Formula Proof and Solved Examples of Mean Median and Mode Relationship
The concept of relation between mean, median and mode is a fundamental pillar in statistics, making it easier to quickly estimate any one measure if the other two are known. This is especially helpful for students during board exams, competitive tests, and real-world data analysis.
What Is Relation Between Mean, Median and Mode?
The relation between mean, median and mode is a statistical formula that connects the three measures of central tendency: mean (average), median (middle value), and mode (most frequent value). This relation is especially useful in statistics for skewed distributions and helps estimate missing values quickly during exams. You will use this concept in data handling, competitive exams, and real-world problems involving average salaries, scores, or other grouped data.
Key Formula for Relation Between Mean, Median and Mode
Here’s the standard formula: Mode = 3 × Median − 2 × Mean
Or, equivalently: Mean − Mode = 3(Mean − Median)
Meaning of Mean, Median and Mode
| Measure | Definition | Example (for data set: 2, 3, 3, 4, 8) |
|---|---|---|
| Mean | The average value of the data. | (2+3+3+4+8)/5 = 4 |
| Median | The middle value when data is ordered. | 3 |
| Mode | The value that appears most frequently. | 3 |
Empirical Relation Between Mean, Median and Mode
The empirical relation between mean, median and mode is used when a frequency distribution is moderately skewed. The formula is:
Mode = 3 × Median − 2 × Mean
If you know any two of the measures, you can use the formula above to find the third measure instantly.
Derivation and Explanation
Here’s a step-by-step derivation of the relation between mean, median and mode, known as Karl Pearson’s empirical formula:
1. Start with the formula: (Mean − Mode) = 3(Mean − Median)2. Expand and rearrange:
3. Bring like terms together:
4. Or, solve for Mode:
This relationship holds good in moderately skewed distributions (not for perfectly symmetrical or highly irregular data).
Solved Example: Applying the Empirical Relation
Question: If the mean of a data set is 12 and the median is 10, find the mode.
1. Write the formula:Mode = 3 × Median − 2 × Mean
2. Substitute values:
Mode = 3 × 10 − 2 × 12
3. Calculate:
Mode = 30 − 24 = 6
Final Answer: The mode is 6.
Application in Skewed Distributions
| Distribution Type | Relation (Order) | Example |
|---|---|---|
| Symmetrical | Mean = Median = Mode | Normal bell curve (e.g., heights of students) |
| Positively Skewed (Tail right) |
Mean > Median > Mode | Wealth distribution, exam marks with outliers |
| Negatively Skewed (Tail left) |
Mean < Median < Mode | Age at retirement, early test scores |
Speed Trick or Vedic Shortcut
During exams, if two out of the mean, median, and mode are given, use the formula instantly:
- To find Mode: Mode = 3 × Median − 2 × Mean
- To find Median: Median = (Mode + 2 × Mean) / 3
- To find Mean: Mean = (3 × Median − Mode) / 2
Try These Yourself
- Given mean = 15, median = 13, find the mode.
- If mode = 16 and mean = 11, what is the median?
- In a data set where mean = median = 18, what is the mode?
- Arrange in order for a positively skewed data: mode, median, mean.
Quick Revision Table: Formulas
| To Find | Formula |
|---|---|
| Mode | 3 × Median – 2 × Mean |
| Median | (Mode + 2 × Mean) ÷ 3 |
| Mean | (3 × Median − Mode) ÷ 2 |
Frequent Errors and Misunderstandings
- Applying the formula to highly skewed or multimodal data (it may not work).
- Confusing terms: mean vs. median vs. mode.
- Incorrect calculation order in rearranged formulas.
- Forgetting to use the ordered (sorted) data for median.
Relation to Other Concepts
Understanding the relation between mean, median and mode helps in mastering all central tendency measures, and connects directly to studying variance and standard deviation, as well as formula tables for statstics. If you want to be thorough for board exams or competitive tests, knowing these connections is crucial.
Classroom Tip
A helpful way to remember: "Mode = 3 × Median − 2 × Mean." Picture the numbers on a scale—mean pulls with outliers, median stands in the middle, and mode is the crowd-favorite. Vedantu’s teachers use simple examples with real-world data (like class test marks) to make this formula click for students.
We explored relation between mean, median and mode—from simple definitions to formula, solved examples, practical shortcuts, and its links with other statistics concepts. To practice more and strengthen your concepts, check out Vedantu’s central tendency questions any time.
FAQs on Relation Between Mean Median and Mode in Statistics
1. What is the relation between mean, median, and mode?
The relation between mean, median, and mode for a moderately skewed distribution is Mode = 3 Median − 2 Mean. This empirical formula is used in statistics when the data is not perfectly symmetrical.
- If the data is symmetrical: Mean = Median = Mode
- If positively skewed: Mean > Median > Mode
- If negatively skewed: Mode > Median > Mean
2. What is the empirical formula relating mean, median, and mode?
The empirical formula is Mode = 3 Median − 2 Mean. It is mainly applicable to moderately skewed distributions in statistics.
- Rearranged form: Mean − Mode = 3(Mean − Median)
- Used when exact mode is difficult to calculate
- Applies approximately, not exactly, in real-life data
3. When is Mean = Median = Mode?
Mean, median, and mode are equal when the distribution is perfectly symmetrical and unimodal. In a normal distribution:
- Mean = Median = Mode
- The curve is symmetric about the center
- There is no skewness
4. How do you derive Mode from Mean and Median?
Mode can be calculated using the formula Mode = 3 Median − 2 Mean. Follow these steps:
- Step 1: Find the Mean
- Step 2: Find the Median
- Step 3: Substitute values into the formula
5. What is the difference between mean, median, and mode?
Mean, median, and mode are three different measures of central tendency used to describe the center of data.
- Mean: Sum of observations ÷ Total number of observations
- Median: Middle value when data is arranged in order
- Mode: Most frequently occurring value
6. What is the relation between mean, median, and mode in a positively skewed distribution?
In a positively skewed distribution, the relation is Mean > Median > Mode. This happens because high extreme values pull the mean to the right.
- Tail of distribution extends to the right
- Mean is most affected by outliers
- Mode remains at the peak
7. What is the relation between mean, median, and mode in a negatively skewed distribution?
In a negatively skewed distribution, the relation is Mode > Median > Mean. This occurs because low extreme values pull the mean to the left.
- Tail of distribution extends to the left
- Mean decreases due to smaller outliers
- Mode stays near the highest peak
8. Can you give an example showing the relation between mean, median, and mode?
Yes, the relation can be verified using an example and the empirical formula. Consider Mean = 20 and Median = 22.
- Using formula: Mode = 3 Median − 2 Mean
- Mode = 3(22) − 2(20)
- Mode = 66 − 40 = 26
9. Why is the empirical relation between mean, median, and mode only approximate?
The empirical relation is approximate because it is based on moderately skewed distributions, not perfectly defined mathematical conditions. It may not hold when:
- The distribution is highly skewed
- The data has multiple modes
- The distribution is irregular
10. How is the relation between mean, median, and mode useful in statistics?
The relation helps estimate one measure of central tendency when the other two are known. It is useful when:
- Mode is difficult to calculate from grouped data
- Checking consistency of calculated values
- Studying skewness of a distribution
