Relationship of mean, median and mode | Derivation

Relationship of mean, median and mode | Derivation

Aditya Raj Anand

Saturday, 4 January 2025

math

The relationship between mean median and mode is the difference between mean and mode is almost equal to the three times of the difference between mean and median. It is also known as the empirical relation between mean median and mode. it is given by,

 Mean − Mode = 3 (Mean − Median)

Empirical relation between mean median and mode

The difference between Mean and mode is equal to the three times the difference between mean and median. that is given by, Mean − Mode = 3 (Mean − Median).

As we understand that the empirical relation between mean median and mode is Mean − Mode = 3 (Mean − Median). but how to get relation to the reference of mean, median and mode. so here below are the relation between mean median and mode taken mean, median, mode as single term.

general relation between mean, median and mode is, Mean − Mode = 3 (Mean − Median).

take Mean as single term:-

Mean − Mode = 3 (Mean − Median)

Soln:-

= Mean − Mode = 3 Mean − 3 Median

= 3 Median − Mode = 3 Mean − Mean

= 3 Median − Mode = 2 Mean

= (3 Median − Mode)/2 = Mean

∴ Mean =  (3/2 Median − 1/2 Mode 

take Median as single term:-

Mean − Mode = 3 (Mean − Median)

soln:-

= Mean − Mode = 3 Mean − 3 Median

= 3 Median − Mode = 3 Mean − Mean

= 3 Median − Mode = 2 Mean

∴ Median = 2/3 Mean + 1/3 Mode

take mode as single term:-

Mean − Mode = 3 (Mean − Median)

Soln:-

= Mean − Mode = 3 Mean − 3 Median

= 3 Median − Mode = 3 Mean − Mean

= 3 Median − Mode = 2 Mean

∴ Mode = 3 Median − 2 Mean

Please note:-

• that empirical relation between mean median and mode for normal distribution (or symmetrical distribution) is mean = median = mode. on the other hand the mean median and mode are equal to each other for normal distribution (or symmetrical distribution).

• The empirical relation between mean median and mode for asymmetrical distribution is mean ≠ mode ≠ median. on the other hand the mean median and mode are not equal to each other for asymmetrical distribution.

Relation between mean median and mode 

The relationship between the mean, median, and mode provides insights into the shape and distribution of data. Here's a breakdown of their relationship:

1. For Symmetric Distributions:

• When a dataset is symmetric (i.e., it has a balanced distribution of values around a central point), the mean, median, and mode tend to be very close to each other.

• Example: In a perfectly symmetric distribution (like a normal distribution), the mean, median, and mode will all be equal.

  Illustration:

  • Mean ≈ Median ≈ Mode

2. For Skewed Distributions:

Right (Positively) Skewed Distribution:

• In a right-skewed distribution (where there are more values on the left side of the distribution and a few larger values on the right), the mean is typically greater than the median, and the median is greater than the mode.

• Example: If most of the data points are small, but there are a few very large values, the mean is pulled in the direction of the larger values.

  Illustration:

  • Mode < Median < Mean

  This suggests that the data has a "tail" on the right side.

Left (Negatively) Skewed Distribution:

• In a left-skewed distribution (where there are more values on the right side and a few smaller values on the left), the mean is typically less than the median, and the median is less than the mode.

• Example: If most of the data points are large, but there are a few very small values, the mean is pulled toward the smaller values.

  Illustration:

  • Mean < Median < Mode

  This suggests that the data has a "tail" on the left side.

3. For Bimodal Distributions:

• In a bimodal distribution (which has two modes), the mean can be influenced by the values of both peaks. The median will fall between the two modes, and the mode will refer to the two most frequent values.

• In such cases, the relationship between the mean, median, and mode can vary greatly depending on the data.

  Illustration (Example with two modes):

• Mode 1 ≠ Mode 2, Median ≠ Mean.

Summary of the General Relationship:

• Symmetric Distribution: Mean ≈ Median ≈ Mode
• Right-Skewed Distribution: Mode < Median < Mean
• Left-Skewed Distribution: Mean < Median < Mode

This relationship helps in identifying the skewness of the data, which is useful for data analysis and interpretation.

Derivation of relation between mean median and mode

Mean − Mode = 3 (Mean − Median)

= Mean − Mode = 3 Mean − 3 Median

= 3 Median = 3 Mean − Mean + Mode

= 3 Median = 2 Mean + Mode

= Median = 2/3 Mean + 1/3 Mode

Hence, the formula of relation between mean median and mode is Median = 2/3 Mean + 1/3 Mode. 

 

Please note that you can also understand mean median and mode by the following examples.

mean median and mode examples

Suppose we want to compare the wage distribution of workers in two factories and determine which factory pays more to its workers. If we compare on the basis of individual workers, we cannot conclude anything. However, if for the given data, we get a representative value that signifies the characteristics of the data, the comparison become easy.

A certain value representative of the whole data and signifying its characteristics is called an average of the data.

Three types of averages are useful for analysing data. They are:-

1. Mean
2. Median
3. Mode

The relation between mean median and mode is a very famous relation and it is also called the empirical relation between mean median and mode as we know above. here is also a complete information about mean median and mode. we all are studying mean median and mode from class 10. but In class 10 we have a little bit idea that what is mean median and mode. here we will discuss about the actual meaning of mean median and mode. like what is mean? what is median? what is mode? in detail. so lets start with Mean.

What is mean?

Mean is that value of a given static data or for a data set, which is the sum of the values divided by the number of values. the arithmetic mean, also called the expected value or average value of any set number.

Mean can be calculated by various method like Direct method, Assumed-Mean method, Step-Deviation method, etc. every student study all these types of method in class 10. 

What is Median?

Median is defined as the middle value for a data set. In statistics the median is the value separating the higher half from the lower half of a given data.

What is Mode?

It is that value of a variate which occurs most often. More precisely, mode is that value of the variable at which the concentration of the data is maximum.

5 plus Example Question Using the Mean, Median and Mode Relationship.

Sure! Here's an example question using the relationship between the mean, median, and mode:

Question:

In a class, the ages of 7 students are as follows: 12, 15, 16, 17, 18, 18, 19.

1. What is the mean age of the students?
2. What is the median age?
3. What is the mode of the ages?

Now, let's use the relationship between mean, median, and mode:

• If the data is roughly symmetric, the mean, median, and mode are often close to each other.
• If the data is skewed, the mean will be pulled in the direction of the skew.

Answer:

1. Mean:
   Sum of ages = 12 + 15 + 16 + 17 + 18 + 18 + 19 = 115
   Mean = 115 / 7 = 16.43 (approximately)

2. Median:
   Since the data is ordered (12, 15, 16, 17, 18, 18, 19), the median is the middle value: 17.

3. Mode:
   The mode is the value that appears most frequently. Here, 18 appears twice, so the mode is 18.

Relationship Insight:

• The mean (16.43), median (17), and mode (18) are close, but the mean is slightly lower than the median and mode, indicating that the data might be slightly skewed to the left (lower values are pulling the mean down).

Certainly! Here are five more examples using the relationship between the mean, median, and mode:

Example 1:

Ages of 9 children in a family: 4, 6, 7, 7, 8, 10, 10, 12, 14.

1. Mean:
   Sum of ages = 4 + 6 + 7 + 7 + 8 + 10 + 10 + 12 + 14 = 78
   Mean = 78 / 9 = 8.67

2. Median:
   Ordered data: 4, 6, 7, 7, 8, 10, 10, 12, 14.
   The middle value is the 5th number: 8.

3. Mode:
   The mode is the number that appears most often: 7 and 10 both appear twice.
   So, the data is bimodal: 7 and 10.

Relationship Insight:
The mean (8.67) is greater than the median (8), suggesting a slight right skew, where higher values (10, 12, 14) are pulling the mean up.

Example 2:

Scores of 6 students in a test: 45, 50, 55, 55, 60, 80.

1. Mean:
   Sum of scores = 45 + 50 + 55 + 55 + 60 + 80 = 345
   Mean = 345 / 6 = 57.5

2. Median:
   Ordered data: 45, 50, 55, 55, 60, 80.
   The median is the average of the 3rd and 4th numbers: (55 + 55) / 2 = 55.

3. Mode:
   The mode is the value that appears most often: 55.

Relationship Insight:
The mean (57.5) is greater than the median (55), suggesting a slight right skew due to the outlier (80).

example 3:

Monthly salaries (in $1000) of 5 employees: 30, 35, 40, 42, 90.

1. Mean:
   Sum of salaries = 30 + 35 + 40 + 42 + 90 = 237
   Mean = 237 / 5 = 47.4

2. Median:
   Ordered data: 30, 35, 40, 42, 90.
   The middle value is the 3rd number: 40.

3. Mode:
   There is no repeated salary, so no mode.

Relationship Insight:
The mean (47.4) is significantly higher than the median (40), indicating a strong right skew, likely due to the outlier salary of 90

Example 4:

Heights (in cm) of 8 people: 150, 160, 160, 165, 170, 170, 180, 200.

1. Mean:
   Sum of heights = 150 + 160 + 160 + 165 + 170 + 170 + 180 + 200 = 1455
   Mean = 1455 / 8 = 181.875

2. Median:
   Ordered data: 150, 160, 160, 165, 170, 170, 180, 200.
   The median is the average of the 4th and 5th numbers: (165 + 170) / 2 = 167.5.

3. Mode:
   The mode is the value that appears most often: 160 and 170.

Relationship Insight:
The mean (181.88) is much higher than the median (167.5), suggesting a strong right skew, which is likely influenced by the outlier of 200.

Example 5:

Number of books read by 10 students: 3, 5, 6, 7, 7, 8, 8, 8, 9, 10.

1. Mean:
   Sum of books = 3 + 5 + 6 + 7 + 7 + 8 + 8 + 8 + 9 + 10 = 77
   Mean = 77 / 10 = 7.7

2. Median:
   Ordered data: 3, 5, 6, 7, 7, 8, 8, 8, 9, 10.
   The median is the average of the 5th and 6th numbers: (7 + 8) / 2 = 7.5.

3. Mode:
   The mode is the value that appears most often: 8.

Relationship Insight:
The mean (7.7) is slightly greater than the median (7.5), indicating that the distribution is slightly right-skewed.T

hese examples demonstrate how the relationship between the mean, median, and mode can provide insights into the distribution of the data, particularly in terms of skewness or symmetry.

What is the mean, median, and mode of 13 16 12 14 19 12 14 13 14?

Mean:-

Sum of the values:

$$13+16+12+14+19+12+14+13+14=117$$

Number of values:
There are 9 values in the dataset.

$$\text{Mean} = \frac{117}{9} = 13$$

So, the mean is 13.

Median:-

arrange the numbers in ascending order and then find the middle number.

Sorted data:

$$12, 12, 13, 13, 14, 14, 14, 16, 19$$

Since there are 9 numbers (an odd number), the median is the number in the middle position, which is the 5th value.

So, the median is 14.

Mode:-

The mode is the number that appears most frequently in the dataset.

From the sorted data, we see that:

• 12 appears 2 times,
• 13 appears 2 times,
• 14 appears 3 times,
• 16 appears 1 time,
• 19 appears 1 time.

Since 14 appears the most (3 times), the mode is 14.

Final Answer:

• Mean = 13
• Median = 14
• Mode = 14

What is the median of 3 7 2 4 7 5 7 1 8 8?

To find the median of the dataset 

3 7 2 4 7 5 7 1 8 8

Sorted data:

1,2,3,4,5,7,7,7,8,8

Since there are 10 values (an even number), the median will be the average of the 5th and 6th values.

• The 5th value is 5
• The 6th value is 7

$$\text{Median}=\frac{5+7}{2}=\frac{12}{2}=6\mathrm{<br>}\mathrm{<br>}$$

The median is 6.

What is the mean, median, and mode of the following data: 5, 10, 10, 12, 13?

Mean:- The mean is calculated by adding all the numbers together and then dividing by the number of values in the dataset.

Sum of the values:

$$5 + 10 + 10 + 12 + 13 = 50$$

Number of values: There are 5 values in the dataset.

$$\text{Mean} = \frac{50}{5} = 10$$

So, the mean is 10.

Median:- 

To find the median, we need to arrange the numbers in ascending order (although they are already in order) and find the middle value.

Sorted data:

$$5, 10, 10, 12, 13$$

Since there are 5 values (an odd number), the median is the middle value, which is the 3rd value in this sorted list.

So, the median is 10.

Mode:-

The mode is the number that appears most frequently in the dataset.

In this case:

• $5$ appears 1 time,
• $10$ appears 2 times,
• $12$ appears 1 time,
• $13$ appears 1 time.

Since 10 appears the most (2 times), the mode is 10.

Final answer 

• Mean = 10
• Median = 10
• Mode = 10

What is the median of 21 62 66 66 79 28 63 48 59 94 19?

Sorted data:

19,21,28,48,59,62,63,66,66,79,94 (Arranged in ascending order)

Now find the middle term,

Since there are 11 numbers (an odd number), the median is the middle value.

The middle value is the 6th value in the sorted list.

$$\text{Median}=62$$

Final answer 

The median is 62.

What is the median of 2 3 4 5 1 2 3 4 6 5?

Sorted data:

Arrange in Ascending order 

$$1,2,2,3,3,4,4,5,5,6$$

Since there are 10 numbers (an even number), the median will be the average of the 5th and 6th values.

The 5th value is $3$, and the 6th value is $4$.

$$\text{Median} = \frac{3 + 4}{2} = \frac{7}{2} = 3.5$$

Median=23+4​=27​=3.5

$$Final\; answer$$

The median is 3.5.

What is the mode of 10, 12, 11, 10, 15, 20, 19, 21, 11, 9, 10?

To find the mode of the dataset $10, 12, 11, 10, 15, 20, 19, 21, 11, 9, 10$, follow these steps:

1. Count the frequency of each number:

• $10$ appears 3 times
• $12$ appears 1 time
• $11$ appears 2 times
• $15$ appears 1 time
• $20$ appears 1 time
• $19$ appears 1 time
• $21$ appears 1 time
• $9$ appears 1 time

2. Identify the most frequent number:

The number that appears the most is 10, which appears 3 times.

Final answer 

The mode is 10.

What is equal to 3 median 2 mean?

median = mode + 2 mean.

3 median 2 mean equal to mode.