measures of central tendency

1. MEASURES OF
CENTRAL TENDENCY-
MODE

2. It is derived
from the
French word
”La Mode”
which means
fashion.
Mode is the
most
fashionable or
a typical
value of a
distribution
because it is
repeated the
highest
number of
times in the
series.
The mode is
by
definition,
the most
commonly
occurring
value.
ORIGIN

3. 1)According to Croxton
and Cowden, “The mode
of a distribution is the
value at the point around
which the items tend to be
most heavily
concentrated”.
2)A.M Tuttle said ,
“Mode is the value which
has the greater frequency
density in its immediate
neighborhood”.
Definition…

4. Types of Model Values
Unimodal Series
• The series of
observations
which contains
only one model
series
Bimodal Series
• The series of
observations
which contains
two modes
• In this the two
modes are the
same value of
greatest
density.
Multimodal
Series
• The series of
observations
which contains
more than two
modes.
• In this the
modes are the
same value of
greatest
density.

5. Types of Modal Series - Examples

6. As per definitions, mode is always the peak, irrespective of the
shape of the curve, symmetric or asymmetric.
⚫ Perfectly symmetrical distribution is the one in which one
mode (uni-mode) is present and the three measures of central
tendency (mean, median and mode) coincide with the highest
point. Eg. Normal (bell-shaped) distribution
⚫ Skewed or asymmetric distribution is the one in which the
data distribution indicates that the three measures (central) are
not the same.

7. Mode in symmetrical distributions

8. Mode in Positively skewed distributions -
curve is skewed to right

9. Mode in Negatively skewed distributions -
curve is skewed to left

10. Applications of Mode
Mode is used by business and commercial management. It is used for the
study of fashions and consumer perceptions.
Merits
Mode is not affected by extreme values. It can be easily understood.
Demerits
a) Mode estimation does not consider all the observations.
b) In bimodal distribution, it is not possible to estimate the mode.
c) It is an unstable measure, because it may change, if sampling fluctuations
are high.
d) The mode cannot be subjected to algebraic treatments.
e) Mode is not a rigidly defined measure, sometimes, grouping table and
analysis table are prepared to find mode, which is laborious.

11. I. Computation of Mode for Individual Series
The sizes of the particles (in a powder) are considered. The mode is calculated for
the data given below. 2, 4, 5, 8, 6, 5, 4, 2, 5, 6, 8, 5, 6, 4, 5, 8, 5, 6, 4 and 5.
In the above data, the number 5 is repeated several times (seven times).
No other number is repeated 5 times. Hence, the mode is 5 m.
II. Computation of Mode for Discrete Series
In case of grouped data, the mode can be estimated, under the discrete and
continuous series.
In the above example here, two highest frequency values are for
particle size 40 and 50.
The first impression will be that the mode lies in value 40 because the
highest frequency concentration of 80 is in it.
But f = 80 may not be true because the neighboring frequencies should
also be considered for mode(max 60+80+70).

12. Preparing Grouping Table
Column I: It has original frequencies and the maximum frequency is marked by bold type
Column II: In this column the frequencies of column I are combined ‘two by two’. (1 and 2;
3 and 4; 5and 6 and so on). Here also the maximum frequency is marked by bold type.
Column III: Here, we leave the first frequency of column I and combine the others in ‘two
by two’. (2 and 3; 4 and 5; 6 and 7 and so on). Again the maximum frequency is marked by
bold type.
Column IV: In this column the frequencies of column I are combined (grouped) in ‘three by
three’. (1, 2 and 3; 4, 5 and 6; 7, 8 and 9 and so on). And again the maximum frequency is
marked by bold type.
Column V: Here we leave the first frequency of the column I and group the others in ‘three
by three’. (2, 3 and 4; 5, 6 and 7; 8, 9 and 10 and so on). Again mark the maximum frequency
by bold type.
Column VI: Now leave the first two frequencies of column I and combine the others in
‘three by three’.(3, 4 and 5; 6, 7 and 8; 9, 10 and 11 and son on). Mark the maximum
frequency by bold type.

13. Preparing Analysis Table
After preparing the grouping table, we prepare the analysis table.
✔ 1) In the table put the column numbers on the left hand
side
✔ 2)various probable value of the mode on the right hand
side.
✔ 3)The values against which frequencies are maximum
marked in the grouping table and are entered by means
of a bar in the relevant ‘box’ corresponding to the
values they represent.
✔ 4)Find total frequency of marked bars

14. Practice Problem) Determine the modal size of particle from the
following data
Size of
Particle
4 5 6 7 8 9 10 11 12 13
Frequen
cy
2 5 8 9 12 14 14 15 11 13
Sol) We find that the value of the x variable 11 has frequency
maximum number of times i.e. 15.
We also notice that the difference between the frequencies of
the values of the variable, on both sides of 15, which are very
close to 11, is very small.
This shows that the values of the variable x are heavily
concentrated on either side of 11.
Therefore, if we find mode just by inspection gives error.

15. GROUPING
TABLE
Size
of
Part
icle(
x)
Freq
uency
(f)
Grouping
Col I
Col
II(1
&2..
)
Col
III
(2&3
..)
Col
IV(1, 2
and 3
Col V
(2, 3
and 4)
Col
VI 3,
4
and
5
4 2
7
15
5 5
13
22
6 8
17
29
7 9
21
35
8 12
26
40
9 14
28
43
10 14
29
40
11 15
26
39
12 11
24
13 13

16. ANALYSIS
TABLE
X
Col No
4 5 6 7 8 9 10 11 12 13
I 1
II 1 1
III 1 1
IV 1 1 1
V 1 1 1
VI 1 1 1
Total
frequenc
y
1 4 5 4 1
From the analysis table it is clear that the value 10 has the
maximum number of bars i.e. 6.
Hence the modal value is 10.

17. II. Computation of Mode for Continuous Series of
Data
The exact value of mode in the case of continuous frequency
distribution can be obtained by the following formulae:

18. Size range 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50
Frequency 12 26 30 18 14
Practice Problem) The particle size range and its distribution are
furnished below. Find the mode.
Size
range(x)
Freque
ncy(f)
Grouping
Col I Col II Col III Col IV Col V Col VI
0 – 10 12 38 68
10 – 20 26 56 74
20 – 30 30 48 62
30 – 40 18 32
40 – 50 14
Sol) Let us prepare Grouping and Analysis table
GROUPING TABLE

19. X
Col No
0 – 10 10 – 20 20 – 30 30 – 40 40 - 50
I 1
II 1 1
III 1 1
IV 1 1 1
V 1 1 1
VI 1 1 1
Total
frequency
1 3 6 3 1
ANALYSIS TABLE
From the analysis table it is clear that the class range 20 -30 has the
maximum number of bars i.e. 6. Hence the modal class is 20 - 30.

20. ⮚ It can be easily understood
⮚ It can be located in some cases by inspection, mode is that point where
there is more concentration of frequencies.
⮚ It is not affected by extreme values
⮚ It represents most frequent value and hence it is very useful in practice
⮚ It can be located in open end distributions
Merits, Demerits and uses of Mode
Merits
❖ The mode is not based on all the observations
❖ The value of the mode cannot be determined in bimodal
distribution
❖ It is unstable measure as it is affected more by sampling fluctuations
❖ It cannot be subjected to algebraic treatments.
❖ It is not rigidly defined measure sometimes as it is necessary to
prepare grouping table and analysis table to find modal class
DeMerits

21. EMPIRICAL RELATION BETWEEN MEAN, MEDIAN AND MODE
A distribution in which mean, median and mode coincide is called a
symmetrical distribution. If the distribution is moderately asymmetrical
then mean, median and mode are connected by the formula:
Size
range
0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80
Frequ
ency
4 6 20 32 33
17 8 2
Practice Problem) The particle size range and its distribution are furnished
below. Find the mode.
Sol) Let us prepare Grouping and Analysis table
Mode = 3 Median – 2 Mean

22. Size
range(x
)
Freque
ncy(f)
Grouping
Col I Col II Col III Col IV Col V Col VI
0 – 10 4
10
30
10 – 20 6
26
58
20 – 30 20
52
85
30 – 40 32
65
82
40 – 50 33
50
58
50 – 60 17
25
27
60 – 70 8
10
70 – 80 2
GROUPING TABLE

23. X
Col No
0 – 10 10 – 20 20 – 30 30 – 40 40 - 50 50 – 60 60 – 70 70 – 80
I 1
II 1 1
III 1 1
IV 1 1 1
V 1 1 1 1 1 1
VI 1 1 1
Total
frequency
1 3 5 5 2 1
ANALYSIS TABLE
This is Bimodal Series. Mode is to be ascertained by applying this formula.
Mode = 3 Median – 2 Mean

24. Size Range(x)
Mid value
(m)
Frequency
(f)
fm cf
0-10 5 4 20 4
10-20 15 6 90 10
20-30 25 20 500 30
30-40 35 32 1120 62
40-50 45 33 1485 95
50-60 55 17 935 112
60-70 65 8 520 120
70-80 75 2 150 122
∑f = 122=n ∑fm = 4820
Calculation of Mean, Median

25. Hence Mode = 3 Median – 2 Mean
Mode = 3(39.7) – 2(39.5)
= 119.1 – 79
= 40.1

26. Thank U