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Relation Between Mean Median and Mode
The relation between mean, median, and mode is very important to understand in statistics and is useful while dealing with similar problems. In the case of a moderately skewed distribution, i.e. in general, the difference between mean and mode is equal to three times the difference between the mean and median. i.e., (Mean  Mode) = 3 (Mean  Median). This can be rearranged as 3 Median = 2 Mean + Mode and this is easy to remember.
1.  Relation Between Mean Median and Mode Formula 
2.  Empirical Relation Between Mean Median and Mode 
3.  FAQs 
Relation Between Mean Median and Mode Formula
Before learning about the relation between mean, median, and mode formula, let us revise the concepts of mean, median, and mode.
 The arithmetic mean refers to the average of a data set of numbers. It can either be a simple average or a weighted average. To calculate a simple average, we add up all the numbers given in the data set and then divide it by the total frequency.
 The median is the middle number of a given data set when it is arranged in either a descending order or ascending order. If there is an odd amount of numbers, the median value is the number that is in the middle whereas if there is an even amount of numbers, the median is the simple average of the middle pair in the dataset. Median is much more effective than a mean because it eliminates the outliers.
 The mode refers to the number that appears the most in a dataset. A set of numbers may have one mode, or more than one mode, or no mode at all.
The formula to define the relation between mean, median, and mode in a moderately skewed distribution is 3 (median) = mode + 2 mean. The proof of the mean, median, mode formula can be understood using Karl Pearson’s formula, which states:
(Mean  Median) = 1/3 (Mean  Mode)
3 (Mean  Median) = (Mean  Mode)
3 Mean  3 Median = Mean  Mode
3 Median = 3 Mean  Mean + Mode
3 Median = 2 Mean + Mode
Empirical Relation Between Mean Median and Mode
We will understand the empirical relation between mean, median, and mode by means of a frequency distribution graph. We can divide the relationship into four different cases:
 In the case of a moderately skewed distribution, Mean – Mode = 3 (Mean – Median).
 In the case of a frequency distribution that has a symmetrical frequency curve, the empirical relation states that mean = median = mode.
 In the case of a positively skewed frequency distribution curve, mean > median > mode.
 In the case of negatively skewed frequency distribution, mean < median < mode.
Observe the graph for each of the abovementioned cases and understand the relation and positioning of mean, median, and mode in the frequency distribution.
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Relation Between Mean, Median and Mode Examples

Example 1: It is given that in a moderately skewed distribution, median = 10 and mean = 12. Using these values, find the approximate value of the mode.
Solution: We know that the relationship between mean, median, and mode in a moderately skewed distribution is 3 median = mode + 2 mean. Let us take mode to be ‘x’. We have been given that the median = 10 and mean = 12. Now, using the relationship between mean, mode, and median we get,
3 × 10 = x + 2 × 12
30 = x + 24
x = 3024
x = 6
Answer: ∴ The value of mode is 6.

Example 2: Find the possible range of median of a positively skewed distribution, if the values of mean and mode are 30 and 20 respectively.
Solution: For a positively skewed frequency distribution, the empirical relation between mean, median, and mode is mean > median > mode. On the basis of this, the range of the median if the mean is 30 and mode is 20 is 30 > median > 20. It means that the median will be greater than 20 and less than 30.
Answer: ∴ 20 < median < 30.

Example 3: If the mean of a symmetrical distribution is 25, find its median and mode.
Solution:
We know that the mean, median, and mode of a symmetrical distribution are all the same.
Hence, median = 25 and mode = 25.
Answer: ∴ median = 25 and mode = 25.
FAQs on Relation Between Mean Median and Mode
What is the Relation Between Mean Median and Mode Class 10?
The most general relationship between mean, median, and mode is defined as 3 median = mode + 2 mean. This relation works for a moderately skewed frequency distribution.
What is the Relationship Between Mean, Median, and Mode for Positive Skewness?
The relationship between mean, median, and mode for positive skewness is mean > median > mode. Here, mean is greater than median and mode, and mode is the smallest value among mean, median, and mode.
What is the Empirical Relation between Mean Median and Mode?
The empirical mean, median, and mode relation is Mean – Mode = 3 (Mean – Median) for moderately skewed frequency distribution.
How are the Mean, Median, and Mode Related?
In general, mean, median, and mode are related with a formula 3 median = 2 mean + mode. If any two values are given, we can find the third value by using this formula.
What is the Relationship Between Mean, Median, and Mode in a Symmetrical Distribution?
In a symmetrical frequency distribution, mode = median = mean. In short, all three values are equal.
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