π Statistics & Probability
Statistics and Probability questions test your ability to organise data, calculate measures of central tendency (mean, median, mode), compute standard deviation, draw and interpret statistical diagrams, and solve probability problems using tree diagrams and Venn diagrams. These are among the most scorable topics in Paper 2, often worth 8-10 marks per question.
Statistics and probability follow very predictable patterns in the exam. If you learn the formulas and practise the standard question types, you can score full marks consistently. The key is to show all working clearly β especially for mean and standard deviation calculations, where partial marks are awarded for correct method even if the final answer has a computational error.
Mean, Median, Mode
Mean is the arithmetic average: sum of all values divided by the count. For frequency tables, use . For grouped data, use mid-values as estimates for each class interval. Median is the middle value when data is arranged in ascending order. Mode is the most frequently occurring value.
πMeasures of Central Tendency
Mean, median, and mode for ungrouped and grouped data.
πWorked Example 1: Mean & Standard Deviation from Frequency Table
Goals scored in 20 matches: 0 (3 times), 1 (7 times), 2 (5 times), 3 (4 times), 4 (1 time). Find the mean and standard deviation.
πWorked Example 2: Mean from Grouped Frequency Table
Marks: 10-19 (4 students), 20-29 (8 students), 30-39 (12 students), 40-49 (6 students). Find the estimated mean.
πWorked Example 1b: Finding Missing Value from Mean
The mean of 6 numbers is 15. Five of the numbers are 12, 18, 14, 11, 20. Find the sixth number.
πWorked Example 1c: Combined Mean
Class A has 20 students with mean mark 65. Class B has 30 students with mean mark 75. Find the combined mean for both classes.
πWorked Example 1d: Effect on Mean When Adding/Removing Values
The mean of 10 numbers is 24. (a) If each number is increased by 5, find the new mean. (b) If each number is multiplied by 3, find the new mean. (c) If a new number 34 is added, find the new mean.
The mean of 5 numbers is 12. If one number (8) is removed, what is the new mean?
Standard Deviation
Standard deviation (SD) measures how spread out the data is from the mean. A small SD means data points are clustered close to the mean; a large SD means they are spread out. The computational formula is generally easier to use than the definitional formula because it avoids calculating each deviation separately.
πMeasures of Spread
Standard deviation and interquartile range to measure how spread out data is.
πWorked Example 3: SD from Raw Data
Find the standard deviation of the data set: 3, 5, 7, 8, 12.
πWorked Example 3b: SD from Grouped Frequency Table
Heights (cm): 150-155 (4 students), 155-160 (10 students), 160-165 (16 students), 165-170 (8 students), 170-175 (2 students). Find the estimated mean and standard deviation.
πWorked Example 3c: Effect of Linear Transformation on SD
A data set has mean 50 and standard deviation 8. Each value is transformed using y = 2x + 5. Find the new mean and standard deviation.
πWorked Example 3d: Comparing Two Data Sets
Test A: mean = 65, SD = 12. Test B: mean = 70, SD = 5. Compare the two tests.
A data set has mean 40 and SD 6. Each value is multiplied by 2 then 10 is added. What is the new SD?
Cumulative Frequency & Box Plots
A cumulative frequency curve (ogive) is drawn by plotting the upper boundary of each class interval against the running total of frequencies. It is used to estimate the median, quartiles, and percentiles. A box-and-whisker plot displays the five-number summary: minimum, Q1, median, Q3, maximum.
πStatistical Diagrams
Types of statistical representations required for O-Level E-Math.
Reading a Cumulative Frequency Curve
Median
Draw horizontal line at on the y-axis. Read across to the curve, then read down to the x-axis.
Lower quartile (Q1)
Draw horizontal line at on the y-axis. Read across and down.
Upper quartile (Q3)
Draw horizontal line at on the y-axis. Read across and down.
Interquartile range (IQR)
. Measures the spread of the middle 50% of the data.
Percentile
To find the th percentile, read at on the y-axis.
πWorked Example 4: Cumulative Frequency
For 80 students, the cumulative frequency curve is drawn. Find the median and interquartile range.
πWorked Example 4b: Drawing a Box-and-Whisker Plot
Minimum = 20, Q1 = 35, Median = 50, Q3 = 65, Maximum = 90.
πWorked Example 4c: Comparing Two Box Plots
School A: Min=30, Q1=45, Med=55, Q3=70, Max=90. School B: Min=40, Q1=60, Med=65, Q3=72, Max=80. Compare the test results.
Histograms and Frequency Density
A histogram is used to display continuous data with unequal class intervals. Unlike a bar chart, the area of each bar (not the height) represents the frequency. The y-axis shows frequency density, calculated as frequency divided by class width.
πWorked Example 4d: Frequency Density
The table shows ages of visitors. 0-10 (20 visitors), 10-25 (45 visitors), 25-30 (30 visitors), 30-50 (40 visitors). Draw a histogram.
From a cumulative frequency curve of 80 values, at what CF do you read the median?
Probability
Probability measures how likely an event is to occur, ranging from 0 (impossible) to 1 (certain). For equally likely outcomes, where n(A) is the number of favourable outcomes and n(S) is the total number of outcomes. You need to master tree diagrams for multi-stage experiments and the complement rule for βat least oneβ problems.
π²Basic Probability
Fundamental probability concepts and rules.
πCombined Events
Addition and multiplication rules for combined events. Tree diagrams and possibility diagrams.
Probability Rules Summary
AND Rule (Multiplication)
for independent events. On a tree diagram, multiply along the branches.
OR Rule (Addition)
for mutually exclusive events. On a tree diagram, add the results of different paths.
Complement Rule
. The easiest way to solve 'at least one' problems is to find and subtract from 1.
Without Replacement
When items are not replaced, both the numerator and denominator change for subsequent draws. Update BOTH!
πWorked Example 5: Probability Without Replacement
A bag has 5 red and 3 blue balls. Two balls are drawn without replacement. Find P(both red) and P(one of each colour).
πWorked Example 6: 'At Least One' Using Complement
A coin is tossed 3 times. Find the probability of getting at least one head.
πWorked Example 7: Two Dice Probability
Two fair dice are thrown. Find P(sum = 7) and P(sum greater than 9).
πWorked Example 8: Independent Events
P(A) = 0.3 and P(B) = 0.4. Events A and B are independent. Find P(A or B).
πWorked Example 9: Conditional Probability Setup
In a box of 10 light bulbs, 3 are defective. Two bulbs are selected at random without replacement. Find the probability that (a) both are defective, (b) exactly one is defective, (c) at least one is defective.
πWorked Example 9b: Venn Diagram Probability
In a class of 40 students, 25 study History (H), 20 study Geography (G), and 10 study both. Find (a) P(H or G), (b) P(H only), (c) P(neither).
Venn diagram showing union, intersection, and complement regions
πWorked Example 9c: Three-Event Probability
Three fair coins are tossed. Find P(exactly 2 heads).
πWorked Example 9d: Probability from a Table
A survey of 100 people records their age group and preferred drink. Under 30: Coffee (25), Tea (15). 30 and over: Coffee (20), Tea (40). A person is chosen at random. Find P(Coffee | Under 30) and P(Under 30 | Coffee).
πWorked Example 9e: Probability with AND and OR (Mutually Exclusive)
A card is drawn from a standard deck of 52 cards. Find (a) P(King or Queen), (b) P(Heart or Diamond), (c) P(King or Heart).
Probability Key Vocabulary
Know these terms precisely for the exam:
- Sample space: The set of all possible outcomes
- Event: A subset of the sample space (a specific outcome or group of outcomes)
- Mutually exclusive: Two events that CANNOT happen at the same time. P(A and B) = 0.
- Independent: One event does not affect the other. P(A and B) = P(A) x P(B).
- Exhaustive: Events that cover ALL possible outcomes. Their probabilities sum to 1.
- Complement: Everything NOT in the event. P(A') = 1 - P(A).
P(A) = 0.3 and P(B) = 0.4 (independent). Find P(A or B).
β οΈCommon Mistakes β Statistics & Probability
Why: AND means multiply. OR means add. "And" = multiplication, "Or" = addition.
Why: Without replacement, both the favourable outcomes and total outcomes change. Update both numerator and denominator.
Pie Charts, Bar Charts, and Stem-and-Leaf Diagrams
While histograms and cumulative frequency curves are the most heavily tested statistical diagrams, you should also be able to read and create pie charts, bar charts, and stem-and-leaf diagrams. These may appear as part of a larger question or as a simple data interpretation exercise.
πWorked Example 10: Pie Chart Calculation
In a survey of 180 students, 60 chose Science, 45 chose Arts, 30 chose Business, and the rest chose Engineering. Calculate the angle for each sector in a pie chart.
πWorked Example 11: Reading a Stem-and-Leaf Diagram
Stem | Leaf: 3 | 2 5 7 8, 4 | 0 1 3 5 6 6 8, 5 | 2 4 7, 6 | 1 3. Key: 3 | 2 means 32.
Common Probability Distributions in Exam Questions
While O-Level does not formally test probability distributions, certain setups recur frequently in exam questions. Being familiar with these patterns helps you solve problems faster.
| Setup | Total Outcomes | Key Considerations |
|---|---|---|
| One fair die | 6 | Outcomes: 1, 2, 3, 4, 5, 6. Each equally likely. |
| Two fair dice | 36 (6 x 6) | List outcomes systematically. Sum = 7 is most common (6 ways). |
| One coin | 2 | H or T. P(H) = P(T) = 0.5. |
| Three coins | List: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. | |
| Cards (52) | 52 | 13 of each suit. 4 of each rank. P(Ace) = 4/52 = 1/13. |
| Balls without replacement | Decreasing | Total decreases by 1 each draw. Update BOTH numerator and denominator. |
πWorked Example 12: Combined Statistics and Probability
The marks of 5 students are: 50, 65, 70, 75, 80. One student is chosen at random. Find (a) the mean mark, (b) P(mark > mean), (c) P(mark within 10 of the mean).
πWorked Example 13: Geometric Probability
A circular dartboard has radius 20 cm. The bullseye is a circle at the centre with radius 4 cm. If a dart lands randomly on the board, find P(hitting the bullseye).
Two dice are thrown. How many outcomes give a sum of 7?
Statistics Scoring Strategy
Statistics questions are usually worth 6-8 marks and are among the most straightforward in Paper 2. Draw neat diagrams, show ALL working for mean and SD calculations, and always label tree diagrams with probabilities that sum to 1 at each branch point. For "at least one" questions, always consider the complement method first.
πMore Worked Examples: Data Representation & Analysis
πWorked Example 14: Box-and-Whisker Plot
The marks for 200 students are: min = 15, Q1 = 38, median = 52, Q3 = 67, max = 95. Find the interquartile range and state the percentage of students who scored more than 67.
πWorked Example 15: Probability with Replacement vs Without
A bag contains 5 red and 3 blue marbles. Two marbles are drawn. Find P(both red) if drawn (a) with replacement, (b) without replacement.
πWorked Example 16: Expected Value
A game costs $2 to play. You roll a fair die: score 6 and win $10, score 5 and win $5, score 1-4 and win nothing. Find the expected profit per game.