Question

Find the mode for the following data:

Class	\[0 - 7\]	\[7 - 14\]	\[14 - 21\]	\[21 - 28\]	\[28 - 35\]	\[35 - 42\]	\[42 - 49\]	\[49 - 56\]
Area	\[26\]	\[31\]	\[35\]	\[42\]	\[82\]	\[71\]	\[54\]	\[19\]

Answer 1

Hint: Mode of a given set of data is the value that has maximum frequency. To find the mode of the given data, we first find out the interval which has maximum frequency. After that we will use the formula for finding mode within an interval.

Complete step-by-step answer:
By observing the given data we can say that the interval \[28 - 35\] has maximum frequency, so the mode of the overall data has to lie within this interval.
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 \[28\], \[{f_1}\] is \[82\], \[{f_0}\] is \[42\], \[{f_2}\] is \[71\] and \[h = 35 - 28 = 7\].
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.
\[
  Mode = l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h \\
\Rightarrow Mode = 28 + \dfrac{{82 - 42}}{{2(82) - 42 - 72}} \times 7 \\
\Rightarrow Mode = 28 + \dfrac{{40}}{{164 - 114}} \times 7 \\
\Rightarrow Mode = 28 + \dfrac{{40}}{{50}} \times 7 \\
\Rightarrow Mode = 28 + 0.8 \times 7 \\
\Rightarrow Mode = 28 + 5.6 \\
\Rightarrow Mode = 33.6 \\
\]
Thus, the value of mode comes out to be \[33.6\], which lies within the interval \[28 - 35\] 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.