Empirical Relationship Between Mean, Median, Mode
The three measures of central tendency — Mean, Median, and Mode — are related by an empirical formula discovered by Karl Pearson. This relationship holds approximately for moderately skewed distributions.
This topic is part of Chapter 14 — Statistics in the CBSE Class 10 syllabus. Students are expected to know this formula and use it to find the third measure when two are known.
The empirical relationship provides a quick shortcut and a way to verify answers after computing Mean, Median, and Mode individually using their respective formulas for grouped data.
What is Empirical Relationship Between Mean, Median, Mode?
Definition: The empirical relationship (Karl Pearson's formula) states that for a moderately skewed frequency distribution:
Equivalently:
- Mode = 3 × Median − 2 × Mean
- Median = (Mode + 2 × Mean) / 3
- Mean = (3 × Median − Mode) / 2
Key facts:
- This is an empirical (observational) relationship, not a mathematical proof.
- It works well for moderately skewed distributions (most real-world data).
- It does NOT hold for highly skewed or bimodal distributions.
- For a perfectly symmetric distribution: Mean = Median = Mode.
Empirical Relationship Between Mean, Median, Mode Formula
Karl Pearson's Empirical Formula:
3 × Median = Mode + 2 × Mean
All three rearranged forms:
| To Find | Formula |
|---|---|
| Mode | Mode = 3 × Median − 2 × Mean |
| Median | Median = (Mode + 2 × Mean) / 3 |
| Mean | Mean = (3 × Median − Mode) / 2 |
Formulas for grouped data (revision):
- Mean = Σf_i x_i / Σf_i (direct method)
- Median = l + [(n/2 − cf) / f] × h
- Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h
Derivation and Proof
Origin of the Empirical Relationship:
- Karl Pearson studied many real-world frequency distributions (heights, weights, test scores, etc.).
- He observed that in moderately skewed distributions, the median always lies between the mean and the mode.
- More precisely, he found that the distance from the mean to the median is approximately one-third of the distance from the mean to the mode.
- Mathematically: Mean − Median ≈ (1/3)(Mean − Mode).
- Multiplying both sides by 3: 3(Mean − Median) = Mean − Mode.
- 3 × Mean − 3 × Median = Mean − Mode.
- Rearranging: Mode = 3 × Median − 2 × Mean.
- Or equivalently: 3 × Median = Mode + 2 × Mean.
Important: This is NOT a theorem with a rigorous proof. It is an empirical observation that holds approximately for most real-world datasets with moderate skewness.
Types and Properties
Type 1: Finding Mode when Mean and Median are given
- Use: Mode = 3 × Median − 2 × Mean.
Type 2: Finding Median when Mean and Mode are given
- Use: Median = (Mode + 2 × Mean) / 3.
Type 3: Finding Mean when Median and Mode are given
- Use: Mean = (3 × Median − Mode) / 2.
Type 4: Verification problems
- Compute all three measures independently and verify that 3 × Median ≈ Mode + 2 × Mean.
Type 5: Skewness interpretation
- If Mean > Median > Mode: positive (right) skew.
- If Mean < Median < Mode: negative (left) skew.
- If Mean = Median = Mode: symmetric distribution.
Methods
Method 1: Using the empirical formula directly
- Identify which two measures are given.
- Substitute into the appropriate rearranged formula.
- Solve for the unknown measure.
Method 2: Compute all three and verify
- Calculate Mean using the direct/assumed mean/step-deviation method.
- Calculate Median using the median formula for grouped data.
- Calculate Mode using the mode formula for grouped data.
- Check: 3 × Median ≈ Mode + 2 × Mean.
When to use which:
- If the problem asks to verify the relationship: compute all three independently first.
- If the problem gives two values and asks for the third: use the formula directly.
- If you computed two measures and want a quick check of the third: use the empirical formula as an estimate.
Solved Examples
Example 1: Finding Mode from Mean and Median
Problem: The mean of a distribution is 24.6 and the median is 26.1. Find the mode using the empirical relationship.
Solution:
Given:
- Mean = 24.6
- Median = 26.1
Using Mode = 3 × Median − 2 × Mean:
- Mode = 3(26.1) − 2(24.6)
- Mode = 78.3 − 49.2
- Mode = 29.1
Answer: The mode is 29.1.
Example 2: Finding Median from Mean and Mode
Problem: The mean and mode of a data set are 30 and 36 respectively. Find the median.
Solution:
Given:
- Mean = 30
- Mode = 36
Using Median = (Mode + 2 × Mean) / 3:
- Median = (36 + 2 × 30) / 3
- Median = (36 + 60) / 3
- Median = 96 / 3 = 32
Answer: The median is 32.
Example 3: Finding Mean from Median and Mode
Problem: The median and mode of a frequency distribution are 25 and 22 respectively. Find the mean.
Solution:
Given:
- Median = 25
- Mode = 22
Using Mean = (3 × Median − Mode) / 2:
- Mean = (3 × 25 − 22) / 2
- Mean = (75 − 22) / 2
- Mean = 53 / 2 = 26.5
Answer: The mean is 26.5.
Example 4: Verifying the Empirical Relationship
Problem: For a frequency distribution, the mean is 45.5, the median is 46, and the mode is 47. Verify the empirical relationship.
Solution:
Given:
- Mean = 45.5, Median = 46, Mode = 47
Verification:
- LHS = 3 × Median = 3 × 46 = 138
- RHS = Mode + 2 × Mean = 47 + 2(45.5) = 47 + 91 = 138
- LHS = RHS = 138
Answer: The empirical relationship 3 × Median = Mode + 2 × Mean is verified.
Example 5: Grouped Data Verification
Problem: The following data gives the marks of 50 students:
| Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|---|---|---|---|---|
| Students | 5 | 8 | 15 | 12 | 10 |
If the mean is 27 and the median is 27.33, find the mode using the empirical formula and verify by the mode formula.
Solution:
Given:
- Mean = 27, Median = 27.33
Using empirical formula:
- Mode = 3 × 27.33 − 2 × 27 = 81.99 − 54 = 27.99 ≈ 28
Verification using mode formula:
- Modal class = 20-30 (highest frequency = 15)
- l = 20, f₁ = 15, f₀ = 8, f₂ = 12, h = 10
- Mode = 20 + [(15 − 8)/(2 × 15 − 8 − 12)] × 10
- Mode = 20 + (7/10) × 10 = 20 + 7 = 27
The empirical value (28) is close to the computed value (27). The small difference is expected — the relationship is approximate.
Answer: Mode ≈ 28 (by empirical formula) and 27 (by mode formula). Values are close, confirming the relationship.
Example 6: Identifying Skewness
Problem: For a distribution, Mean = 40, Median = 38, Mode = 34. Determine the type of skewness and verify the empirical relationship.
Solution:
Given:
- Mean = 40, Median = 38, Mode = 34
Skewness:
- Mean > Median > Mode (40 > 38 > 34)
- This indicates a positively skewed (right-skewed) distribution.
Verification:
- 3 × Median = 3 × 38 = 114
- Mode + 2 × Mean = 34 + 80 = 114
- LHS = RHS. Verified.
Answer: The distribution is positively skewed. The empirical relationship holds.
Example 7: Mean Unknown with Grouped Data
Problem: The mode of the following distribution is 36 and the median is 33.5. Find the mean using the empirical formula.
| Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|---|---|---|---|---|
| Frequency | 4 | 8 | 10 | 12 | 6 |
Solution:
Given:
- Mode = 36, Median = 33.5
Using Mean = (3 × Median − Mode) / 2:
- Mean = (3 × 33.5 − 36) / 2
- Mean = (100.5 − 36) / 2
- Mean = 64.5 / 2 = 32.25
Answer: The mean is 32.25.
Example 8: Negative Skew Example
Problem: For a distribution, Mean = 50, Mode = 56. Find the median and determine the type of skewness.
Solution:
Given:
- Mean = 50, Mode = 56
Finding Median:
- Median = (Mode + 2 × Mean) / 3
- Median = (56 + 100) / 3 = 156/3 = 52
Skewness:
- Mode > Median > Mean (56 > 52 > 50)
- This indicates a negatively skewed (left-skewed) distribution.
Answer: Median = 52. The distribution is negatively skewed.
Example 9: Symmetric Distribution Check
Problem: If the mean of a distribution is 42 and the mode is also 42, find the median. What can you conclude about the distribution?
Solution:
Given:
- Mean = 42, Mode = 42
Finding Median:
- Median = (Mode + 2 × Mean) / 3
- Median = (42 + 84) / 3 = 126/3 = 42
Conclusion:
- Mean = Median = Mode = 42
- The distribution is perfectly symmetric.
Answer: Median = 42. The distribution is symmetric.
Example 10: Finding Missing Value in Frequency Table
Problem: In a frequency distribution, the mean is 28, the mode is 34, and the median is unknown. One of the class frequencies is missing. Use the empirical relationship to estimate the median, which can then help narrow down the missing frequency.
Solution:
Given:
- Mean = 28, Mode = 34
Finding Median:
- Median = (Mode + 2 × Mean) / 3
- Median = (34 + 56) / 3 = 90/3 = 30
This estimated median of 30 indicates that the median class is the class interval containing the 30-mark. This helps in setting up the median formula equation to find the missing frequency.
Answer: The estimated median is 30.
Real-World Applications
Quick Estimation:
- When computing all three measures is time-consuming, compute any two and estimate the third.
Data Validation:
- After computing Mean, Median, and Mode separately, verify using the empirical formula. A large discrepancy may indicate a calculation error.
Understanding Skewness:
- The relative positions of Mean, Median, and Mode reveal the shape of the distribution.
- Positive skew: Mean > Median > Mode (tail on right).
- Negative skew: Mean < Median < Mode (tail on left).
Economics and Social Sciences:
- Income distributions, exam score distributions, and population data are commonly analysed using these three measures and their relationship.
Key Points to Remember
- 3 × Median = Mode + 2 × Mean (Karl Pearson's empirical formula).
- Rearranged: Mode = 3 × Median − 2 × Mean.
- Rearranged: Median = (Mode + 2 × Mean) / 3.
- Rearranged: Mean = (3 × Median − Mode) / 2.
- This is an empirical (approximate) relationship, not an exact identity.
- It works best for moderately skewed distributions.
- For symmetric distributions: Mean = Median = Mode.
- Positive skew: Mean > Median > Mode.
- Negative skew: Mean < Median < Mode.
- Use this formula to verify your computed values or to quickly estimate a missing measure.
Practice Problems
- The mean and median of a distribution are 35 and 33 respectively. Find the mode.
- The mode and mean are 42 and 36. Find the median.
- The median and mode of a data set are 52 and 58. Find the mean.
- For a distribution: Mean = 22.5, Median = 24, Mode = 27. Verify the empirical relationship.
- The mean of a distribution is equal to its median. What is the mode?
- If the mode of a distribution is 60 and the median is 55, find the mean and determine the type of skewness.
- The mean marks of 50 students is 48. The median is 50. Estimate the mode. What type of skewness does this indicate?
- A teacher computed Mean = 65, Median = 68, Mode = 74 for exam scores. Verify the empirical relationship and state the skewness.
Frequently Asked Questions
Q1. What is the empirical relationship between mean, median, and mode?
The empirical relationship is: 3 × Median = Mode + 2 × Mean. It can be rearranged to find any one measure when the other two are known.
Q2. Who discovered this relationship?
Karl Pearson, a British mathematician and statistician, discovered this empirical relationship through observations of many frequency distributions.
Q3. Is this formula exact or approximate?
It is approximate. It holds well for moderately skewed distributions but may not be accurate for highly skewed or multimodal distributions.
Q4. When does Mean = Median = Mode?
When the distribution is perfectly symmetric (e.g., a normal distribution or bell curve), all three measures of central tendency are equal.
Q5. How do you determine skewness from these three values?
If Mean > Median > Mode: positively skewed. If Mean < Median < Mode: negatively skewed. If all three are equal: symmetric.
Q6. Can this formula give a negative mode?
Mathematically, yes, if the data values are negative. But for typical Class 10 data (marks, heights, weights), the values are non-negative. A negative result would indicate incorrect input values.
Q7. Is this formula in the CBSE Class 10 syllabus?
Yes. The NCERT textbook states this relationship and includes problems where students use it to find unknown measures or verify computed values.
Q8. Why is the median always between the mean and mode for skewed data?
The median is resistant to extreme values (outliers), while the mean is pulled toward them. The mode is where data clusters most. For moderately skewed data, the median naturally falls between the other two.
Q9. Can the empirical formula be used for ungrouped data?
Yes, the formula applies to both grouped and ungrouped data as long as the distribution is moderately skewed. However, for ungrouped data, direct computation of all three measures is usually straightforward.
Q10. What if the data has two modes (bimodal)?
The empirical formula does not work reliably for bimodal distributions. It is designed for unimodal, moderately skewed data.
Related Topics
- Mean of Grouped Data
- Median of Grouped Data
- Mode of Grouped Data
- Mean (Arithmetic Average)
- Statistics - Collection and Presentation
- Frequency Distribution Table
- Histogram of Grouped Data
- Frequency Polygon
- Mean of Ungrouped Data
- Median of Ungrouped Data
- Cumulative Frequency Distribution
- Ogive (Cumulative Frequency Curve)
- Mean by Assumed Mean Method
- Mean by Step-Deviation Method
