit weighting for this exam.How to study IB Mathematics: Analysis and Approaches Unit 4
Start with the organizing decision
Before solving, restate the decision in operational terms: Define the population or random variable and verify the model conditions before calculating or interpreting a probability.. Your first written move should be to Identify the reference group, support, distribution conditions, and the statistic or probability the question requests..
Practice the same idea in several representations
Rotate through a data table, histogram, cumulative-frequency graph, or box plot, a probability table or distribution curve, a conditional-probability table with the reference group marked. Use each representation to practice Sampling, Histograms, Cumulative Frequency, and Box Plots, Measures of Central Tendency and Dispersion, Linear Correlation and Regression Lines, Probability: Combined Events, Conditional Probability, and Independence, Discrete Random Variables and the Binomial Distribution, The Normal Distribution, Standardization, and Inverse Normal, Bayes' Theorem (HL), Continuous Random Variables, Probability Density Functions, and Variance (HL), and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for claiming equal means imply equal distributions; using a distribution command without checking its conditions; conditioning on the entire sample instead of the named group. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official IB subject guide / subject brief for the 2027 administration.
Where to go next
Return to the IB Mathematics: Analysis and Approaches to see every unit and plan your revision.