Empirical Relation Between Mean, Median, and Mode: Formula & Stepwise Example
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: Formula, Derivation & Examples
1. What is the empirical relationship between mean, median, and mode for a Class 10 student?
For the CBSE Class 10 syllabus, the empirical relationship is a key formula that connects the three measures of central tendency for moderately skewed data. The formula is: Mode ≈ 3 Median – 2 Mean. This practical formula is very useful for finding one measure when the other two are given, especially in objective or short-answer questions in board exams.
2. How do the values of mean, median, and mode compare in different types of distributions?
The comparison between mean, median, and mode helps identify the skewness of a dataset, a core concept in statistics:
Symmetrical Distribution: In a perfectly symmetrical (bell-shaped) curve, all three values are identical: Mean = Median = Mode.
Positively Skewed Distribution: The tail of the graph extends to the right. Here, the mean is pulled towards the higher values, resulting in: Mean > Median > Mode.
Negatively Skewed Distribution: The tail of the graph extends to the left. The mean is pulled towards the lower values, so: Mean < Median < Mode.
3. Can you provide an example of how to use the empirical formula to estimate the mode?
Certainly. Imagine a dataset where the Mean is 25 and the Median is 28. To find the approximate value of the mode, you can apply the empirical formula:
Mode ≈ 3 Median – 2 Mean
By substituting the given values:
Mode ≈ (3 × 28) – (2 × 25)
Mode ≈ 84 – 50
Mode ≈ 34
Thus, the estimated value of the mode is 34.
4. Why is the relationship 'Mode ≈ 3 Median – 2 Mean' considered an 'empirical' one?
It is called an 'empirical' relationship because it is not derived from a universal mathematical proof or theorem. Instead, it is based on observation and experimental data. The statistician Karl Pearson observed that for many moderately asymmetrical (skewed) distributions, this approximation holds remarkably true. It describes a common pattern found in real-world data but is not a fundamental law, which is why it has limitations.
5. How does a single extreme value (outlier) impact the mean, median, and mode differently?
An extreme outlier affects each measure of central tendency very differently:
Mean: It is highly sensitive to outliers. Since every value is used in its calculation, a single very high or very low value can significantly pull the mean in that direction.
Median: It is resistant to outliers. The median is the middle value of an ordered dataset, so an extreme value at either end does not change its position.
Mode: It is generally unaffected by outliers, as it only represents the most frequently occurring value. An outlier is usually not the most common value.
6. In which situations does the empirical formula for mean, median, and mode fail or give inaccurate results?
The empirical formula provides a poor estimate and may fail in several cases, including:
For highly skewed distributions, where the data is stretched too far in one direction.
In bimodal or multimodal distributions, which have two or more distinct modes.
With datasets that contain extreme outliers that significantly distort the mean.
For very small datasets where a clear distribution shape isn't well-defined.
7. Why is it often better to use the median instead of the mean for real-world data like national income?
Understanding this relationship is crucial for interpreting data correctly. For example, in economics, a country's income is often reported using the median income because it is resistant to outliers. A few billionaires (extreme high-value outliers) can make the mean income seem very high and not representative of a typical citizen. If the mean income is much higher than the median, it indicates significant income inequality (a positively skewed distribution). The median provides a more accurate picture of the 'central' or typical income.
8. Is there a formal mathematical proof for the relationship between mean, median, and mode?
No, there is no universal mathematical proof for the formula Mode ≈ 3 Median – 2 Mean. This is precisely why it is called an empirical relationship. It was established by the statistician Karl Pearson through observation of numerous real-world datasets that were moderately skewed. It is an approximation that works well under specific conditions (unimodal, moderately asymmetrical distributions) but is not a fundamental mathematical theorem.
9. What does it signify if a dataset's mean is less than its median?
If the mean of a dataset is less than the median, it indicates that the data is negatively skewed (or skewed to the left). This happens when there are a few unusually low values that pull the mean down. The median, being the middle value, is less affected by these low outliers. This relationship (Mean < Median) is a key indicator of a distribution where the bulk of the data points are concentrated at the higher end, with a tail stretching towards the lower values.
