Measure of mode in biostatic

1. THE MODE:
The mode is defined as that value which occurs most
frequently in a set of data i.e. it indicates the most common
result.
EXAMPLE:
Suppose that the marks of eight students in a particular test
are as follows:
2, 7, 9, 5, 8, 9, 10, 9
Obviously, the most common mark is 9. In other words,
mode = 9.

2. MODE IN CASE OF RAW DATA
PERTAINING TO A CONTINUOUS VARIABLE
In case of a set of values (pertaining to a continuous
variable) that have not been grouped into a frequency
distribution (i.e. in case of raw data pertaining to a
continuous variable), the mode is obtained by counting the
number of times each value occurs.
Let us consider an example. Suppose that the
government of a country collected data regarding the
percentages of revenues spent on Research & Development
by 49 different companies, and obtained the following
figures:

3. Percentage of Revenues Spent on
Research and Development
Company Percentage Company Percentage
1 13.5 14 9.5
2 8.4 15 8.1
3 10.5 16 13.5
4 9.0 17 9.9
5 9.2 18 6.9
6 9.7 19 7.5
7 6.6 20 11.1
8 10.6 21 8.2
9 10.1 22 8.0
10 7.1 23 7.7
11 8.0 24 7.4
12 7.9 25 6.5
13 6.8 26 9.5
EXAMPLE

4. Company Percentage Company Percentage
27 8.2 39 6.5
28 6.9 40 7.5
29 7.2 41 7.1
30 8.2 42 13.2
31 9.6 43 7.7
32 7.2 44 5.9
33 8.8 45 5.2
34 11.3 46 5.6
35 8.5 47 11.7
36 9.4 48 6.0
37 10.5 49 7.8
38 6.9
Percentage of Revenues Spent on
Research and Development

5. DOT PLOT
The horizontal axis of a dot plot contains a scale for
the quantitative variable that we are wanting to represent.
The numerical value of each measurement in the data
set is located on the horizontal scale by a dot. When data
values repeat, the dots are placed above one another,
forming a pile at that particular numerical location.
4.5 6 7.5 9 10.5 12 13.5
R&D

6. 4.5 6 7.5 9 10.5 12 13.5
R&D
Xˆ= 6.9
Dot Plot
As is obvious from the above diagram, the value 6.9 occurs 3
times whereas all the other values are occurring either once
or twice.
Hence the modal value is 6.9.
Also, this dot plot shows that almost all of the R&D
percentages are falling between 6% and 12%, most of the
percentages are falling between 7% and 9%.

7. We will be interested to note that
mode is such a measure that can be
computed even in case of nominal
and ordinal levels of measurements.

8. For example
The marital status of an adult can be
classified into one of the following
five mutually exclusive categories:
Single, married, divorced, separated
and widowed.

9. Nominal scale is that where a certain
order exists between the groupings.
For example:
Speaking of human height, an adult
can be regarded as tall, medium or
short.

10. A company has developed five
different bath oils, and, in order to
determine consumer-preference, the
company conducts a market survey.

11. Number of Respondents favouring
various bath-oils
0
100
200
300
400
No.ofRespondents
I II III IV V
Mode
Bath oils

12. The largest number of respondents
favaoured bath-oil NO.II, as
evidenced by the bar-chart.
Thus, we can say that Bath-oil No.II is
the mode.

13. THE MODE IN CASE OF A DISCRETE FREQUENCY
DISTRIBUTION:
In case of a discrete frequency distribution,
identification of the mode is immediate; one simply finds that
value which has the highest frequency.
Example:
An airline found the
following numbers of
passengers in fifty flights of a
forty-seater plane.
No. of Passengers
X
No. of Flights
f
28 1
33 1
34 2
35 3
36 5
37 7
38 10
39 13
40 8
Total 50
Highest Frequency fm = 13
occurs against the X value 13.
Hence:
Mode = = 39Xˆ

14. THE MODE IN CASE OF THE FREQUENCY
DISTRIBUTION OF A CONTINUOUS VARIABLE:
In case of grouped data, the modal group is easily
recognizable (the one that has the highest frequency).
At what point within the modal group does the mode lie?

15.    
hx
ffff
ff
lX
mm
m
21
1ˆ



Mode:
Where
l = lower class boundary of the modal class,
fm = frequency of the modal class,
f1 = frequency of the class preceding the
modal class,
f2 = frequency of the class following modal
class, and
h = length of class interval of the modal class

16. EPA MILEAGE RATINGS
Class
Limit
Class
Boundaries
Frequency
30.0 – 32.9 29.95 – 32.95 2
33.0 – 35.9 32.95 – 35.95 4
36.0 – 38.9 35.95 – 38.95 14
39.0 – 41.9 38.95 – 41.95 8
42.0 – 44.9 41.95 – 44.95 2
Total 30

17. Class
Limits
Class
Boundaries
No. of
Cars
30.0 – 32.9 29.95 – 32.95 2
33.0 – 35.9 32.95 – 35.95 4 = f1
36.0 – 38.9 35.95 – 38.95 14 = fm
39.0 – 41.9 38.95 – 41.95 8 = f2
42.0 – 44.9 41.95 – 44.95 2
EPA MILEAGE RATINGS

18. It is evident that the third class is the modal class.
The mode lies somewhere between 35.95 and 38.95.
In order to apply the formula for the mode, we
note that fm = 14, f1 = 4 and f2 = 8.
Hence we obtain:

19.    
825.37
875.195.35
3
610
10
95.35
3
814414
414
95.35Xˆ









Hence, we obtained:

20. 0
2
4
6
8
10
12
14
16
29.95
32.95
35.95
38.95
41.95
44.95
Miles per gallon
NumberofCars
X
Y

21. 0
2
4
6
8
10
12
14
16
28.45
31.45
34.45
37.45
40.45
43.45
46.45
Miles per gallon
NumberofCars
X
Y
The frequency polygon of the same distribution was:

22. 0
2
4
6
8
10
12
14
16
28.45
31.45
34.45
37.45
40.45
43.45
46.45
Miles per gallon
NumberofCars
X
Y
Frequency curve was as indicated by the dotted line in the following figure:

23. Xˆ = 37.825
0
2
4
6
8
10
12
14
16
28.45
31.45
34.45
37.45
40.45
43.45
46.45
Miles per gallon
NumberofCars
X
Y
In this example, the mode is 37.825, and if we locate this value on the X-axis,
we obtain the following picture:

24. Example
The following table contains the ages
of 50 managers of child-care centers
in five cities of a developed country.

25. Ages of a sample of managers
of Urban child-care centers
42 26 32 34 57
30 58 37 50 30
53 40 30 47 49
50 40 32 31 40
52 28 23 35 25
30 36 32 26 50
55 30 58 64 52
49 33 43 46 32
61 31 30 40 60
74 37 29 43 54
Convert this data into Frequency Distribution and
find the modal age.

26. Frequency Distribution of
Child-Care Managers Age
Class Interval Frequency
20 – 29 6
30 – 39 18
40 – 49 11
50 – 59 11
60 – 69 3
70 – 79 1
Total 50

27. Mode:
Where
l = lower class boundary of the modal class,
fm = frequency of the modal class,
f1 = frequency of the class preceding the
modal class,
f2 = frequency of the class following modal
class, and
h = length of class interval of the modal class
   
hx
ffff
ff
lX
mm
m
21
1ˆ




28. Hence, the mode is given by
   
18 6ˆ 29.5 10
18 6 18 11
12
29.5 10
12 7
120
29.5
19
29.5 6.3 35.8
X

  
  
  

 
  

29. Mode:
X
79.559.549.539.529.519.5
20
15
10
5
0
69.5
Ages of Managers
Y
No.ofManagers
ˆ 35.8X 
Mode = 35.8

30. PROPERTIES
OF THE MODE
•The mode is easily understood and easily ascertained in case
of a discrete frequency distribution.
•It is not affected by a few very high or low values.
The question arises, “When should we use the mode?”
The answer to this question is that the mode is a valuable
concept in certain situations such as the one described below:

31. Suppose the manager of a men’s clothing store is asked about
the average size of hats sold. He will probably think not of the
arithmetic or geometric mean size, or indeed the median size.
Instead, he will in all likelihood quote that particular size
which is sold most often. This average is of far more use to him
as a businessman than the arithmetic mean, geometric mean or
the median.
The modal size of all clothing is the size which the businessman
must stock in the greatest quantity and variety in comparison
with other sizes.
On the other hand, sometimes a frequency distribution
contains two modes in which case it is called a bi-modal
distribution as shown below:
EXAMPLE

32. f
0 X
THE BI-MODAL FREQUENCY
DISTRIBUTION