Measures of Central Tendency in Statistics

Central tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence of large datasets in a single, representative number.

The representative value of a data set, generally the central value or the most occurring value that gives a general idea of the whole data set is called Measure of Central Tendency.

Central Tendency

The three most commonly used measures of central tendency are mean, median, and mode.

Mean

Mean in general terms is used for the arithmetic mean of the data, but other than the arithmetic mean there are geometric mean and harmonic mean as well that are calculated using different formulas. Here in this article, we will discuss the arithmetic mean.

Mean for Ungrouped Data

Arithmetic mean (\bar{x}) is defined as the sum of the individual observations (x_i) divided by the total number of observations N. In other words, the mean is given by the sum of all observations divided by the total number of observations.

\bold{\bar{x} = \dfrac{\sum x_i}{N}}

OR

Mean = Sum of all Observations ÷ Total number of Observations

Example: If there are 5 observations, which are 27, 11, 17, 19, and 21, then the mean (\bar{x}) is given by

\bar{x} = (27 + 11 + 17 + 19 + 21) ÷ 5
⇒ \bar{x}= 95 ÷ 5
⇒ \bar{x} = 19

Mean for Grouped Data

Mean (\bar{x}) is defined for the grouped data as the sum of the product of observations (x_i) and their corresponding frequencies (f_i) divided by the sum of all the frequencies (f_i).

\bold{\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}}

Example: If the values (x_i) of the observations and their frequencies (f_i) are given as follows:

then Arithmetic mean (\bar{x}) of the above distribution is given by

\bar{x} = (4×5 + 6×10 + 15×8 + 10×7 + 9×10) ÷ (5 + 10 + 8 + 7 + 10)
⇒ \bar{x} = (20 + 60 + 120 + 70 + 90) ÷ 40
⇒ \bar{x} = 360 ÷ 40
⇒ \bar{x} = 9

Related Resources,

• Mean Using Direct Method
• Shortcut Method for Arithmetic Mean
• Mean Using Step Deviation Method

Types of Mean

Mean can be classified into three different class groups, which are

• Arithmetic Mean
• Geometric Mean
• Harmonic Mean

Arithmetic Mean: The formula for the Arithmetic Mean is given by 

\bold{\bar{x} = \dfrac{\sum x_i}{N}}

Where,

• x₁, x₂, x₃, . . ., x_n are the observations, and
• N is the number of observations.

Geometric Mean: The formula for Geometric Mean is given by 

\bold{\text{G.M.} = \sqrt[n]{x_1\cdot x_2\cdot x_3\cdot \ldots \cdot x_n}}

Where,

• x₁, x₂, x₃, . . ., x_n are the observations, and
• n is the number of observations.

Harmonic Mean: The formula for Harmonic Mean is given by 

\bold{\text{H. M. } = \frac{n }{1/x_1 + 1/x_2 +\ldots + 1/x_n}}

OR

\bold{\text{H. M. } = \frac{n }{\sum (1/x_i)}}

Where,

• x₁, x₂, . . ., x_n are the observations, and
• n is the number of observations.

Properties of Mean (Arithmetic)

There are various properties of Arithmetic Mean, some of which are as follows:

• The algebraic sum of deviations from the arithmetic mean is zero, i.e., \bold{\sum{(x_i - \bar{x})} = 0}.
• If it \bold{\bar{x}}is the arithmetic mean of observations, and a is added to each of the observations, then the new arithmetic mean is given by \bold{\bar{x'} =\bar{x}+a}
• If \bold{\bar{x}}is the arithmetic mean of observations, and a is subtracted from each of the observations, then the new arithmetic mean is given by \bold{\bar{x'} =\bar{x}-a}
• If \bold{\bar{x}} is the arithmetic mean of observations, and a is multiplied by each of the observations, then the new arithmetic mean is given by \bold{\bar{x'} =\bar{x}\times a}
• If \bold{\bar{x}} is the arithmetic mean of observations, and each of the observations is divided by a, then the new arithmetic mean is given by \bold{\bar{x'} =\bar{x}\div a}

Median

Median of any distribution is that value that divides the distribution into two equal parts such that the number of observations above it is equal to the number of observations below it. Thus, the median is called the central value of any given data, either grouped or ungrouped.

Median of Ungrouped Data

To calculate the Median, the observations must be arranged in ascending or descending order. If the total number of observations is N, then there are two cases

Case 1: When N is Odd

Median = Value of observation at [(n + 1) ÷ 2]^th Position

When N is odd the median is calculated as shown in the image below.

Case 2: When N is Even

Median = Arithmetic mean of Values of observations at (n ÷ 2)^th and [(n ÷ 2) + 1]^th Position

When N is even the median is calculated as shown in the image below.

Example 1: If the observations are 25, 36, 31, 23, 22, 26, 38, 28, 20, 32, then the Median is given by 

Arranging the data in ascending order: 20, 22, 23, 25, 26, 28, 31, 32, 36, 38

N = 10 which is even then

Median = Arithmetic mean of values at (10 ÷ 2)th and [(10 ÷ 2) + 1]th position
⇒ Median = (Value at 5th position + Value at 6th position) ÷ 2
⇒ Median = (26 + 28) ÷ 2
⇒ Median = 27

Example 2: If the observations are 25, 36, 31, 23, 22, 26, 38, 28, 20, then the Median is given by

Arranging the data in ascending order: 20, 22, 23, 25, 26, 28, 31, 36, 38

N = 9 which is odd then
Median = Value at [(9 + 1) ÷ 2]^th position
⇒ Median = Value at 5^th position
⇒ Median = 26

Median of Grouped Data

Median of Grouped Data is given as follows:

\bold{Median =l+ \dfrac{N/2 - c_f}{f} \times h}

Where,

• l is the lower limit of the median class,
• n is the total number of observations,
• c_f is the cumulative frequency of the preceding class,
• f is the frequency of each class, and 
• h is the class size.

Example: Calculate the median for the following data.

Solution:

Create the following table for the given data.

Class	Frequency	Cumulative Frequency
10 - 20	5	5
20 - 30	10	15
30 - 40	12	27
40 - 50	8	35
50 - 60	5	40

As n = 40 and n/2 = 20,
Thus, 30 - 40 is the median class.
l = 30, c_f = 15, f = 12, and h = 10

Putting the values in the formula  \bold{Median =l+ \dfrac{N/2 - c_f}{f} \times h}
Median = 30 + (20 - 15)/12) × 10
⇒ Median = 30 + (5/12) × 10
⇒ Median = 30 + 4.17
⇒ Median = 34.17

So, the median value for this data set is 34.17

Mode

Mode is the value of that observation which has a maximum frequency corresponding to it. In other, that observation of the data occurs the maximum number of times in a dataset.

Mode of Ungrouped Data

Mode of Ungrouped Data can be simply calculated by observing the observation with the highest frequency. Let's see an example of the calculation of the mode of ungrouped data.

The mode of the data set is the highest frequency term in the data set, as shown in the image added below.

Example: Find the mode of observations 5, 3, 4, 3, 7, 3, 5, 4, 3.

Solution:

Create a table with each observation with its frequency as follows:

xi	5	3	4	7
fi	2	4	2	1

Since 3 has occurred a maximum number of times i.e. 4 times in the given data;

Hence, Mode of the given ungrouped data is 3.

Mode of Grouped Data

Formula to find the mode of the grouped data is:

\bold{Mode = l +\left [\dfrac{f_1-f_0}{2f_1-f_0-f_2}\right]×h}

Where,

• l is the lower class limit of modal class, 
• h is the class size, 
• f₁ is the frequency of modal class, 
• f₀ is the frequency of class that proceeds the modal class, and 
• f₂ is the frequency of class that succeeds the modal class.

Example: Find the mode of the dataset which is given as follows.

Solution:

As the class interval with the highest frequency is 40-50, which has a frequency of 16. Thus, 40-50 is the modal class. 

Thus, l  = 40 , h = 10 , f₁ = 16 , f₀ = 12 , f₂ = 10 

Plugging in the values in formula \bold{Mode = l +\left [\dfrac{f_1-f_0}{2f_1-f_0-f_2}\right]×h}             , we get

Mode = 40 + (16 - 12)/(2 × 16 - 12 - 10) × 10
⇒ Mode = 40 + (4/10)×10
⇒ Mode = 40 + 4
⇒ Mode = 44

Therefore, the mode for this set of data is 44.

Measures of Central Tendency: Formulas Table

The formulas for calculating mean, median, and mode vary slightly depending on whether the data is ungrouped or grouped. The table below summarizes the key formulas for each measure of central tendency:

the

Empirical Relation Between Measures of Central Tendency

The three central tendencies are related to each other by the empirical formula, which is given as follows:

2 × Mean + Mode = 3 × Median

This formula is used to calculate one of the central tendencies when two other central tendencies are given.

Solved Question on Measures of Central Tendency

Question 1: Find the mean of the following numbers: 12, 18, 25, 30, 15

Solution:

Mean = Sum of all Observations ÷ Total number of Observations
Mean = 12 + 18 + 25 + 30 + 15 / 5
Mean = 100 / 5 = 20

Question 2 : Find the median of the following numbers: 5, 8, 12, 3, 10

Solution:

Arrange in order: 3, 5, 8, 10, 12
Middle value = 8

Question 3 :Find the mode of the following data: 6, 8, 6, 9, 10, 6, 11.

Solution:

6 appears most frequently (3 times)

Question 4 :Find the mean of the following data:

Solution:

Create the following table for the given data.

Class Interval	Frequency (f)	Midpoint(x)	f × x
10-20	5	15	75
20-30	8	25	200
30-40	12	35	420
40-50	16	45	720
50-60	9	55	495
Total	50		1910

\bold{\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}}
\bold{\bar{x} = \dfrac{1910}{50}} = 38.2

Question 5 :Find the Median of the following data:

Solution:

Create the following table for the given data.

Class Interval	Frequency (f)	Cumulative Frequency (CF)
10-20	5	5
20-30	10	15
30-40	12	27
40-50	8	35
50-60	5	40
Total	40

• Total frequency, N = 40, so N/2 = 20
• Median class = 30–40

\bold{Median =l+ \dfrac{N/2 - c_f}{f} \times h}
Where:
L = 30, F = 15, f = 12, h = 10
\bold{Median =30 + \dfrac{20 - 15}{12} \times 10}= 34.16

Question 6 :Find the Mode of the following data:

Solution:

Create the following table for the given data.

Class Interval	Frequency (f)
0-10	3
10-20	7
20-30	12
30-40	15 (Highest frequency)
40-50	10
50-60	5

• Median class = 30–40
• L = 30
• f₁ = 15
• f₀ = 12
• f₂ = 10

\bold{Mode = l +\left [\dfrac{f_1-f_0}{2f_1-f_0-f_2}\right]×h}
\bold{Mode = 30 +\left [\dfrac{15-12}{2(15)-12-10}\right]×10}
{Mode = 30 +\left [\dfrac{3}{8}\right]×10}
Mode = 30 + 3.75
Mode = 33.75

Unsolved Question on Measures of Central Tendency

Question 1: Find the mean of the following numbers: 10, 18, 30, 45, 15, 30.

Question 2 : Find the median of the following numbers: 4, 8, 6, 3, 15.

Question 3 :Find the mode of the following data: 5, 8, 8, 9, 10, 6, 8, 11.

Question 4 :Find the mean of the following data:

Question 5 :Find the Median of the following data:

Question 6 :Find the Mode of the following data: