Question

Find the mode of the following data:

Marks	Below \[10\]	Below \[20\]	Below \[30\]	Below \[40\]	Below \[50\]
No. of students	\[8\]	\[20\]	\[45\]	\[58\]	\[70\]

Answer 1

Hint: To find the mode from the data, we will first have to convert the given data into interval type data. Then we will find the interval with the highest frequency. After that we can simply apply the formula for finding mode in an interval of data.

Complete step-by-step answer:
Now, it is given that student scoring marks below \[10\] are \[8\], and then the number of students scoring below \[20\] marks are \[20\]. Using these conditions, we can form an interval of students scoring marks from \[10\] to \[20\].
In these classes the number of students will be the number of students scoring below \[8\] subtracted from the number of students scoring below \[20\]. The reason why we are subtracting this is because the students who score below \[8\] are obviously scoring below \[20\], so when we make an interval of students scoring between \[10\] to \[20\], we have to subtract the student scoring marks below \[10\] as they will be repeated.
So the number of students scoring between \[10\] to \[20\] are \[20 - 8 = 12\] students.
We will form similar classes of the same class width, that is \[10\].
Number of students scoring between \[20\] to \[30\] will be given by \[45 - 20 = 25\] students.
Number of students scoring between \[30\] to \[40\] will be given by \[58 - 45 = 13\] students.
Number of students scoring between \[40\] to \[50\] will be given by \[70 - 58 = 12\] students.
On observing the number of students in each interval, we can easily say that the class \[20 - 30\] is the modal class.
Now to find the exact value of mode, we will use the formula for finding mode within an interval.
The formula of mode is \[Mode = l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h\] .
In this formula \[l\] represent the lower limit of class into consideration, \[{f_1}\] represent the frequency of the class into consideration, \[{f_0}\] represent the frequency of the class just before the class into consideration, \[{f_2}\] represents the frequency of the class next to the class into consideration and \[h\] represent the class width.
Thus, by observing the data, we can say that \[l\] is equal to \[20\], \[{f_1}\] is \[25\], \[{f_0}\] is \[12\], \[{f_2}\] is \[13\] and \[h = 30 - 20 = 10\].
Now, as we have the values of all the variables, we can substitute these values in the formula and find the mode of the given data.
\[
  Mode = l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h \\
  Mode = 20 + \dfrac{{25 - 12}}{{2(25) - 12 - 13}} \times 10 \\
\Rightarrow Mode = 20 + \dfrac{{13}}{{50 - 25}} \times 10 \\
\Rightarrow Mode = 20 + \dfrac{{13}}{{25}} \times 10 \\
\Rightarrow Mode = 20 + \dfrac{{13}}{5} \times 2 \\
\Rightarrow Mode = 20 + \dfrac{{26}}{5} \\
\Rightarrow Mode = 20 + 5.2 \\
\Rightarrow Mode = 25.2 \\
\]
Thus, the value of mode comes out to be \[25.2\], which lies within the interval \[20 - 30\] that we had considered.

Note: Mode can be found easily, if the data is discrete value, where you have to simply point out the value which has maximum frequency or is repeated maximum number of times. However, we have to use a formula for finding the mode when data is given in the form of intervals or classes. Here, the data was not given in the form of interval that is why first had to convert it, if it directly given in the form of intervals, then there is no need to convert the data into any other form.