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Relationship Between Mean, Median & Mode

Introduction

Statistics में Mean, Median और Mode central tendency के measures हैं - यानी ये dataset के centre या typical value को दर्शाते हैं। Symmetric distribution में ये तीनों बराबर होते हैं, लेकिन skewed data में इनका आपस का relation बदल जाता है।

यह pattern इसलिए महत्वपूर्ण है क्योंकि इससे एक empirical relationship के ज़रिए तीन में से किसी एक value (अक्सर Mode) का अनुमान लगाया जा सकता है, जब बाकी दो दी हों।

Pattern: Relationship Between Mean, Median & Mode

Pattern

मुख्य concept: Moderately skewed data के लिए empirical formula है - Mode = 3 × Median - 2 × Mean.

जब Mode को सीधे निकालना मुश्किल हो और Mean व Median ज्ञात हों, तब यह relation बहुत काम आता है।

Step-by-Step Example

Question

किसी dataset का mean 30 है और median 25 है। Empirical formula का उपयोग करके mode निकालें।

Solution

  1. Step 1: दिए गए values लिखें

    Mean = 30, Median = 25

  2. Step 2: Empirical relationship लिखें

    Mode = 3 × Median - 2 × Mean

  3. Step 3: Values substitute करें

    Mode = 3 × 25 - 2 × 30 = 75 - 60 = 15

  4. Final Answer:

    Mode = 15

  5. Quick Check:

    क्योंकि Mean > Median, distribution positively skewed होगा (Mode < Median) - सही है! ✅

Quick Variations

1. कभी-कभी Mode दिया होता है और Mean या Median निकालना होता है।

2. Formula को इस तरह भी लिखा जा सकता है:

  • 1. Mean = (3 × Median - Mode) ÷ 2
  • 2. Median = (2 × Mean + Mode) ÷ 3
3. Skewness पहचानने में helpful: अगर Mean > Median > Mode → Positive skewness.

Trick to Always Use

  • Step 1: दिए गए दो measures को साफ़-साफ़ लिखें।
  • Step 2: Formula apply करें: Mode = 3 × Median - 2 × Mean.
  • Step 3: Mean या Median निकालना हो तो उसी अनुसार formula rearrange करें।

Summary

Summary

In the Relationship Between Mean, Median & Mode pattern:

  • Symmetric (normal) data में तीनों बराबर होते हैं।
  • Skewed data के लिए empirical formula: Mode = 3 × Median - 2 × Mean.
  • Mean > Median > Mode → Positive skewness.
  • Mode > Median > Mean → Negative skewness.
  • Data interpretation में missing values तुरंत निकालने के लिए यह relation बहुत मदद करता है।

Practice

(1/5)
1. The mean of a dataset is 40 and the median is 35. Find the mode using the empirical relationship.
easy
A. 25
B. 30
C. 35
D. 45

Solution

  1. Step 1: Recall the empirical formula

    Mode = 3 × Median - 2 × Mean.

  2. Step 2: Substitute values

    Mode = 3 × 35 - 2 × 40 = 105 - 80 = 25.

  3. Final Answer:

    Mode = 25 → Option A.

  4. Quick Check:

    Mean (40) > Median (35) so Mode should be < Median for positive skew - 25 is consistent ✅

Hint: Use Mode = 3×Median - 2×Mean directly when Mean and Median are known.
Common Mistakes: Mixing up Mean and Median positions in the formula.
2. If Mean = 60 and Mode = 50, find the Median using the empirical relation.
easy
A. 55
B. 56.67
C. 58.67
D. 52

Solution

  1. Step 1: Rearrange the empirical formula for Median

    Mode = 3×Median - 2×Mean → Median = (2×Mean + Mode) ÷ 3.

  2. Step 2: Substitute values

    Median = (2×60 + 50) ÷ 3 = (120 + 50) ÷ 3 = 170 ÷ 3 = 56.67.

  3. Final Answer:

    Median = 56.67 → Option B.

  4. Quick Check:

    Median (56.67) lies between Mode (50) and Mean (60) - logical ✅

Hint: Use Median = (2×Mean + Mode)/3 when Mean and Mode are known.
Common Mistakes: Dividing by 2 instead of 3 after rearrangement.
3. The Mode of a distribution is 20 and the Mean is 25. Find the Median.
medium
A. 21.50
B. 22.50
C. 23.33
D. 24.00

Solution

  1. Step 1: Use the empirical formula

    Mode = 3×Median - 2×Mean → rearrange to Median = (Mode + 2×Mean) ÷ 3.

  2. Step 2: Substitute values

    Median = (20 + 2×25) ÷ 3 = (20 + 50) ÷ 3 = 70 ÷ 3 = 23.33.

  3. Final Answer:

    Median ≈ 23.33 → Option C.

  4. Quick Check:

    Mode (20) < Median (23.33) < Mean (25) - indicates positive skew, consistent ✅

Hint: Rearrange to Median = (Mode + 2×Mean)/3 when Mode and Mean are given.
Common Mistakes: Subtracting 2×Mean instead of adding in the rearranged form.
4. The Median and Mode of a series are 50 and 60 respectively. Find the Mean.
medium
A. 43
B. 50
C. 53.33
D. 45.00

Solution

  1. Step 1: Rearranged formula for Mean

    From Mode = 3×Median - 2×Mean → Mean = (3×Median - Mode) ÷ 2.

  2. Step 2: Substitute values

    Mean = (3×50 - 60) ÷ 2 = (150 - 60) ÷ 2 = 90 ÷ 2 = 45.00.

  3. Final Answer:

    Mean = 45.00 → Option D.

  4. Quick Check:

    Mode (60) > Median (50) > Mean (45) - distribution is negatively skewed, consistent ✅

Hint: Use Mean = (3×Median - Mode)/2 for Mean when Median and Mode are known.
Common Mistakes: Forgetting to divide the final result by 2.
5. If the Mean of data is 80 and Mode is 90, find the Median.
medium
A. 83.33
B. 83.30
C. 82.50
D. 85

Solution

  1. Step 1: Use the rearranged formula

    Median = (2×Mean + Mode) ÷ 3.

  2. Step 2: Substitute values

    Median = (2×80 + 90) ÷ 3 = (160 + 90) ÷ 3 = 250 ÷ 3 = 83.33.

  3. Final Answer:

    Median ≈ 83.33 → Option A.

  4. Quick Check:

    Median (83.33) lies between Mean (80) and Mode (90) - reasonable ✅

Hint: Median = (2×Mean + Mode)/3 when Mean and Mode are given.
Common Mistakes: Using the original formula without rearranging properly.

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