Difference Between Mean, Median and Mode

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Mean, Median, and Mode are the three measures of central tendency in Statistics. Central Tendency is a single value that describes the entire set of grouped or ungrouped data.

Mean is the arithmetic average of the given set of data values.
Median is the positional average of a dataset denoting the middlemost number or central value.
Mode is the value that has the highest frequency in a given set of values.

Difference between Mean, Median, and Mode lies in the different ways of calculating the data. Mean involves mathematical calculation whereas the median, and the mode can be calculated by simple observation of the data values. Mean is used to calculate the average value, Median is used to calculate the middle value, and Mode helps to calculate the most repeated value.

Table of Content

What is Mean?
What is Median?
What is Mode?
Difference Between Mean, Median and Mode
Relation Between Mean, Median and Mode
Mean, Median and Mode Solved Examples
Things to Remember
Sample Questions

Key Terms: Mean, Median, Mode, Measures of Central Tendency, Mean Formula, Mode Formula, Median Formula, Dataset

What is Mean?

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Mean is the mathematical average of a given set of grouped or ungrouped data.

It is calculated by dividing the sum of all the values by the number of values.
It is the ratio of the sum of all observations to the total number of observations.
Mean value represents the average value and may not be equal to any of the data values.
Mean is denoted by the symbol ‘x̄’.

Mean, Median and Mode Detailed Video Explanation

Mean Formula

Mean Formula is expressed as follows:

x̄ = (x₁+ x₂+ … + x_n)/n

It can also be written as:

x̄ = Σx_i/n

Where

x₁, x₂, … , x_n: Data Values
n: Total Number of Values.
Σx_i: Sum of all Observations

Solved Example

Example: Determine the Mean of the given values: 6, 7, 5.

Solution: The data values are 6, 7, and 5.

Sum of Observations (Σx_i) = 6 + 4 + 2 = 18

Number of Observations = 3

Using Mean Formula, we get

x̄ = Σx_i/n

x̄ = (18)/3 = 6

Thus, the mean of the given values is 6.

What is Median?

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Median of a group of observations is referred to as the value which equally divides the group into two parts.

Median is the value that exceeds and is exceeded by the same number of observations.
It helps determine the middle value in an observation.
It is the middlemost value in a dataset arranged in ascending order.
Median is Positional Average as the data value in the middle of the set is the median.

Read More:

Relevant Concepts
Mode Formula	Difference between Mean and Median	Mean of Grouped Data
Frequency Distribution Table	Class Mark	Statistics Formula
Mode of Grouped Data	Difference Between Average and Mean	Arithmetic Mean Formula

Median Formula

On the basis of the total number of data values, Median Formula is given as:

Median Formula When n is Odd
Median Formula When n is Even

Median Formula When n is Odd

Median Formula, when the number of observations is odd is:

Median = [(n + 1)/2]^th Term

Median Formula When n is Even

Median Formula, when the number of observations is even is:

Median = [(n/2)^th Term + ((n/2) + 1)^th Term]/2

Here, n refers to the number of observations.

Solved Example

Example: Determine the median for the data: 3, 3, 4, 6, 6, and 14.

Solution: The data values are 3, 3, 4, 6, 6, and 14.

On arranging in ascending order, we get 3, 3, 4, 6, 6, and 14.

Number of Observations, n = 8 (Even Number).

Using Median Formula When n is Even, we get

Median = [(n/2)^th term + ((n/2) + 1)^th term]/2

Median = [(6/2)^th term + ((6/2) + 1)^th term]/2

= (3^th term + 4^th term)/2 = (4 + 6)/2 = 10/2 = 5

Therefore, the median of 3, 3, 4, 6, 6, and 14 is 5.

Read More: Statistics Important Questions

What is Mode?

Mode of a set of observations is the value of the observations that occur the most number of times.

Mode is the most repeated value in a given set of observations.
It is also referred to as Modal value.
It has the highest bar if presented in a histogram or a chart form.
If there is no repeated number in a given set of values, then no Mode will exist for that series.

Solved Example

Example: What is the Mode of the given observation: {2, 1, 4, 4, 4, 6, 8, 4, 1, 8, 4, 4, 9, 4, 4}?

Solution: Given values are 2, 1, 4, 4, 4, 6, 8, 4, 1, 8, 4, 4, 9, 4, and 4.

Mode of {2, 1, 4, 4, 4, 6, 8, 4, 1, 8, 4, 4, 9, 4, 4} is 4.

This is because 4 has occurred the most number of times in the given observation

Difference Between Mean Median and Mode

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Mean, Median, and Mode are three important measures of central tendency that are co-related to each other.

Mean: Average of a Set of Observations
Median: Middlemost Value of a Set of Observations
Mode: Most Frequently Occurring Value of a Set of Observations

Difference Between Mean Median and Mode is given as follows:

Parameter	Mean	Median	Mode
Meaning	Mean is defined as the arithmetic average of a given set of data.	Median is the value of the middle-most observation obtained after arranging the data in ascending order.	Mode is the value the occurs the most number of times in an observation.
Formula	Mean = Sum of Observations/Total Number of Observations	Median Formula when n is Odd: M = [(n+1)/2]^thTerm	Most Frequently Occurring Observation or Value
Formula	Mean = Sum of Observations/Total Number of Observations	Median Formula when n is Even: M = [(n/2)^th Term + ((n/2) + 1)^th Term]/2	Most Frequently Occurring Observation or Value
Preference	Mean is preferred when the data is normally distributed.	Median is preferred when the data distribution is irregular.	Mode is preferred when the distribution of data is nominal.
Calculation	To calculate the mean, add all the values in the dataset and divide them by the number of observations.	To calculate the median, arrange the data in ascending and descending order, check the number of observations, and apply the formula accordingly.	Mode of a set of observations is the value of the observations that occur the most number of times.

Relation Between Mean Median and Mode

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In statistics, there exists a relation between mean, median, and mode, for a moderately skewed distribution. There is an “empirical relationship” between Mean Median and Mode. Relation Between Mean Median and Mode states that

Mode is equal to the difference between thrice the median and twice the mean.

It can be expressed mathematically as:

Mode = 3 Median – 2 Mean

The relationship is used to calculate the different values of mean, median, and mode.

Solved Example

Example: In a moderately skewed distribution, the Median of the dataset is 20 and the Mean is 22.5. What is the Mode?

Solution: Given that,

Mean = 22.5
Median = 20
Mode =?

Usng the Relation Between Mean Median and Mode

Mode = 3 Median – 2 Mean

Mode = 3 x 20 + 2 x 22.5 = 60 - 45

Mode = 15

Thus, the mode is 15.

Mean, Median and Mode Solved Examples

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Here are a few solved examples on Mean, Median, and Mode for a better understanding of the three measures of central tendency.

Example 1: Consider a set of values as 4, 2, 1, 6, 5, 3, 7, 1, 10, 9, and 8. What will be the Mean, Median, and Mode?

Solution: Given values are 4, 2, 1, 6, 5, 3, 7, 1, 10, 9, and 8.

Mean: Using the Mean Formula,

Mean = Sum of Observations/Total Number of Observations

Mean = (1+1+2+3+4+5+6+7+8+9+10)/10 = 56/11 = 5.09

Median: To calculate the Median, we need to arrange the values in ascending order,

1,1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Using the Median Formula for Odd values

Median = [(n+1)/2]^thTerm = [(11+1)/2]^th=12/2^th Term = 6^th Term = 5

Mode: 1 is the mode of the given data set as it is the most repeated value.

Example 2: What is the value of the mode of the data, if the mean is 12 and the median is 10?

Solution: Given that,

Mean = 12
Median = 10
Mode =?

The relation between Mean, Median, and Mode is

Mode = 3 Median – 2 Mean

Mode = 3 x 10 – 2 x 12 = 30 – 24

Mode = 6

Therefore, the mode of the given set of values is 6.

Check More:

Related Topics
Types of Data in Statistics	Graphical Representation of Data and Central Tendency	Graphical Representation of Data
Frequency Distribution Formula	Cumulative Frequency Distribution	Mean Absolute Deviation

Things to Remember

Mean, Median, and Mode are the three measures of central tendency in the discipline of Statistics.
Mean is defined as the average value of a given set of data.
Median is the middlemost value of an arranged set of data.
Mode is the most frequently occurring value in a given set of data.
Mean Formula is given as x̄ = Σx_i/n.
Median Formula (For Odd Number of Observations) is M = [(n+1)/2]^thTerm.
Median Formula (For Even Number of Observations) is M = [(n/2)^th Term + ((n/2) + 1)^th Term]/2.
Relation between Mean, Median, and Mode can be expressed as Mode = 3 Median – 2 Mean.

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Sample Questions

Ques. A batsman scored the following numbers of runs in six innings: 36, 35, 50, 46, 60, and 55. Calculate the mean runs scored. (3 Marks)

Ans. The runs scored by the batsman are 36, 35, 50, 46, 60, and 55.

Using the Mean Formula,

Mean = Sum of All Observation/Total Number of Observations

Mean Runs Scored = 36 + 35 + 50 + 46 + 60 + 55/6

Mean Runs = 282/6 = 47

Thus, the mean runs scored by the batsman in the six innings is 47.

Ques. Find the mean of the first 6 natural numbers. (3 Marks)

Ans. The first 6 natural numbers are 1, 2, 3, 4, 5, 6.

Using the Mean Formula,

Mean = Sum of All Observation/Total Number of Observations

Mean = (1 + 2 + 3 + 4 + 5 + 6)/6

Mean = 21/6 = 3.5

Thus, the mean of first six natural numbers is 3.5.

Ques. Which among the Mean, Median and Mode have the highest value for given data values: 13, 17, 22, 37, 56, 17, 29, and 34? (3 Marks)

Ans. Given data values are 13, 17, 22, 37, 56, 17, 29, and 34.

Arranging the values in ascending order, we get 13, 17, 17, 22, 29, 34, 37, and 56.

Mean = (13 + 17 + 17 + 22 + 29 + 34 + 37 + 56)//8 = 225/8 = 28.12

Median = (2 + 29)/2 = 25.5

Mode = 17

Thus, the mean has the maximum value.

Ques. Find the median of the data 23, 33, 48, 13, 15, 26, and 37. (3 Marks)

Ans. The observations are given as 23, 33, 48, 13, 15, 26, and 37.

Arrange the data in ascending order we get,

13, 15, 23, 26, 33, 37, 48

Here, n = 7 which is odd.

Using Median Formula When n is Odd,

Median = [(n + 1)/2]^th term

Median = Value of [(7 + 1)/2]^thobservation = Value of 4^th observation = 26.

Thus, the median is 26.

Ques. Find the median of the data 38, 32, 43, 44, 47, 26, 40, 46, and 33. (3 Marks)

Ans. The data values are 38, 32, 43, 44, 47, 26, 40, 46, and 33.

Arrange the data in ascending order we get,

26, 32, 33, 38, 40, 43, 44, 46, 47

Here, n = 9, which is odd.

Using Median Formula When n is Odd,

Median = [(n + 1)/2]^th term

Median = Value of [(9 + 1)/2]^thobservation = Value of 5^th observation = 40.

Thus, the median is 40.

Ques. Find the Mode of the data if it is given that the Mean is 3 and the Median is 1. (2 Marks)

Ans. Given that,

Mean = 3
Median = 1

The empirical relationship between Mean and Median is given as:

Mode = 3 Median – 2 Mean.

Mode = 9- 2 = 7

Thus, the Mode is 7.

Ques. Find the Mode of the Data: 1, 1, 2, 4, 2, 1, 2, 2, 4. (1 Mark)

Ans. Arrange the numbers with the same values together,

1,1, 1, 2, 2, 2, 2, 3, 4, 4

Clearly, 2 occurs the maximum number of times, so 2 is the mode of the given data.

Ques. The median of the observations 11, 12, 14, 17, x+2, x+4, 31, 32, 35, 41, arranged in ascending order is 24. Find the value of x. (3 Marks)

Ans. Given values are 11, 12, 14, 17, x+2, x+4, 31, 32, 35, and 41.

Here, n = 10 which is even.

If n is even, Using the Median Formula When n is Even,

Median = [(n/2)^th term + ((n/2) + 1)^th term]/2

Median = [(10/2)^th term + ((10/2) + 1)^th term]/2

24 = (5^th term + 6^th term)/2

24 = (x+2) + (x+4)/2

24= 2x + 6/2

24 = x+3

x = 21

Thus, the value of x is 21.

Ques. Find the median of the data: 20, 26, 60, 49, 36, 32, 31, 33, 52. If 26 is replaced by 53, what will be the new median? (3 Marks)

Ans. The data values are 20, 26, 60, 49, 36, 32, 31, 33, and 52.

Arranging the data in ascending order, we get

20, 26, 31, 32, 33, 36, 49, 52, 60

Here, n = 9 which is odd. Using Median Formula When n is Odd,

Median = [(n + 1)/2]^th term

Median = Value of (9+1/2)^th observation = Value of 5^th observation = 33

Hence, the median is 33.

When 26 is replaced by 53 the data in ascending order is:

20, 31, 32, 33, 36, 49, 52, 53, 60

Median = Value of (9+1/2)^th observation = Value of 5^th observation = 36.

Ques. Determine the Mean value of the given numbers: 12, 14, 10. (3 Marks)

Ans. Given numbers are 12, 14, and 10.

Sum of Observations = 12 + 14 + 10 = 36

Number of Observations = 3
Using the Mean Formula,

Mean = Sum of All Observation/Total Number of Observations

Mean Value = 36 / 3 = 12

Thus, the mean value of the numbers is 12.

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Difference Between Mean, Median and Mode

What is Mean?

Mean Formula

Solved Example

What is Median?

Median Formula

Median Formula When n is Odd

Median Formula When n is Even

Solved Example

What is Mode?

Solved Example

Difference Between Mean Median and Mode

Relation Between Mean Median and Mode

Solved Example

Mean, Median and Mode Solved Examples

Things to Remember

Sample Questions

CBSE X Related Questions

1.
The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.

2.
Find the sums given below :
\(7 + 10\frac 12+ 14 + ....... + 84\)
\(34 + 32 + 30 + ....... + 10\)
\(–5 + (–8) + (–11) + ....... + (–230)\)

4.
If 3 cot A = 4, check whether \(\frac{(1-\text{tan}^2 A)}{(1+\text{tan}^2 A)}\) = cos² A – sin²A or not

6.
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\frac{(\text{1 + tan² A})}{(\text{1 + cot² A})} = (\frac{\text{1 - tan A }}{\text{ 1 - cot A}})^²= \text{tan² A}\)

Similar Mathematics Concepts

Comments

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Difference Between Mean, Median and Mode

What is Mean?

Mean Formula

Solved Example

What is Median?

Median Formula

Median Formula When n is Odd

Median Formula When n is Even

Solved Example

What is Mode?

Solved Example

Difference Between Mean Median and Mode

Relation Between Mean Median and Mode

Solved Example

Mean, Median and Mode Solved Examples

Things to Remember

Sample Questions

CBSE X Related Questions

1. The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.

2. Find the sums given below :\(7 + 10\frac 12+ 14 + ....... + 84\)\(34 + 32 + 30 + ....... + 10\)\(–5 + (–8) + (–11) + ....... + (–230)\)

3. The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare themMonthly consumption (in units) Number of consumers65 - 85 485 - 1055105 - 12513125 - 14520145 - 16514165 - 1858185 - 2054

4. If 3 cot A = 4, check whether \(\frac{(1-\text{tan}^2 A)}{(1+\text{tan}^2 A)}\) = cos2 A – sin2 A or not

6. Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\frac{(\text{1 + tan² A})}{(\text{1 + cot² A})} = (\frac{\text{1 - tan A }}{\text{ 1 - cot A}})^²= \text{tan² A}\)

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.

2.
Find the sums given below :
\(7 + 10\frac 12+ 14 + ....... + 84\)
\(34 + 32 + 30 + ....... + 10\)
\(–5 + (–8) + (–11) + ....... + (–230)\)

4.
If 3 cot A = 4, check whether \(\frac{(1-\text{tan}^2 A)}{(1+\text{tan}^2 A)}\) = cos² A – sin²A or not

6.
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\frac{(\text{1 + tan² A})}{(\text{1 + cot² A})} = (\frac{\text{1 - tan A }}{\text{ 1 - cot A}})^²= \text{tan² A}\)