Relation Between Mean Median Mode: Formula and Solved Example

Mean Median Mode Relation Overview

Understanding the mean median mode relation is crucial for dealing with similar issues in statistics. In the case of a moderately skewed distribution, or generally speaking, the difference between the mean and median is equal to three times the difference between the mean and median.

Mean Median Mode Relation: Mean

Mean has been learned by most of the people in middle school math as the norm. The mean is what you get by combining all the values and dividing the total by the number of values, given a set of values. You have a set of values called the population X₁,......,X_nwritten in math notation.

The mean number is very helpful. It describes the group's property. It is important to understand that the mean is not an entity-in reality, there can be anything whose value matches the mean, but the mean is a summarized representation of the population.

Example: Calculate the average value of the following numbers: 6, 7, and 5.

Solution: To calculate the value of Mean,

The given observations' total is equal to 6 + 4 + 2 = 18.

Therefore, we must divide the sum by the total number of values in order to determine the mean value.

3 values in total make up the question in this case.

Consequently, Mean Value = 18 / 3 = 6

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Mean Median Mode Relation: Median

The median frequently serves as a better proxy for the average group member. The median will be the number in the middle of a list if all the values are taken and arranged in ascending order. A characteristic that all group members share is the median. The mean might not be particularly close to the quality of any group member, according to the value distribution. The mean is also subject to skewing; even one value that differs noticeably from the rest of the group can significantly alter the mean. The median gives you a central group member without the skew effect that outliers introduce. The median value in a normal distribution represents a representative sample of the population.

Example: Calculate the median between the numbers 42, 16, and 36.

Solution:The number 16 is the median of the numbers 42, 16, and 36 because it falls in the middle when the numbers are written as (42, 16, 36). and, Median helps determine the middle value in an observation.

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Mean Median Mode Relation: Mode

The group's most prevalent member is the mode. It doesn't matter which value is the most prevalent; the mode is always the largest or smallest in the group. Because they are typically the least meaningful, the majority of these three median measures are also the least used. However, it can be useful occasionally. The median, mean, and mode will all be the same if your data is both regular and flawless. This is practically unimaginable and never occurs in real life.

Example: What is the given observation's mode value? {2, 1, 4, 4, 4, 6, 8, 4, 1, 8, 4, 4, 9, 4, 4}

Solution: The given observation's mode value is 4. Its values are 2, 1, 4, 4, 4, 6, 8, 4, 1, 8, 4, 4, 9, 4. This is due to the fact that 4 occurred the most frequently during the course of the observation.

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Mean Median Mode Relation Formula

The equation 3 (median) = mode + 2 mean can be used to define the mean median mode relation in a moderately skewed distribution. Karl Pearson developed the following formula, which can be used to understand the proof of the mean, median, and mode formula:

(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

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Empirical Relationship between Mean, Median and Mode

When dealing with a distribution that is moderately skewed, the difference between the mean and mode can be approximated as being three times the difference between the mean and median. This relationship between the empirical mean, median, and mode can be expressed as:

Mean – Mode = 3 (Mean – Median)

Or

Mode = 3 Median – 2 Mean

Either of these two ways of expressing equations can be used as per convenience since, by expanding the first representation, we get the second one as shown below:

Mean – Mode = 3 (Mean – Median)

Mean – Mode = 3 Mean – 3 Median

By rearranging the terms,

Mode = Mean – 3 Mean + 3 Median

Mode = 3 Median – 2 Mean

However, we can define the relation between mean, median and mode for different types of distributions, as explained below:

Mean Median Mode Relation with Frequency Distribution

·Frequency Distribution with a Symmetrical Frequency Curve

When a frequency distribution graph displays a symmetrical frequency curve, the values of the mean, median, and mode will be identical.

Mean = Median = Mode

·For Positively Skewed Frequency Distribution

For a frequency distribution that is positively skewed, the mean will always be greater than the median, and the median will always be greater than the mode.

Mean > Median > Mode

·For Negatively Skewed Frequency Distribution

If a frequency distribution has a negative skew, then the mean will always be less than the median, and the median will always be less than the mode.

Mean < Median < Mode

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Difference between Mean Median Mode

The following table has all the information about the difference between the mean, mode, and median on the following basis:

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Hints to remember the Difference Between Mean Median Mode

Do you struggle to understand the mean, median, and mode differences? Here are a few pointers that might be useful.

1. French for fashionable, "à la mode" also describes a common method of serving ice cream. The most well-liked or fashionable number in a set is called "Mode." The word MOde is also like MOst.
2. The "mean" one is the one that requires you to perform mathematical operations (adding and dividing all the numbers).
3. Both "Middle" and "Median" have the same number of letters. 

Things to Remember

1. The empirical relationship between mean, median, and mode defines mode as the difference between three times the median and two times the mean. 
2. Karl Pearson's formula is used to express how the Mean, Median, and Mode are related to one another.
3. The middle number, if there are an odd number of numbers, is the median value.
4. The median is the simple average of the middle pair in the dataset when there are an equal number of numbers.
5. Because it removes the outliers, a median is much more useful than a mean.
6. There could be one mode, several modes, or no modes at all for a given set of numbers.
7. Mean, median, and mode all attempt to condense a dataset into a single number that can be used to represent a "typical" data point.

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Sample Questions using mean median mode relation

Question 1: The mean is 22.5 and the median is 20, in a moderately skewed distribution. Find the mode's approximation using these values. 

Solution: Given,

Mean = 22.5

Median = 20

Mode = x

Using the correlation between mean, mode, and median that we have discovered,

(Mean – Mode) = 3 (Mean – Median)

So,

22.5 – x = 3 (22.5 – 20)

22.5 – x = 7.5

∴x = 15

So, Mode = 15.

Question 2: In a distribution that is moderately skewed, the median value is 10 and the mean value is 12. Find the approximate value of the mode using these values. 

Solution: We know that in a moderately skewed distribution, the relationship between the mean, median, and mode is 3 median = mode + 2 mean. Let's assume that mode is "x." It has been stated that the mean is 12 and the median is 10. Now, using the relationship between mean, mode, and median, we get,
3 × 10 = x + 2 × 12
30 = x + 24
x = 30-24
x = 6
Therefore, the value of mode is 6.

Question 3: If the mean and mode are 30 and 20, respectively, then determine the potential range of the median for a positively skewed distribution. 

Solution: The empirical relationship between the mean, median, and mode for a frequency distribution that is positively skewed is mean > median > mode.Based on this, the range of the median is 30 > median > 20 if the mean is 30 and the mode is 20. The median will therefore be higher than 20 and lower than 30.

Question 4: If the mode is 35.3 and the mean is 30.5, then use an empirical formula to find the median of the data.

Answer: Mode = 3(Median) – 2(Mean)

35.3 = 3(Median) – 2(30.5)

35.3 = 3(Median) – 61

96.3 = 3 Median

Median = 96.33 = 32.1

Question 5: Given that the mean and median of a moderately skewed distribution are 10 and 12, respectively. Find the mode's approximation using these values. 

Answer: We are aware that in a moderately skewed distribution, the relationship between the mean, median, and mode is 3 median = 2 mode + 2 mean. Let's assume that "x" is the mode. It has been stated that the mean is 12 and the median is 10. Now that we have the mean, mode, and median relationship, we can:

3 × 10 = x + 2 × 12

30 = x + 24

x = 30-24

x = 6

Therefore, the value of mode is 6.