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1. MEASURES OF CENTRAL TENDENCY
DrSanjay Patil
Arithmetic Mean, Median & Mode.

2. MEASURES OF CENTRAL TENDENCY M
In statistics, a central tendency is a central value
or a typical value for a probability distribution.
• It is occasionally called an average or just the
center of the distribution.
• The most common measures of central tendency
are the arithmetic mean, the median and the
mode.
• Measures of central tendency are defined for a
population(large set of objects of a similar nature)
and for a sample (portion of the elements of a
population).

3. DEFINITIONS M
A measure of central tendency is a typical
value around which other figures gather” M
“An average stand for the whole group of which
it forms a part yet represents the whole
M
It is a single number of value which can be
considered typical in a set of data as a whole.

4. MEASURES OF CENTRAL TENDENCY M
Measures of central tendency are also usually
called as the averages.
• They give us an idea about the concentration of
the values in the central part of the distribution.
• The following are the 3 measures of central
tendency that are in common use:
• (i) Arithmetic mean, (ii) Median, (iii) Mode,

5. IMPORTANCE OF CENTRAL TENDENCY • To find representative value
• To make more concise data
• To make comparisons
• Helpful in further statistical analysis

6. MEAN M
• The MEAN of a set of values or
measurements is the sum of all the
measurements divided by the number of
measurements in the set.
M
• The mean is the most popular and widely
used. It is sometimes called the arithmetic
mean.

7. MEAN M
Mean for Ungrouped data
M
If we get the mean of the sample, we call it the
sample mean and it is denoted by (read “x
bar”).
• If we compute the mean of the population, we
call it the parametric or population mean,
denoted by µ (read “mu”).

8. EXAMPLE M
e.g.1. Calculate arithmetic mean for the
following series to know the mean size of
reaction of tuberculin test performed on 10
boys.
n
X
X
S
=
8
1080
10
12,11,10,9,8,8,7,7,5,3
= = =

9. ARITHMETIC MEAN IN
DECRETE
SERIES:
M
In discrete series observations are arranged
in asecendingorder according to the size of
item. Arithmetic mean is calculated by
multiplying size of item with corrosponding
frequencies. Add all such multiplication or
products and divide by the number of
observations in the series which will give the
value of arithmetic mean.

10. EXAMPLE M
Calculations: e.g. Following table gives the heights
in centimeters of 60 school children find the
avaragehight.
M
Hight
in cm (X)
No. of students (f)
(
fx
)
M
140121680
M
145101450
M
150081200
M
155162480
M
160091440
M
16505825
M
∑f 60∑
fx9075
2.151
ffx
X=
S
S
=

11. MEAN CHARACTERISTICS • It measures stability. M
Mean is the most stable among other measures of
central tendency because every score contributes
to the value of the mean.
• It may easily affected by the extreme scores.
• It may not be an actual score in the distribution.
• It can be applied to interval level of measurement.
• It is very easy to compute.

12. MEDIAN • Median is what divides the scores in the
distribution into two equal parts.
• It is also known as the middle score or the
50th percentile.
• Fifty percent (50%) lies below the median value
and 50% lies above the median value.

13. MEDIAN M
TheMEDIAN, denoted Md, is the middle value
of the sample when the data are ranked in
order according to size.
• it defined as “ The median is that value of the
variable which divides the group into two equal
parts, one part comprising of all values greater,
and the other, all values less than median”

14. MEDIAN M
1. Arrange the scores (from lowest to highest or
highest to lowest). Median of Ungrouped Data
M
2. Determine the middle most score in a
distribution if n is an odd number and get the
average of the two middle most scores if n is an
even number.

15. EXAMPLE M
1) Median in ungrouped series:
M
e.g. Calculate median for the given series -
M
90, 95, 102, 108, 96, 98, 120, 118, 115
M
Ascending order- 90, 95, 96, 98, 102, 108, 115, 118,
120 M
M
Median = Size of
M
M
M
M
102 is the median. (5 thitem.)
th
2
1n+
item
2
)1n(
th
+
item 5
2
19
th
=
+

16. MEDIAN IN DISCRETE SERIES: M
In discrete series observations are arranged
in ascending order according to size of item.
Cumulative frequency is worked out and size of
item corropsondingwithcumulative frequency is
median size.

17. EXAMPLE M
e.g. 1.Calculate median for the following discrete se ries.
Size of item (x)
Frequency (f)
Cumulative Frequency (
fx
)
3 4 4
6 7 11
9 10 21
12 12 33
15 6 39
18 3 42
21 2 44
total 44

18. D
Median = Size of
(
)
item
2
1n
th
+
item 5.22
2
1 44
th
=
+
22.5 lies in cumulative frequency group of
33 with corresponding size of item 12.
Median = 12.

19. MEDIAN
• It may not be an actual observation in the
data set.
• It is not affected by extreme values because
median is a positional measure. Properties
of the Median
• It can be applied in ordinal level.
• The exact midpoint of the score distribution
is desired. When to Use the Median
• There are extreme scores in the distribution.

20. MEDIAN M
It may not be an actual observation in the data
set.
• It is not affected by extreme values because
median is a positional measure. Properties of
the Median
• It can be applied in ordinal level.
• The exact midpoint of the score distribution is
desired. When to Use the Median
M
There are extreme scores in the distribution.

21. MODE M
The MODE, denoted Mo, is the value which
occurs most frequently in a set of
measurements or values.
M
In other words, it is the most popular value in a
given set.
M
Definition: “the mode of a distribution is the
value at the point armed with the item tend to
most heavily concentrated. It may be regarded
as the most typical of a series of value”

22. MODE M
Example 2: In a crash test, 11 cars were tested
to determine what impact speed was required
to obtain minimal bumper damage. Find the
mode of the speeds given in miles per hour
below. 24, 15, 18, 20, 18, 22, 24, 26, 18, 26,
24 Solution: Ordering the data from least to
greatest, we get: 15, 18, 18, 18, 20, 22, 24,
24, 24, 26, 26 Answer: Since both 18 and 24
occur three times, the modes are 18 and 24
miles per hour.

23. EXAMPLE M
Mode in ungrouped data:
M
In ungrouped data mode is located by eye inspection.
M
The size of item which has the greatest frequency.
M
e.g. Calculate mode for the following.
M
1) 10, 12, 15, 18, 18, 16, 15, 18, 20
M
The size of item 18 appears for maximum number item s
M
Mode = 18
M
2) 15, 18, 19, 21, 15, 18, 20, 24, 25.
M
This series is a bimodal series having two values mo de i.e.
15, 18

24. PROPERTIESOF MODE • It is used when you want to find the value
which occurs most often.
• It is a quick approximation of the average.
• It is an inspection average.
• It is the most unreliable among the three
measures of central tendency because its
value is undefined in some observations.

25. CONCLUSION • A measure of central tendency is a measure that te lls
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 divide d
by the number of numbers in a set of data. This is
also known as average.
• Median is the number present in the middle when
the numbers in a set of data are arranged in
ascending or descending order. If the number of
numbers in a data set is even, then the median is t he
mean of the two middle numbers.
• Mode is the value that occurs most frequently in a
set of data.

26. RELATION M
Relations Between the Measures of Central
Tendency
• In symmetrical distributions, the median and
mean are equal For normal distributions, mean
= median = mode
• In positively skewed distributions, the mean is
greater than the median
• In negatively skewed distributions, the mean is
smaller than the median

27. D
Thank you
D
Jay Research