2.5 Data Handling, Probability & Word Problems
Key Takeaways
- Measures of central tendency summarise data: Mean (sum divided by count), Median (middle value when ordered), Mode (most frequent value), and Range (difference between maximum and minimum values).
- Data interpretation requires extracting quantitative values accurately from bar charts, line graphs, dual-axis charts, tables, and pie charts using angle proportions (Value = (Sector Angle ÷ 360°) × Total).
- Theoretical probability measures event likelihood on a scale from 0 to 1, calculated as P(Event) = Favourable Outcomes ÷ Total Outcomes.
- Systematic multi-step word problem solving relies on the RUCSAC model (Read, Understand, Choose operation, Solve, Answer, Check).
- Problem-solving heuristics such as drawing bar diagrams, working backwards, making organized lists, and estimation prevent calculation errors and verify answer reasonableness.
2.5 Data Handling, Probability & Word Problems
The final section of Chapter 2 covers statistical averages, chart interpretation, probability concepts, and multi-step word problem heuristics. GL Assessment 11+ tests place heavy emphasis on extraction of data from complex charts and solving non-routine word problems.
Measures of Central Tendency & Data Spread
Statistical measures summarize a set of data into key representative values.
| Measure | Definition | Calculation Method |
|---|---|---|
| Mean | The arithmetic average | Add all values together and divide by total number of values |
| Median | The middle value | Order data from smallest to largest; select middle term (or average of two middle terms) |
| Mode | The most frequent value | Count frequencies; identify the value that occurs most often |
| Range | The spread of data | Subtract the minimum value from the maximum value ($\text{Max} - \text{Min}$) |
Missing Value Mean Problems
A common 11+ question type asks candidates to find a missing data point when given the mean.
Worked Example: The mean of five numbers is $14$. Four of the numbers are $9$, $16$, $12$, and $18$. Find the fifth number.
- Find Total Sum: $\text{Total Sum} = \text{Mean} \times \text{Count} = 14 \times 5 = 70$.
- Sum Known Numbers: $9 + 16 + 12 + 18 = 55$.
- Subtract to Find Missing Value: $70 - 55 = \mathbf{15}$.
Graphical Data Interpretation
1. Bar Charts & Line Graphs
Always check axis scales carefully! On line graphs, interpolate values between marked gridlines by determining the value per small division.
2. Pie Chart Sector Calculations
A pie chart represents data as sectors of a circle totalling $360^\circ$.
Worked Example: A pie chart shows the favourite pets of $180$ students. If the sector for 'Dogs' has a central angle of $140^\circ$, how many students chose dogs?
- Fraction of pie chart = $\frac{140}{360} = \frac{7}{18}$.
- Number of students = $\frac{7}{18} \times 180 = 7 \times 10 = \mathbf{70\text{ students}}$.
Probability Foundations
Probability quantifies the likelihood of an event occurring, expressed on a scale from $0$ (impossible) to $1$ (certain).
Complementary Events
The probability of an event not happening is $1$ minus the probability that it does happen:
Worked Example: A bag contains $5$ red marbles, $7$ blue marbles, and $8$ green marbles. What is the probability of randomly picking a marble that is NOT blue?
- Total marbles = $5 + 7 + 8 = 20$.
- Favourable outcomes (not blue = red + green) = $5 + 8 = 13$.
- $P(\text{Not blue}) = \mathbf{\frac{13}{20}}$.
Systematic Multi-Step Word Problem Heuristics
Multi-step word problems require a structured analytical approach to prevent avoidable errors.
The RUCSAC Framework
- Read the question carefully twice.
- Understand the key information and underline given numbers.
- Choose the correct operation(s) and strategy.
- Solve step-by-step showing neat written workings.
- Answer the specific question asked (check units!).
- Check answer reasonableness using estimation.
Key Problem-Solving Strategies
Strategy 1: Working Backwards
When a problem describes a sequence of changes applied to an initial unknown amount, start at the final result and apply inverse operations step by step.
Worked Example: Maya spends $\frac{1}{3}$ of her pocket money on a book and $£4$ on snacks. She then spends half of her remaining money on a bus ticket. If she has $£5$ left, how much pocket money did she start with?
- Step 1 (Bus Ticket): After bus ticket, she has $£5$. Since she spent half her remaining money on the ticket, before the bus ticket she had $£5 \times 2 = £10$.
- Step 2 (Snacks): Before spending $£4$ on snacks, she had $£10 + £4 = £14$.
- Step 3 (Book): She spent $\frac{1}{3}$ of her money on a book, leaving $ rac{2}{3}$ of her initial money. Therefore, $£14$ equals $\frac{2}{3}$ of her total pocket money.
- Step 4 (Initial Money): If $\frac{2}{3} = £14$, then $\frac{1}{3} = £7$, and initial money $\frac{3}{3} = 3 \times £7 = \mathbf{£21}$.
Strategy 2: Drawing Bar Models
Bar models provide a visual representation of relationships between parts and wholes, making complex fraction and ratio word problems easy to solve.
The mean of five numbers is 14. Four of the numbers are 9, 16, 12, and 18. What is the fifth number?
A pie chart represents the favourite pets of 180 students. If the sector for 'Dogs' has a central angle of 140°, how many students chose dogs?
A bag contains 5 red marbles, 7 blue marbles, and 8 green marbles. If one marble is drawn at random, what is the probability that it is NOT blue?
Maya spends 1/3 of her pocket money on a book and £4 on snacks. She then spends half of her remaining money on a bus ticket. If she has £5 left, how much pocket money did she start with?