Question

Calculate mode, given the following distribution of data:

Class Interval	4−8	8−12	12−16	16−20	20−24	24−28	28−32	32−36	36−40
Frequency	10	12	16	14	10	8	17	5	4

Answer 1

Grouping table and subsequent analysis table are as follows:

Grouping table:



Analysis Table:

Column	Class interval corresponding to highest frequencies in the grouping table
	4 − 8	8 − 12	12 − 16	16 − 20	20 − 24	24 − 28	28 − 32	32 − 26	36 − 40
I II III IV V VI	✓	✓ ✓ ✓	✓ ✓ ✓ ✓ ✓	✓ ✓ ✓	✓		✓
Total	1	3	5	3	1	0	1	0	0

Analysis table shows that of all the class intervals, 12 − 16 has the highest frequency of 5. It has got the maximum (✓) marks. Accordingly, 12 − 16 is the model class interval of the series.
lower limit of the modal group (l₁)= 12
f₁= frequency of the modal class=16
f₀= frequency of class preceding the modal class (8-12)=12
f₂= frequency of class following the modal class (16-20)=14
i=class interval=16-12=4

Z can be calculated by applying the following formulae:

$\mathrm{Z}=l_1=\frac{f_1-f_0}{2f_1-f_0-f_2}\times i$
$$\begin{gathered} \mathrm{or},\mathrm{Z}=12+\frac{16-12}{2\left(16\right)-12-14}\times 4 \\ \mathrm{or},\mathrm{Z}=12+\frac{4\times 4}{32-26} \\ \mathrm{or},\mathrm{Z}=12+\frac{16}{6} \\ \mathrm{or},\mathrm{Z}=12+2.67 \\ \Rightarrow \mathrm{Z}=14.67 \end{gathered}$$

Hence, the mode (Z) of the above series= 14.67.