This presentation discusses various measures of central tendency including the mean, median, mode, harmonic mean and geometric mean. It provides definitions and formulas for calculating each measure along with their merits and demerits. The mean is the sum of all values divided by the number of values and can be affected by outliers. The median is the middle value when values are arranged from lowest to highest. The mode is the most frequent value. Harmonic and geometric means are other types of averages.

1. PRESENTATION OF SNM
► Fiston roll no. 25
► MARWADI UNIVERSITY
► B.CHEMICAL ENG, 4TH SEM

2. CONTENT:
► CENTRAL TENDENCIES
► MEAN
► MODE
► MEDIAN
► HORMONIC MEAN AND
GEOMETRIC MEAN
► THEIR MERTS AND DIMERTS
► THEIR APPLICATIONS

3. MEASURES OF CENTRAL
TENDANCY:
• Statisticians have developed four
different methods for measuring
central tendency;
1.The Mean
2.The Median
3.The Mode
4.HARMONIC MEAN AND

4. Continue
..
• A measure of central tendency is
a single value that attempts to
describe a set of data by
identifying the central position
within that set of data. As such,
measures of central tendency
are sometimes called measures
of central location.

5. Continue
..
• They are
also
classed as
summary statistics.
• The
mean
called
the
likely
the
average)
measure
(often
is
most
of
centr
al
tendency
that you are most familiar with,
but there are others, such as
the median and the mode.

6. Continue
..
• The mean, median and
mode,harmonic and geometric mean
are all valid measures of central
tendency, but under different
conditions, some measures of
central tendency become more
appropriate to use than others.
• In the following sections, we will look
at the mean, median and mode, and
learn how to calculate them and
under what conditions they are most
appropriate to be used.

7. 1. Mean
The mean is the sum of the value of each observation in
a dataset divided by the number of observations. This is
also known as the arithmetic average.
The mean, often called the average, of a numerical set of
data, is simply the sum of the data values divided by the
number of values. The mean is the balance point of a
distribution. The calculations for the mean of a sample and
the total population are done in the same way. However, the
mean of a population is constant, while the mean of a sample
varies from sample to sample

8. Continued
The mean of a sample or a population is computed
by adding all of the observations and dividing by the
number of observations.
Population mean = μ = ΣX / N OR Sample mean
= x = Σx / n
where ΣX is the sum of all the population
observations, N is the number of population
observations, Σx is the sum of all the sample
observations, and n is the number of sample
observations.

9. Continued
When statisticians talk about the mean of a
population, they use the Greek letter μ to
refer to the mean score. When they talk
about the mean of a sample, statisticians
use the symbol x to refer to the mean score.

10. Arihtmetic Mean (group-data) :
Formula:
A.M=ΣfX/Σf
where
X = Individual
value
f = Frequency
The symbol of Σ is pronounced as sigma and is used to
represent the sum of number.

11. Ungrouped data
Ungrouped data is the opposite of grouped data
with only one possible answer.
For example: The ages of 200 people
entering a park on a Saturday afternoon. The
ages are:
27, 8, 10, 49 etc.

12. Arithmetic Mean (ungroup-
data)
Formula:
Mean = sum of elements / number of
elements
= a1+a2+a3+.....+an/n
Arithmetic Mean = ΣX/n
where
X = Individual value
n = Total number of values

13. Advantages of Mean:
•
•
•
It is easy to understand & simple calculate.
It is based on all the values.
It is easy to understand the arithmetic
average even if some of the details of the
data are lacking.

14. Disadvantages of Mean:
•
•
•
•
It is affected by extreme values.
It cannot be calculated for open end classes.
It cannot be located graphically
It gives misleading conclusions.

15. 2.The Median
• The second measure of central
tendencies
• The goal of the median is to
locate the midpoint of the
distribution.
• There are no specific symbols
or notions to identify the
median

16. Conti.
…
• The median is simply identified by
the word median.
• In addition, the definition and the
computations for the median are
identical for a sample and for a
population.

17. The
Definition
• If the scores in a distribution
are listed in order from
smallest to largest
• The median is the midpoint of
the list
• More specifically, the median is
the point on the measurement
scale below which 50% of the
scores in the distribution are
located.

18. Finding the Median
for most
distribution
• The scores are divided into
equal-sized group.
• We are not locating the
midpoint from highest to lowest
X values
• To find the median, list the
scores in order from smallest
to largest

19. Conti.
…
• Begin with the smallest score
and count the score as you
move up the list
• The median is the first point you
reach that is greater than of
50% of the score in the
distribution
• The median can be equal to a
score in the list or it can be a
point between two scores

20. Conti.
…
• Notice that the median is not
algebraically defined (there is
no equation for computing the
median)
• Means that there is a degree of
subjectivity in determining the
exact value

21. 4.GEOMETRIC MEAN

22. HARMONIC MEAN

23. EMPIRICALRELATIONBETWEENMEDIANANDMODE

24. Merits
1. Simplicity
– It is very simple measure of central
tendency
effect
of
1. Free fromthe
extreme values
2. Real value
• Representative
compared to
value
as
arithmeticmean
average, the value of which may
not exist in the series at all.

25. Demerits
Unrealistic
– When the median is located somewhere
between the two middle values, it remains
only an approximate measure, not a
precise value.

26. Cont.
…
Lack of representative character
– limited representative character as it is
not based on all the items in the series.
Lack of algebraic treatment
– Arithmetic mean is capable of further
algebraic treatment, but median is not.

27. 3.Mode
Mode is the most frequent
value or score in the distribution.
It is defined as that value of the item in
a series.
It is denoted by the capital letter Z.
Highest point of the
frequencies distribution curve.

28. Merits of Mode :
•
•
•
•
•
Mode is readily
comprehensible and easily
calculated
It is the best representative of data
It is not at all affected by extreme
value.
The valueof mode can
also be determined
graphically.
It is usually an actual value of an

29. Demerits of
Mode
•
•
It is not based on
all observations.
It is not capable of
further
mathematical
manipulation.
• Mode
is
extent
affected to a
great by
sampling
•
fluctuations.
Choice of grouping has
great influence on the
value of mode.

30. Conclusion:
A measure of central tendency is a
measure that tells us where the
middle of a bunch of data lies.
Mean is the most common measure
of central tendency. It is simply the
sum of the numbers divided by the
number of numbers in a set of data.
This is also known as average.

31. Continue…
► Median is the number present
in the middle when the numbers
in a set of data are arranged in
ascending or descending order..
► Mode is the value that occurs
most frequently in a set of data.

32. THANK YOU
FOR YOUR
INTENTION
!!.