IB Diploma Mathematics: Analysis & Approaches SL · Statistics & Probability

Mini-Lesson

Statistics & Probability

This mini-lesson covers SL Statistics & Probability: data & sampling, measures of centre and spread, correlation & regression, probability (including conditional probability and tree diagrams), discrete random variables, and the binomial and normal distributions.

Answer as you go and collect ⭐ stars. Press Start when ready.

Data

Sampling & types of data

Statistics starts with data collected from a sample of a population. Data may be discrete (counted), continuous (measured) or categorical.

Good sampling: random, systematic, stratified or quota methods aim to represent the population fairly and avoid bias.

Quick check

Complement

?The probability it rains tomorrow is 0.3. What is the probability it does not rain?

Centre & spread

Measures of centre and spread

Summarise data with a centre and a spread:

Mean = sum ÷ count; median = middle value; mode = most frequent.
Spread: range = max − min; interquartile range = Q₃ − Q₁; standard deviation σ.

Worked example

Data 4, 8, 6, 10, 2: mean = 30/5 = 6; ordered 2,4,6,8,10 gives median = 6.

Calculate

Mean

✎Find the mean of the data set 4, 8, 6, 10, 2.

Hint: (4 + 8 + 6 + 10 + 2) ÷ 5 = 30 ÷ 5.

Correlation

Correlation & regression

Bivariate data may show a linear trend. Pearson's r measures the strength and direction of linear correlation, with −1 ≤ r ≤ 1.

The least-squares regression line y on x is used to predict y from x — reliable only within the data range (interpolation).

Quick check

Independence

?If events A and B are independent, then P(A ∩ B) equals:

Calculate

Tree diagram

✎A bag holds 3 red and 2 blue counters. Two are drawn without replacement. Find P(both red).

Hint: (3/5) × (2/4) = 6/20.

Probability

Probability basics

For equally likely outcomes, P(A) = favourable ÷ total, and 0 ≤ P(A) ≤ 1.

P(A′) = 1 − P(A)the complement; P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Sort it

What kind of data is it?

Tap a variable, then its data type.

🔢 Discrete

📏 Continuous

🏷️ Categorical

Conditional

Combined & conditional probability

Independent events: P(A ∩ B) = P(A)P(B). Otherwise use conditional probability:

P(A | B) = P(A ∩ B) / P(B)

Worked example — tree diagram

Bag of 3 red, 2 blue, draw two without replacement: P(both red) = (3/5)(2/4) = 6/20 = 0.3.

Calculate

Expected value

✎X ~ B(20, 0.25). Find E(X).

Hint: E(X) = np = 20 × 0.25.

Quick check

Correlation range

?Pearson's correlation coefficient r always satisfies:

Random variables

Discrete random variables

A discrete random variable X takes values with probabilities that sum to 1. Its expected value (mean) is:

E(X) = Σ x·P(X = x)the long-run average outcome

Calculate

Binomial probability

✎X ~ B(10, 0.3). Find P(X = 2) to 3 decimal places.

Hint: ₁₀C₂ (0.3)² (0.7)⁸ = 45 × 0.09 × 0.7⁸.

Match it

Match each statistic to its meaning

Tap a statement on the left, then its matching partner on the right.

Statistic

Meaning

Distributions

Binomial & normal distributions

Binomial X ~ B(n, p): n independent trials, two outcomes, constant p.

P(X = r) = ₙCᵣ pʳ(1−p)ⁿ⁻ʳ · E(X) = np

The normal distribution is continuous, symmetric and bell-shaped. Standardise with z = (x − μ)/σ, then use the standard normal to find probabilities.

Calculate

Normal probability

✎X ~ N(50, 5²). Find P(X < 55) to 3 decimal places. (z = (55−50)/5 = 1.)

Hint: standardise to P(Z < 1) and read the standard normal.

Quick check

Is it binomial?

?Which situation is modelled by a binomial distribution?

Strategy

Reading the question

Identify the data type and whether events are independent. For "at least" or "at most" binomial questions use the complement. For normal problems, sketch the bell and shade.

Exam habit: a probability must lie in [0, 1] and Pearson's r in [−1, 1] — check before writing your answer.

Recap

The big ideas to know

Data: discrete / continuous / categorical; sample fairly

Centre & spread: mean, median, mode; range, IQR, σ

Probability: P(A′)=1−P(A); P(A|B)=P(A∩B)/P(B)

Binomial: P(X=r)=ₙCᵣpʳ(1−p)ⁿ⁻ʳ, E(X)=np

Normal: standardise z=(x−μ)/σ then use the bell

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