Mean, Median & Mode

🎯 Learning Objective

Find the three measures of central tendency and know which one each question is really asking for.

💡 Concept

• Mean (average) = Sum of all values ÷ Number of values
• Median = the MIDDLE value after arranging the data in order
• For odd count → the single middle term; for even count → average of the two middle terms
• Mode = the value that occurs MOST often
• Empirical relation: Mode = 3 × Median − 2 × Mean

🧮 Key Formulas

Mean = Σx ÷ n

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Median = middle value (data sorted)

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Mode = most frequent value

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Mode = 3 Median − 2 Mean

✏️ Easy Example

Q. Find the mean of 4, 8, 6, 10, 12.

1. Sum = 4 + 8 + 6 + 10 + 12 = 40
2. Count = 5
3. Mean = 40 ÷ 5

Answer: 8

🇮🇳 Real-Life Example

A shoe shop watches three numbers: the mean sale value, the median customer's spend, and the mode — the size sold most often. The mode is what he restocks first.

📝 Exam-Level Example

Q. Find the median of 7, 3, 9, 5, 11.

1. Arrange in order: 3, 5, 7, 9, 11
2. Count = 5 (odd) → middle (3rd) term

Answer: 7

📝 Exam-Level Example

Q. Find the median of 4, 6, 8, 10.

1. Already sorted; count = 4 (even)
2. Two middle terms = 6 and 8
3. Median = (6 + 8) ÷ 2

Answer: 7

📝 Exam-Level Example

Q. Find the mode of 2, 3, 3, 5, 3, 7, 5.

1. Count each: 3 appears 3 times, 5 appears twice, others once
2. Most frequent = 3

Answer: 3

🪄 Memory Trick

Never find the median without sorting first — that single skipped step is the most common statistics mistake in exams.

⚠️ Common Mistakes

• ❌ Finding the median without sorting the data first
• ❌ For even count, taking one middle value instead of averaging two
• ❌ Confusing mode (most frequent) with mean

🏆 Exam Tips

• ✅ Mean uses every value; median and mode do not
• ✅ If all values are different, there is no mode

📌 Summary

• Mean = sum ÷ count
• Median = middle value (sort first!); even count → average the two middle
• Mode = most frequent value
• Mode = 3 Median − 2 Mean