This document discusses various measures of central tendency and dispersion used in statistics. It defines average, mean, median and mode as the main measures of central tendency. It provides formulas and methods to calculate arithmetic mean, weighted arithmetic mean, median and mode for both discrete and continuous data sets. The document also introduces absolute measures of dispersion like range, mean deviation, quartile deviation, standard deviation and relative measures of dispersion like coefficient of range, coefficient of mean deviation and coefficient of variation.

1. SRI KRISHNA COLLEGE OF TECHNOLOGY
SCHOOL OF MANAGEMENT
BMS
Measures of central tendency
Mr. R SARAVANAN
Assistant Professor

2. AVERAGE
• “Average is an attempt to find one single figure to describe whole of
figures”
• - Clark
• “An average value is a single value within the range the range of the
data that is used to represent all of the values in the series. Since an
average is somewhere within the range of the data, it is also called a
measure of central value”
• - Croxton & Cowden

3. Measures of
central
tendency
Simple Average
Median
Mean
Simple
Arithmetic Mean
Weighted
Arithmetic Mean
Mode
Special Average
Harmonic Mean
Geometric Mean

4. Measures of central tendency
• A single value which can be considered as typical or representative of a set of
observations and around which the observation cab be considered as center is called
an “ Average” or centre of location.
• Since such typical value tends to lie centrally within a set of observations when
arranged according to magnitudes, averages are called measures of central
tendency.

5. Measures of central tendency
• In the study of a population with respect to one in which we are interested we may get
a large number of observation.
• It is not possible to grasp any idea about the characteristics when we look at all the
observations.
• So it is better to get one number for one group.
• That number must be a good representative one for all the observation to give a clear
picture of that characteristic. Such representative number can be a central value for all
these observation.

6. Characteristics for a good average
It should be easy to understand and compute
It should be based on all items in the data
It should be capable of further algebraic treatment
It should not affected by extreme observation

7. ARITHMETIC MEAN
Arithmetic mean is the most important among the all
averages. The arithmetic mean of a given set in
individual series is their sum divided by the number of
observation.
The arithmetic average may be defined as
the sum of aggregate of a series of items
divided by their number.
-W I King The arithmetic mean of a series of items is
obtained by adding the values of the items
and dividing by the number of items.
- Croxton & Cowden

8. Calculation of Arithmetic Mean
Short cut Method
Direct Method
Ungrouped or raw data

9. Calculation of Arithmetic Mean
Short cut Method
Direct Method
Discrete series & Continuous series
Where
f= the frequency of individual class
N = Total frequencies
Where
f= the frequency of individual class
A = Any value in x
N = Total frequencies

10. Calculation of Arithmetic Mean
Step Deviation Method
Continuous series

11. Weighted Arithmetic Mean
• One of the limitations of the arithmetic mean is that it gives equal importance (weight)
to all the items in the series.
• But there are cases where relative importance of all the items are not equal.
• Weighted arithmetic mean is the correct tool for measuring the central tendency of the
given observation in such a cases.
• Here, the term weight stands for the relative importance of different items or
observation.

12. Weighted Arithmetic Mean

13. Weighted Arithmetic Mean
• Weighted mean should be applied under following situation
a) When the importance of each item in the frequency distribution is not same
b) The number of items in different groups of the data is widely different
c) In the calculation of ratios, percentages or rates a weighted average is more
representative.
d) When the change in the size of items is accompanied by the change in relative
proportion of the number of items.

14. MEDIAN
• Median is defined as the value of the middle item when the data are arranged in an
ascending or descending order of magnitude.
• Thus, in an ungrouped frequency distribution if the “n” values are arranged in
ascending or descending order of magnitude, the median is the middle value if ‘n’ is
odd.
• When ‘n’ is even, the median is the mean of the two middle value.
The median is that value of the variable which divides the group
into two equal parts, one part comprising all values greater than
and the other all values lesser than the median.
- Connor’s

15. Calculation of Median – Raw data

16. Median – Continuous series

17. MODE
• The mode refers to that value in a distribution, which occur most frequently. It is an
actual value, which has the highest concentration of items in and around it.
• The mode is the most frequently occurring value in a set of observation. Date may
have two modes.
• In this case, we say the data are bimodal, and set of observation with more than two
modes are referred to as multimodal.
The value of the variable which occurs most
frequently in a distribution is called the mode
- Kenney and Keeping

18. Continuous data
xC
l
Mode
2
1
1







19. Calculation of mode – continuous
data
f1 = frequency of the modal class
f0 = frequency of the class preceding the modal class
f2 = frequency of the class succeeding the modal class
C = the difference between upper limit and lower limit of modal class
c
l
Mode f
f
f
f
f


 


2
0
1
0
1
2
1

20. Measures of dispersion
• Dispersion or variation indicates spread of data or variation of data from the central
tendency.
Methods of measures of dispersion:
• In measuring dispersion, it is imperative to know the amount of variation (absolute
measure) and the degree of variation (relative measures).
• There are two kinds of measures of dispersion, namely
a) Absolute measures of dispersion
b) Relative measures of dispersion

21. Measures of
dispersion
Absolute
Measure
Range Mean Deviation
Quartile
Deviation
Standard
Deviation
Relative
measure
Coefficient of
Range
Coefficient of
mean deviation
Coefficient of
Quartile
deviation
Coefficient of
variation

22. Range and coefficient of range
• Range is the simplest possible measure of dispersion and is defined as the different
between the largest and smallest values of the variables.
• It is based only on two values, not on the entire data set.
Range = L – S
L = Largest Value
S = Smallest Value