Mixed Statistics Problems

🎯 Learning Objective

Combine the ideas: combined mean, effect of changing data, and the mean–median–mode relation.

💡 Concept

• Sum = Mean × n — the master link, exactly like averages
• Combined mean of two groups = (n₁·x̄₁ + n₂·x̄₂) ÷ (n₁ + n₂)
• Add the same k to EVERY value → mean rises by k, but range and SD stay the same
• Multiply every value by k → mean and SD both become k times
• Empirical relation: Mode = 3 Median − 2 Mean (find the third from the other two)

🧮 Key Formulas

Sum = Mean × n

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Combined mean = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂)

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Mode = 3 Median − 2 Mean

✏️ Easy Example

Q. The mean of 6 numbers is 20. If one number, 15, is replaced by 45, find the new mean.

1. Old sum = 20 × 6 = 120
2. New sum = 120 − 15 + 45 = 150
3. New mean = 150 ÷ 6

Answer: 25

🇮🇳 Real-Life Example

A teacher merging two sections' marks can't just average the two averages — Section A (20 kids, 60) and B (30 kids, 70) combine to 66, not 65, because B has more students pulling the mean up.

📝 Exam-Level Example

Q. Section A has 20 students with average 60 marks; Section B has 30 students with average 70 marks. Find the combined average.

1. Total of A = 20 × 60 = 1200
2. Total of B = 30 × 70 = 2100
3. Combined = (1200 + 2100) ÷ (20 + 30) = 3300 ÷ 50

Answer: 66 marks

📝 Exam-Level Example

Q. For a distribution, the mean is 30 and the median is 28. Find the mode.

1. Mode = 3 Median − 2 Mean
2. = 3 × 28 − 2 × 30 = 84 − 60

Answer: 24

📝 Exam-Level Example

Q. The average of 5 numbers is 12. If 3 is added to each number, what is the new average?

1. Adding the same value to every number shifts the mean by that value
2. New average = 12 + 3

Answer: 15

🪄 Memory Trick

Combined mean is a WEIGHTED average — the bigger group pulls the answer toward its own average. Never average two averages directly.

⚠️ Common Mistakes

• ❌ Averaging two group averages directly instead of weighting by size
• ❌ Adding k to each value and changing the SD (SD stays the same)
• ❌ Wrong signs in Mode = 3 Median − 2 Mean

🏆 Exam Tips

• ✅ Always convert to totals first (Sum = Mean × n)
• ✅ Add-to-each shifts the mean but never the spread (range, SD)

📌 Summary

• Sum = Mean × n is the master link
• Combined mean weights by group size, not a plain average
• Add k to each → mean +k, spread unchanged
• Mode = 3 Median − 2 Mean