Probability & Statistics
Understand box plots, quartiles, interquartile range, and conditional probability to analyse data and chance events.
Box Plots & Quartiles
A box plot (box-and-whisker plot) is a visual summary of a data set showing five key values: the minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum.
Five-Number Summary
Min
Smallest value
Q1
25th percentile
Q2
Median (50th)
Q3
75th percentile
Max
Largest value
Interquartile Range (IQR) = Q3 − Q1. This measures the spread of the middle 50% of data.
Example: Data set: 3, 5, 7, 8, 12, 14, 18, 21, 25
Min = 3, Max = 25, Median (Q2) = 12 (middle value)
Q1 = 6 (median of lower half: 3, 5, 7, 8 → (5+7)/2 = 6)
Q3 = 19.5 (median of upper half: 14, 18, 21, 25 → (18+21)/2 = 19.5)
IQR = 19.5 − 6 = 13.5
Conditional Probability
Conditional probability is the probability of an event occurring given that another event has already occurred. We write it as P(A|B) — “the probability of A given B.”
Conditional Probability Formula
P(A|B) = P(A ∩ B) P(B)
Example: Two-way table
In a class of 30 students: 18 play sport, 10 play music, and 5 play both.
P(Music | Sport) = P(Music ∩ Sport) / P(Sport) = (5/30) / (18/30) = 5/18
So the probability a sport-playing student also plays music is 5/18 ≈ 0.278.
Independent & Dependent Events
Two events are independent if the occurrence of one does not affect the probability of the other. If P(A|B) = P(A), then A and B are independent.
Independent Events
P(A and B) = P(A) × P(B)
Example: Rolling a die then flipping a coin.
Dependent Events
P(A and B) = P(A) × P(B|A)
Example: Drawing cards without replacement.
Knowledge Check
Test your understanding of probability and statistics concepts.
Question 1
For the data set 4, 7, 9, 11, 15, 18, 22, what is the median?
Question 2
If Q1 = 12 and Q3 = 28, what is the interquartile range (IQR)?
Question 3
In a box plot, which part represents the middle 50% of the data?
Question 4
A bag has 5 red and 3 blue balls. You draw one ball, then draw another without replacement. What is P(both red)?
Question 5
P(A) = 0.4 and P(B) = 0.3. If A and B are independent, what is P(A and B)?
Question 6
In a two-way table, 60 out of 200 students study both Science and Maths. If 100 students study Maths, what is P(Science | Maths)?
Question 7
A box plot has a long whisker on the right side. What does this tell us about the data?
Question 8
P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.2. Are events A and B independent?
Question 9
For the data set 2, 5, 8, 10, 12, 15, 17, 20, what is Q1?
Question 10
A test has a 90% accuracy rate. If 5% of the population has a disease, what additional information do you need to find P(disease | positive test)?
Key Concepts Summary
- ●Box plots display the five-number summary: min, Q1, median, Q3, max.
- ●IQR = Q3 − Q1 measures the spread of the middle 50%.
- ●Conditional probability: P(A|B) = P(A ∩ B) / P(B).
- ●Events are independent if P(A ∩ B) = P(A) × P(B).
- ●Two-way tables and tree diagrams help organise conditional probability problems.