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 ShortCut Method:
\(\overline{X}= A + \frac{\sum fd}{\sum f} \)
Where,
f  Frequency 
d  X – A 
A  Assumed Mean 
c. StepDeviation 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 
 Frequency Distribution or Continuous Series:
 Direct Method:
\(\overline{X}= \frac{\sum fm}{\sum f} \)
Where,
f  Frequency 
m  MidValues 
 Assumed Mean or ShortCut Method:
\(\overline{X}= A + \frac{\sum fd}{\sum f} \)
Where,
f  Frequency 
d  m – A 
A  Assumed Mean 
 StepDeviation 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.

Individual Series

When the number observations are odd
\(M = Size of \frac{(N + 1)}{2}^{th} term \)
Where,
N  Number of observations 

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 

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.

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

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

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

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 
f_{1}  Frequency of the modal class 
f_{0}  Frequency of the class just preceding the modal class 
f_{2}  Frequency of the class just succeeding the modal class 
Solved Example
 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  Midvalues (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
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