Solution:

Let us form the frequency distribution table of the given data as follows:

Clearly, 2 is the number of wickets taken by the bowler in the maximum number (i.e., 3) of matches. So, the mode of this data is 2.


In a grouped frequency distribution, it is not possible to determine the mode by
looking at the frequencies. Here, we can only locate a class with the maximum
frequency, called the modal class. The mode is a value inside the modal class, and is
given by the formula:

Mode ` = l + ( (f_1 - f_0 )/( 2 f_1 - f_0 - f_2) ) xx h`

where l = lower limit of the modal class,
h = size of the class interval (assuming all class sizes to be equal),
`f_1` = frequency of the modal class,
`f_0` = frequency of the class preceding the modal class,
`f_2` = frequency of the class succeeding the modal class.

Let us consider the following examples to illustrate the use of this formula.

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Solution:

Here the maximum class frequency is 8, and the class corresponding to this
frequency is 3 – 5. So, the modal class is 3 – 5.

Now

modal class` = 3 – 5`, lower limit (l ) of modal class = 3, class size (h) = 2
frequency `( f_1 )` of the modal class = 8,
frequency `( f_0 )` of class preceding the modal class = 7,
frequency `( f_2)` of class succeeding the modal class = 2.
Now, let us substitute these values in the formula :

Mode `= l + ( (f_1 -f_0 )/( 2 f_1 - f_0 - f_2 )) xx h`

`= 3+ ( (8-7)/( 2 xx 8 -7 -2) ) xx 2 = 3 +2/7 = 3.286`

Therefore, the mode of the data above is `3.286`.

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Solution:

Refer to Table 14.3 of Example 1. Since the maximum number of students
(i.e., 7) have got marks in the interval `40 - 55`, the modal class is `40 - 55`. Therefore,

the lower limit ( l ) of the modal class = 40,

the class size `( h) = 15`,

the frequency `( f_1 )` of modal class = 7,

the frequency `( f_0)` of the class preceding the modal class = 3,

the frequency `( f_2 )` of the class succeeding the modal class = 6.

Now, using the formula:

Mode ` = l + ( (f_1 -f_0)/( 2 f_1 - f_0 - f_2 ) ) xx h` ,

we get Mode ` = 40 + ( (7-3)/(14-6-3) ) xx 15 = 52`

So, the mode marks is 52.

Now, from Example 1, you know that the mean marks is 62.

So, the maximum number of students obtained 52 marks, while on an average a
student obtained 62 marks.

`text (Remarks : )`

1. In Example 6, the mode is less than the mean. But for some other problems it may
be equal or more than the mean also.

2. It depends upon the demand of the situation whether we are interested in finding the
average marks obtained by the students or the average of the marks obtained by most
of the students. In the first situation, the mean is required and in the second situation,
the mode is required.

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