sher khan

1. MEASURES OF
CENTRAL TENDENCY
Presented by :-
Sher Khan
M.Sc. Ill Sem.
Department of Zoology
Shia P.G. College, Lucknow

2. “It is a sort of average or typical value of the
items in the series and its function is to
summarise the series in terms of this average
value”
Central tendency may be considered as synonym of
average. Averages, are generally the central part of
the distribution and therefore they are also called the
measures of central tendency.
The measures of central tendency is defined as :-
Introduction

3. Types of Measures of Central Tendency :-
There are usually three basic measures of central tendency. These
are :
(1)Mathematical average
(2)Average of position
(3)Measures of partition values
1. Mathematical average :-Average represented purely in
mathematical values are known as mathematical average. It is of
three types :-
(i) Arithmetic mean
(ii) Geometric mean
(iii) Harmonic mean.

4. 2. Average of position :- Mean exhibited by
position is called average of position. It is of
two types:-
(i) Median
(ii) Mode.
3.Measures of partition value:- It is measures of
location. It divides the total observations by an
imaginary line into two or more parts
expressed in percentage.

5. MATHEMATICAL AVERAGE
1. Arithmetic mean:- Average obtained
arithmetically is collect Arithmetic mean.
Arithmetic mean can be obtained both
ungrouped data and grouped data.
(i) Ungrouped data :- If the values of N items are
X1, X2, X3, Xn...... Be the value of variate X, then
simple arithmetic mean ( ) is obtained by dividing
the sum of the values of all the items by the total
number of observations

6. Arithmetic mean can be obtained by following
formula:-

7. (ii) Grouped data :- When data is presented in frequency
distribution then mean can be obtained by two methods.
(a) Discrete series
(b) continuous series
(a) Discrete series :- If data is in frequency distribution but not
in class interval, then it is called as discrete series.

8. Example

9. (b) Continuous series:- In this case the arithmetic mean is
calculated after taking into consideration of mid-points of
various classes

10. Example

11. MERITS AND DEMERITS OF ARITHMETIC MEAN
Merits :-
• It is rightly defined and is an easy and ideal
measures of central tendency.
•It covers all the observations and is easy to
calculate.
•It is affected least by fluctuation of sampling. In
other words arithmetic mean is a stable average.
•Arithmetic mean provides base of many other
methods of statistics.

12. Demerits :-
Obtained mean in a series may not be represented
by any observation.
It is very much affected by extreme observation.
By eliminating even a single series, calculation
becomes unreal.
It cannot be determined by inspection nor can be
represented graphically.
In extremely skewed distribution arithmetic mean is
not representative of the distribution.

13. 2. GEOMETRIC MEAN
The geometric mean is defined as the Nth
root of the product of n observations.

14. Example

15. MERITS AND DEMERITS OF GEOMETRIC MEAN
Merits:-
• It is based on all the observations.
•It is rigidly defined.
•It is capable of further algebraic
manipulation.
•It is not much affected by fluctuation of
sampling.
•It is particularly useful in dealing with
ratios, rates and percentages.

16. Demerits :-
•It cannot be used when any of the
quantities are zero or negative.
•It is difficult to calculate and interpret.
•It may come out to be a value which is
not existing in the series.

17. 3. HARMONIC MEAN
The Harmonic mean is defined as the “reciprocal of
the arithmetic mean of the reciprocals of the given
values” . For e.g. , reciprocal of 5 is 1/5, reciprocal
of 9 is 1/9 and so on. If variables are expressed in
ratios or rates, the proper average to be used is
Harmonic mean.

18. Example

19. MERITS AND DEMERITS OF HARMONIC
MEAN
Merits :-
• It is rightly defined.
•It is based on all observations of a series.
•It gives greater weightage to the smaller
items.
•It is useful to study the rate of respiration,
rate of pulse, heart beat etc. In unit time.
•It is not much affected by sampling
fluctuations.

20. Demerits:-
•It is not easy to calculate and understand.
•It cannot be calculated if one value is zero.
•It cannot be calculated if negative and
positive values are given in a series.
Relationship between Arithmetic mean,
Geometric mean and Harmonic mean
AM>GM>HM.

21. AVERAGE OF POSITION
MEDIAN:-
The value of the middle most observation,
when the data are arranged in ascending or
descending order of magnitude, is called the
median of the data.

23. Example

24. MERITS AND DEMERITS OF MEDIAN
Merits :-
•If found directly, it represents an actual item.
•The values of only the middle items are required
to be known.
•It is easy to calculate.
•It can also used in qualitative measures.
•It eliminates the effects of extreme items, since
they are not taken into account in its calculations,
except for arranging the data in increasing or
decreasing order.

25. Demerits:-
•Arithmetic explanation of median is not possible.
•To obtain data must be kept in ascending order
or descending order.
•It gives equal importance to all series.
•It is not very useful in further analysis, because it
is difficult to handle mathematically.

26. MODE
Mode is that value which is repeated maximum
times in a series. In other words we can say that
the mode is that value which has the maximum
frequency.

27. Example

28. MERITS AND DEMERITS OF MODE
Merits :-
•It avoids the effects extreme items.
•Often it can be ascertained by mere inspection.
•Only the values occurring with high frequencies are
required to be known for its determination. All
values need not be known.
•Bi-modal distribution may give a good indication of
the heterogeneity of a population.

29. Demerits:-
•It is not well defined and is rarely used for
higher life science researchers.
•Arithmetic explanation of mode is not
possible.
•Sometimes it is indefinite.
•It becomes difficult in multi-modal
distribution.
•It is not based on all the observation of a
series.

30. Relationship between Mean, Median,
and Mode
There is empirical relationship between Mean,
Median and Mode of a series of items . If distribution
of item values be symmetrical then the Mean,
median and Mode coincides, otherwise the distance
between the Mode and Median is usually twice the
distance the Median and the Mean. Thus:-
Mode-Median=2(Median-Mean)
Or, Mode-Mean=3(Median-Mean)

31. Thank You