🎯 Data and Statistics Problem Types
📊 Problem Pattern Catalog
Type 1: Finding Measures of Center
Pattern: Calculate mean, median, or mode of a data set Key Formula: Mean = sum/count, Median = middle value, Mode = most frequent
Worked Example:
Find the mean, median, and mode of: 2, 4, 4, 6, 8, 10, 10, 10
Solution: Mean: $\frac{2+4+4+6+8+10+10+10}{8} = \frac{54}{8} = 6.75$
Median: Data is already ordered. With 8 values, median is average of 4th and 5th: $\frac{6+8}{2} = 7$
Mode: 10 appears most frequently (3 times)
Type 2: Weighted Averages
Pattern: Find average when different values have different weights Key Formula: $\frac{\sum(w_i \cdot x_i)}{\sum w_i}$
Worked Example:
A student’s grades are: Test 1 (80, weight 2), Test 2 (90, weight 3), Final (85, weight 5). Find the weighted average.
Solution: Weighted average = $\frac{2 \cdot 80 + 3 \cdot 90 + 5 \cdot 85}{2 + 3 + 5} = \frac{160 + 270 + 425}{10} = \frac{855}{10} = 85.5$
Type 3: Probability Calculations
Pattern: Find probability of simple events Key Formula: $P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$
Worked Example:
A bag contains 3 red, 4 blue, and 5 green marbles. What’s the probability of drawing a red marble?
Solution: Total marbles = 3 + 4 + 5 = 12 Favorable outcomes = 3 (red marbles) Probability = $\frac{3}{12} = \frac{1}{4}$
Type 4: Complementary Probability
Pattern: Find probability of the opposite event Key Formula: $P(\text{not A}) = 1 - P(\text{A})$
Worked Example:
The probability of rain tomorrow is 0.3. What’s the probability of no rain?
Solution: $P(\text{no rain}) = 1 - P(\text{rain}) = 1 - 0.3 = 0.7$
Type 5: Range and Spread
Pattern: Find range or analyze how spread out data is Key Formula: Range = Maximum - Minimum
Worked Example:
Find the range of: 15, 22, 18, 25, 20, 19
Solution: Maximum = 25, Minimum = 15 Range = 25 - 15 = 10
🔍 Problem-Solving Strategy
- Identify what measure you need to find
- Organize the data if necessary
- Use the appropriate formula or method
- Check your calculation with estimation
- Consider the context of the problem
⚠️ Common Mistakes
- Not ordering data before finding median
- Forgetting to divide by total weight in weighted averages
- Miscounting outcomes in probability
- Confusing mean and median in calculations
- Missing the complement in probability problems