Mean Median Mode Formula

In our day to day life, we often hear that the average height of students of class 10 is 5 ft. and 5 inches or the average marks of 12^th Board exams were 75%, etc. But have you ever wondered how did this average get calculated? It is with the help of the Measures of Central Tendency. A Measure of Central Tendency refers to a single or individual value that defines the manner in which a group of data assembles around a central value. In other words, one single value describes the behavior pattern of the whole data group. Let’s start learning the three measures of central tendency i.e. Mean Median Mode Formula.

Mean Median Mode Formula

 What is Mean?

The mean of a series of data is the value equal to the sum of the values of all the observations divided by the number of observations. It is the most commonly used measure of central tendency. Also, it is very easy to calculate. We denote Mean by \(\overline{X}\).

Mean Formula

1. Individual Series

\( \overline{X} = \frac{Sum of all the values of the observations}{No. of observations} \)

2. Discrete Series

a. Direct Method:

\(\overline{X}= \frac{\sum fx}{\sum f} \)

Where,

f	Frequency
x	Values

b. Assumed Mean or Short-Cut Method:

\(\overline{X}= A + \frac{\sum fd}{\sum f} \)

Where,

f	Frequency
d	X – A
A	Assumed Mean

 c. Step-Deviation Method:

\(\overline{X}= A + \frac{\sum fd’}{\sum f} \times C \)

Where,

f	Frequency
A	Assumed Mean
d’	\(\frac{(X – A)}{C}\)
C	Common factor

 

3. Frequency Distribution or Continuous Series:
4. Direct Method:

\(\overline{X}= \frac{\sum fm}{\sum f} \)

Where,

f	Frequency
m	Mid-Values

 

1. Assumed Mean or Short-Cut Method:

\(\overline{X}= A + \frac{\sum fd}{\sum f} \)

Where,

f	Frequency
d	m – A
A	Assumed Mean

 

1. Step-Deviation Method:

\(\overline{X}= A + \frac{\sum fd’}{\sum f} \times C \)

Where,

f	Frequency
A	Assumed Mean
d’	\(\frac{(m – A)}{C} \)
C	Common factor

Weighted Arithmetic Mean Formula:

\(\overline{X}= \frac{\sum WX}{\sum W} \)

Where,

X	Values
W	Weights

What is the Median?

Median is the central or the middle value of a data series. In other words, it is the mid value of a series that divides it into two parts such that one half of the series has the values greater than the Median whereas the other half has values lower than the Median. For the calculation of Median, we need to arrange the data series either in ascending order or descending order.

1. Individual Series

2. When the number observations are odd

\(M = Size of  \frac{(N + 1)}{2}^{th} term \)

Where,

N	Number of observations

 

1. When the number observations are even:

\(M = \frac{\left \{Size of \frac{(N + 1)}{2}^{th} item+ Size of (\frac {N}{2} +1)^{th} item\right \}}{2} \)

Where,

N	Number of observations

 

2. Discrete Series:

\(Median = Size of  \frac{(N + 1)}{2}^{th} term \)

Where,

N	\(\sum f \)

In this case, the value corresponding to the cumulative frequency just greater than the value obtained after applying the above formula is the Median of the series.

3. Frequency Distribution or Continuous Series:

Firstly, we need to calculate the Median class by applying the following formula:

Median class = \(\frac{N}{2} \)

\(M = \frac{l}{2} + \frac{h}{f} \left [ \frac{N}{2} – c.f. \right ] \)

Where,

l	Lower limit of the Median class
h	Size of the median class
f	Frequency of the median class
N	Sum of frequencies
c.f.	Cumulative frequency of the class just preceding the median class

What is Mode?

Mode refers to the value that occurs a most or the maximum number of times in a data series.

Mode formula

1. Individual Series:

We find the mode of an individual series by simply inspecting it and finding the item that occurs maximum number of times.

2. Discrete Series:

The Mode of a discrete series is the value of the item that has the highest frequency.

3. Frequency Distribution or Continuous Series:

Firstly, we need to find out the Modal class. Modal class is the class with the highest frequency. Then we apply the following formula for calculating the mode:

Mode = l +  h    \( \frac{f_1 -f_0}{(2 f_1 -f_0 -f_2)} \)

Where,

L	lower limit of the modal class
f1	Frequency of the modal class
f0	Frequency of the class just preceding the modal class
f2	Frequency of the class just succeeding the modal class

Solved Example

1. Calculate the Average marks of the following series using Direct Method.

Marks	No. of students
0 – 10	10
10 – 20	30
20 – 30	70
30 – 40	50
40 – 50	20

Solution:

Marks	Mid-values (m)	No. of students (f)	fm
0 – 10	5	10	50
10 – 20	15	30	450
20 – 30	25	70	1750
30 – 40	35	50	1750
40 – 50	45	20	900
		\(\sum f = 180 \)	\(\sum fm = 4900 \)

\( \overline{X}= \frac{\sum fm}{\sum f} \)

\(\frac{4900}{180} \)

= 27.22