Question

IQ of $50$ students was recorded as follows:

IQ score	$80-90$	$90-100$	$100-110$	$110-120$	$120-130$	$130-140$
No. of students	$6$	$9$	$16$	$13$	$4$	$2$

Compute the mode of the data.

Answer 1

Step 1: Finding different values for mode formula

Given table:

IQ score	$80-90$	$90-100$	$100-110$	$110-120$	$120-130$	$130-140$
No. of students	$6$	$9$	$16$	$13$	$4$	$2$

From the above table, we can see that the class $100-110$ has a maximum frequency, so the modal class is $100-110$.

For determining the mode we will use the formula:

$\mathrm{Mode}=l+\left(\frac{{\mathrm{f}}_1-{\mathrm{f}}_0}{2{\mathrm{f}}_1-{\mathrm{f}}_0-{\mathrm{f}}_2}\right)\times h$

Where, $l$ = Lower limit of modal class $(100-110)=100$

$h$ = Size of the class-interval $=10$

$f_1$ = Frequency of the modal class $=16$

$f_0$ = Frequency of the class preceding the modal class $=9$

$f_2$ = Frequency of the class succeeding the modal class $=13$

Step 2: Finding the mode

Substituting the values in the mode formula:

$$\begin{gathered} Mode=100+\left(\frac{16-9}{2\times 16-9-13}\right)\times 10 \\ =100+\left(\frac{7}{10}\right)\times 10 \\ =100+7 \end{gathered}$$

$\therefore Mode=107$

Hence, the mode of the data is $107$.