Question

Find the mode from the following:

Mid-Value	12	18	24	30	36	42	48
Frequency	20	12	8	24	16	8	12

Answer 1

To find the mode from the given data, we first need to convert the mid-values into class intervals and then apply the formula for the mode of grouped data.

1. Find the class intervals

The difference between consecutive mid-values is constant: 18 – 12 = 6. So, the class width (h) is 6.

To find the lower and upper limits of each class, we subtract and add half of the width `(6/2 = 3)` to each mid-value.

Class Interval	Mid-Value	Frequency (f)
9 – 15	12	20
15 – 21	18	12
21 – 27	24	8
27 – 33	30	24 (Highest Frequency)
33 – 39	36	16
39 – 45	42	8
45 – 51	48	12

2. Identify the modal class

The highest frequency is 24, which corresponds to the class interval 27 – 33.

Lower limit (L) of the modal class = 27

Frequency of modal class (f₁) = 24 

Frequency of preceding class (f₀) = 8 

Frequency of succeeding class (f₂) = 16 

Class width (h) = 6

3. Calculate the mode

We use the formula:

Mode = `L + ((f_1 - f_0)/(2f_1 - f_0 - f2)) xx h`

Substituting the values:

Mode = `27 + ((24 - 8)/(2(24) - 8 - 16)) xx 6`

Mode = `27 + (16/(48 - 24)) xx 6`

Mode = `27 + (16/24) xx 6`

Mode = `27 + (2/3) xx 6`

Mode = 27 + (2 × 2)

Mode = 27 + 4

Mode = 31

The mode of the given data is 31.