Range, Mean Deviation & Standard Deviation

🎯 Learning Objective

Measure how spread out data is using range, mean deviation and standard deviation.

💡 Concept

• These measure DISPERSION — how spread out the data is around its centre
• Range = Largest value − Smallest value (the simplest measure)
• Mean deviation = average of the absolute gaps from the mean = Σ|x − mean| ÷ n
• Variance = Σ(x − mean)² ÷ n; Standard deviation (SD) = √Variance
• Larger SD → more spread / less consistent data

🧮 Key Formulas

Range = Max − Min

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Mean deviation = Σ|x − x̄| ÷ n

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SD = √(Σ(x − x̄)² ÷ n)

✏️ Easy Example

Q. Find the range of 12, 5, 20, 8, 15.

1. Largest = 20, Smallest = 5
2. Range = 20 − 5

Answer: 15

🇮🇳 Real-Life Example

Two batsmen both average 40. The one with the smaller standard deviation is the 'consistent' player who scores 38, 42, 40 rather than 5, 80, 35 — selectors trust low SD.

📝 Exam-Level Example

Q. Find the mean deviation of the data 2, 4, 5, 6, 8.

1. Mean = (2 + 4 + 5 + 6 + 8) ÷ 5 = 25 ÷ 5 = 5
2. Absolute gaps from 5: 3, 1, 0, 1, 3 → sum = 8
3. Mean deviation = 8 ÷ 5

Answer: 1.6

📝 Exam-Level Example

Q. Find the standard deviation of the same data 2, 4, 5, 6, 8.

1. Mean = 5 (from above)
2. Squared gaps: 9, 1, 0, 1, 9 → sum = 20
3. Variance = 20 ÷ 5 = 4
4. SD = √4

Answer: 2

🪄 Memory Trick

SD is just: gaps from mean → square them → average (that's variance) → square root. Do it in that fixed order every time.

⚠️ Common Mistakes

• ❌ Forgetting the final square root and reporting variance as SD
• ❌ Not using absolute values in mean deviation (gaps can't be negative here)
• ❌ Dividing by (n−1) instead of n at this level

🏆 Exam Tips

• ✅ Range uses only two values; SD uses all of them
• ✅ Variance = SD² — if a question gives variance, just take its root

📌 Summary

• Range = Max − Min
• Mean deviation = Σ|x − mean| ÷ n
• Variance = Σ(x − mean)² ÷ n; SD = √Variance
• Bigger SD = more spread / less consistent