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 relationship between mean, median, and mode?
The mean, median, and mode are measures of central tendency describing the center of a dataset. The mean is the average, the median is the middle value when data is ordered, and the mode is the most frequent value. Their relationship varies depending on the data's distribution: in a symmetrical distribution, they are equal; in a skewed distribution, they differ, with the mean being pulled towards the tail of the skew.
2. What is the empirical formula relating mean, median, and mode?
The empirical relationship, applicable to moderately skewed distributions, is expressed as: Mode = 3 × Median – 2 × Mean. This formula provides an approximation and may not hold precisely for all datasets. It is crucial to understand its limitations and the context in which it's applicable.
3. How is the empirical formula derived?
The formula's derivation typically involves considerations of skewness and the relative positions of mean, median, and mode in a skewed distribution. While rigorous mathematical proofs vary, the core idea is to express the relationship between the measures of central tendency within a specific distribution model.
4. How do mean, median, and mode relate in a positively skewed distribution?
In a positively skewed distribution, the mean is greater than the median, which is greater than the mode (Mean > Median > Mode). The long tail on the right pulls the mean towards higher values.
5. How do mean, median, and mode relate in a negatively skewed distribution?
In a negatively skewed distribution, the mean is less than the median, which is less than the mode (Mean < Median < Mode). The long tail on the left pulls the mean towards lower values.
6. What if I only know two of the three values (mean, median, mode)?
If you know any two of the three values (mean, median, mode) and the distribution is approximately symmetrical or moderately skewed, you can estimate the third value using the empirical formula: Mode = 3 × Median – 2 × Mean. Remember this is an approximation; the accuracy depends on the data's characteristics.
7. When does the empirical formula fail to provide an accurate estimate?
The empirical formula's accuracy diminishes in several cases: highly skewed distributions, bimodal or multimodal distributions, and datasets with extreme outliers. In these situations, the formula may not provide a reliable estimate of the missing value. Other statistical methods might be more appropriate.
8. Are there any other relationships between mean, median, and mode besides the empirical formula?
Yes, in a perfectly symmetrical distribution, the mean, median, and mode are all equal. This is a fundamental property illustrating the central tendency in perfectly balanced datasets.
9. How are mean, median, and mode used in real-world applications?
These measures are widely used in various fields. For instance, in finance, the mean helps analyze average returns; in healthcare, the median might represent the typical recovery time; and in marketing, the mode might indicate the most popular product.
10. Can the relationship between mean, median, and mode be visualized graphically?
Yes, using histograms or frequency distribution curves, you can visually inspect the relative positions of the mean, median, and mode and gain insights into the data's distribution and skewness. The positions of these central tendency values help illustrate the shape and balance of the data distribution.
11. What are the limitations of using only mean, median, and mode to describe a dataset?
While mean, median, and mode provide information about the center, they do not fully describe the dataset's characteristics. Measures of dispersion (like variance and standard deviation) are also essential to understand data spread and variability.
12. How can I improve my understanding of the relationships between mean, median and mode?
Practice solving problems with different types of datasets, both symmetrical and skewed. Visualizing data through histograms and frequency curves enhances understanding. Working through detailed examples and derivations will build a stronger foundation in this concept.
