This document provides an overview of measures of central tendency in statistics, focusing on the arithmetic mean. It defines the arithmetic mean as the sum of all values divided by the total number of items. It discusses different types of series (individual, discrete, continuous) and methods to calculate the arithmetic mean for each type, including the direct method, short-cut method, and step deviation method. The document provides examples of calculating the arithmetic mean using these different methods for an individual series of student marks.

1. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Overview:
● Meaning
● Objective and functions of averages
● Requisites of a measures of central tendency
● Arithmetic Mean
● Charlier’s accuracy Check
● Properties of arithmetic Mean
● Combined Mean
● Corrected Mean
● Weighted mean
● Median
● Quartiles
● Mode

2. Meaning:
● Measure of central tendency is a single value, which is used to represent an entire
set of data.
● Also known as ‘Average’ or ‘Measures of Location’.
● The following 3 principal measures are widely used in statistical analysis:
● Arithmetic Mean;
● Median;
● Mode;
● A measure of central tendency is a summary statistic that represents the central
point or typical value of a dataset.
● These measures indicate where most values in a distribution fall and are also
referred to as the central location of a distribution.

3. ● To present huge data in summarised form: It is very difficult to grasp large number
of numerical figures.Averages summaries such data into a single figure, which
makes easier to understand and remember.
● To make Comparison easier: Averages are very helpful for making comparative
studies as they reduce the mass of statistical data to a single figure.
● To help in Decision-making: Average provide such values, which becomes a
guidelines for decision makers.Most of the decisions to be taken in research or
planning are based on the average value of certain variables.

4. Objective and Functions of Averages:
● To know about universe from a sample: Averages also help in obtaining an idea
of a complete universe by means of sample data. The averages of a sample
presents a clear picture of the averages of the population.
● To trace precise relationship: Average becomes essential when it is desired to
establish relationship between different groups in Quantitative terms.
● Base for computing other measures: Averages offer a base for computing various
other measures like dispersion, Skewness,kurtosis that help in many other phases
of statistical analysis.

5. Measures of Central Tendency:
Types of Averages
MathematicalAverages Commercial Averages
Positional Averages
- Arithmetic Mean ( X )
- GeometricMean (G)
- HarmonicMean (H)
- Median
(M or Me)
- Mode (Z)
- Moving Average
- Progressive
Average
- Composite
Average

6. Requisites of a Measure of Central Tendency:
Rigidly defined:
● An average should be clear and rigid so that there is no confusion and there is one
and only one interpretation.
● Preferably, it should be defined by an algebraic formula, so that the average
computed from a set of data by anybody remains the same.
Based on all the observations:
● Average should be calculated by taking into consideration each and every item of
the series. If it is not based on all the items, it cannot be said to be representative
of the whole group.
● It should be least affected by fluctuations of sampling.

7. It should be least affected by fluctuations of sampling:
● An average should possess sampling stability.
● If we take two or more independent random samples of the same size from a
given population and compute the average for each, then the values so obtained
from different samples should not differ much from one another.
Capable of further Algebraic Treatment:
● Average should be capable of further mathematical and statistical analysis to
expand its utility.

8. Easy to understand and calculate:
● The value of an average should be computed by using a simple method without
reducing its accuracy and other advantages.
Not affected much by Extreme values:
● The value of an average should not be affected much by extreme values. If one or
two very small or very large items unduly affect the average, the average value
may not truly represent characteristics of the entire set of data.

9. Arithmetic Mean:
● Arithmetic Mean is a simple average of all items in a series.
● It is the simplest measure of central tendency.Thearithmetic mean of a series
is called ‘Mean’.
● Definition: Arithmetic Mean or Mean is the number which is obtained by
adding the values of all the items of a series and dividing the total by the
number of items.

10. Arithmetic Mean:
● Formula:
Arithmetic mean is generally written as X.
X = X1+ X2+X3+...........Xn = ∑ X
N N
X1+ X2+X3+...........Xn are the values of different items in the series.
N = total number of items
∑ is a sign called sigma.It refers to the sum total of the values of different
items in the series.

11. Types of Arithmetic Mean:
● Arithmetic mean is of two types:-
● Simple Arithmetic Mean: In it, all items of a series are given equal
importance.
● Weighted Arithmetic Mean: In it, different items of a series are
assigneddifferent weights in accordance with their relative
importance.

12. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Methods of Calculating Simple Arithmetic Mean:
● We know, there are types of statistical series:
● Individual Series
● Discrete Series
● Continuous Series
Arithmetic mean may be calculated with respect to these series using different
methods.

13. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Individual Series:
● Individual series is the series in which items are listed singly, i.e.each item is given
a separate value.It means, individual series do not have frequencies of the items.
Calculation of Arithmetic Mean in Individual Series.
● Mean is computed by taking the sum of all observations and dividing the
sum by the number of observations in the set. There are 3 methods to
calculate arithmetic mean of individual series:
● Direct Method;
● Short-Cut Method;
● Step Deviation Method;

14. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Discrete Series:
● In case of discrete series, values of variable shows the repetitions,i.e., frequencies
are given corresponding to different values of variables.
● The total number of observations,i.e.,N=Sum total of frequency = ∑f.
● Arithmetic mean in a discrete series can be computed by applying:
● Direct Method;
● Short-Cut Method;
● Step Deviation Method;

15. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Continuous Series:
● In case of continuous series(grouped frequency distribution), the value of a
variable is grouped in various class-intervals(like 10-20,20-30,etc)along with their
respective frequencies.
● The process of the calculation of arithmetic average in a continuous series is same
as in case of a discrete series.
● The mid-points of the various class-intervals are used to replace the class-interval.
● In the continuous series also, the following three methods are used to calculate
arithmetic mean:
● Direct Method;
● Short-Cut Method;
● Step Deviation Method;

16. Direct Method:
Individual Series:
● According to this method, all the units are added and then their total is divided by the number of
items and the quotient becomes the arithmetic mean.
Steps of Direct Method:-
1. Let the items(observations) be X1+ X2+X3+...........Xn.
2. Add up the values of all the items and obtain the total,i.e, ∑X.
3. Find out total number of items in the series, i.e., N.
4. Divide total value of all items(∑X) by total number of items (N); i.e. X = ∑X/N
{Where , X= Arithmetic Mean;
∑X = Sum of all the values of items;
N = Total number of items}

17. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Direct Method:
Individual Series:
Example 1. The marks obtained by 10 students in a subject are:
Calculate Arithmetic Mean by Direct Method
Students A B C D E F G H I J
Marks 85 60 50 75 55 40 55 70 45 65

18. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Direct Method:
Individual Series:
Solution: Total Marks(∑X)=600 Marks; Total number of Students(N)=10
Arithmetic Mean(X) = ∑X = 600/10 = 60 marks
Ans: Arithmetic Mean = 60 marks
The direct method is generally used when there are few items and the size of the figures
is small.If it is not so, there would be considerable difficulty in the calculation of mean.
To remove this difficulty, a ‘Short-Cut Method’ is used.

19. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Short-CutMethod:
Individual Series:
● Under this method, any figure is assumed as the mean and deviations are
calculated from this assumed mean.
● The need for Short-Cut Method arises when there are large number of
observations or it is difficult to compute arithmetic mean by direct method.
● This method is also called ‘Assumed Mean Method’.

20. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Steps of Short-Cut Method:
● Let the items (observations) be X1+ X2+X3+...........Xn.
● Decide any item of the series as assumed mean(A).
● Calculate the deviations(d) of items from assumed mean(A),i.e. Deduct A
from each items of the series,i.e., X-A
● Take the sum total of deviations and denote it as ∑d.
● Find out the total number of items in the series,i.e., N.

21. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Steps of Short-Cut Method:
● Apply the following formula: X = A+ ∑d
N
{Where, X = Arithmetic mean;
A = Assumed mean;
d = X-A,i.e., deviations of variables from assumed mean;
∑d = ∑(X-A), i.e., sum of deviations of variables from assumed mean;
N=total number of items}

22. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Short-Cut Method:
Individual Series:
Example: Calculate the arithmetic mean of the marks (individual series) by the Short-Cut
Method(Assumed Mean Method)
Students A B C D E F G H I J
Marks 85 60 50 75 55 40 55 70 45 65

23. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Short-Cut Method:
Individual Series:
Solution:
Calculation of Arithmetic Mean(Short-Cut Method)
Students Marks
(∑X)
d=X-A
(A=50)
A 85 +35
B 60 +10
C 50 0
D 75 +25
E 55 +5
Students Marks
(∑X)
d=X-A
(A=50)
F 40 -10
G 55 +5
H 70 +20
I 45 -5
J 65 +15
N=10 ∑X=600 ∑d=100

24. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Short-Cut Method:
Individual Series:
Solution: Arithmetic Mean (X) = A + ∑d = 50 + 100= 60 marks
N 10
Ans: Arithmetic Mean = 60 marks

25. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Step Deviation Method:
Individual Series:
● Step Deviation Method further simplifies the short-cut method.
● In this method, deviations from assumed mean are divided by a common factor(C)
to get step deviations. Then, these step deviations are used to calculate the value
of arithmetic mean.

26. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Step Deviation Method:
Steps of Step Deviation Method:
● Let the items(observations) be X1+ X2+X3+...........Xn.
● Decide any item of the series as assumed mean(A).
● Calculate the deviations(d) of items from assumed mean(A), i.e., deduct A
from each item of the series.
● Find out common factor (C) from d and calculate d’ (step deviations) which is
d/C.
● Take the sum total of step deviation(d’) and denote it as ∑d’.
● Find out the total number of items in the series,i.e.,N.

27. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Step Deviation Method:
● Apply the following formula: X = A + ∑d’ x C
N
{Where, X = Arithmetic mean;
A = Assumed mean;
d = X-A,i.e., deviations of variable from assumed mean;
d’ = X-A/C ,i.e. Step deviations(deviations divided by common factor);
∑d’ = Sum of Step Deviations;
C= Common Factor;
N=Total number of items}

28. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Step Deviation Method:
Individual Series:
Example 1. The marks obtained by 10 students in a subject are:
Calculate Arithmetic Mean by Step Deviation Method
Students A B C D E F G H I J
Marks 85 60 50 75 55 40 55 70 45 65

29. CLASS 11th ECONOMICS - CHAPTER - INTRODUCTION
Step Deviation Method:
Individual Series:
Solution: Calculate of Arithmetic Mean(Step Deviation Method)
Students Marks
(∑X)
d=X-A
A=50
d’ = X-A
C
C=5
A 85 +35 +7
B 60 +10 +2
C 50 0 0
D 75 +25 +5
E 55 +5 +1

30. Step Deviation Method:
Individual Series:
Solution: Calculate of Arithmetic Mean(Step Deviation Method)
Students Marks
(∑X)
d=X-A
A=50
d’ = X-A/C
C=5
F 40 -10 -2
G 55 +5 +1
H 70 +20 +4
I 45 -5 -1
J 65 +15 +3
N=10 d’=20

31. Step Deviation Method:
Individual Series:
Solution: Arithmetic Mean(X) = A + d’ x C = 50 + 20 x 5 =60 marks
N 10
Ans: 60 Marks