Question

Explain the calculation method of the mode in continuous or indiscrete series with the help of examples.

Answer 1

We will solve the question by first finding the class-interval.

It is clear by the inspection that the mode-class is 17.5-22.5, because its frequency is highest. Now Z is calculated by the following formula :

Example 5. 

Find out the mode from the following data :

Class Interval	10-19	20-29	30-39	40-49	50-59	60-69	70-79
Frequency	10	12	18	30	16	6	8

Solution: 

Firstly we will convert inclusive series into exclusive series before solving the question. After that the mode will be calculated by the following way :

Class 39.5 – 49.5 has highest frequency and it is the mode-class.

Example 6. 

Find out the mode of the following frequency distribution :

Solution: 

In this series the frequency ‘13’ comes twice. Thus, it is not possible to set modeclass by inspection. Therefore, the mode will be determined by grouping method.

Determination of mode by grouping method :

Analytical Table :

It is clear by the analytical table that the mode-class is 80-90.

Answer 2

In the continuous or indiscrete series, the mode class is first found by either inspection or grouping method. After this, the mode is calculated with the help of the following formula:

Alternative Formula : If the value of mode from the above formula comes out of the boundaries of the mode class, then use the following alternative formula.

Meaning of the symbols used in formula :

L₁ = Lower limit of mode class 

D₁ = Difference between the mode class and its preceding mode-class 

D₂ = Difference between the mode class and its succeeding mode-class

= Class-Interval 

f₁ = frequency of mode-class 

f₀= frequency of class preceding mode class 

f₂= Frequency of class succeeding mode class

Z = Mode

It has to be kept in mind that the class interval of the series is the same. If not, then they have to be made the same. Also, the series must be exclusive. The inclusive series has to be changed to exclusive series. Similarly, if the middle point is given then class intervals have to be computed.

Example 1. 

Find out the mode from the following series :

Obtained Marks	0-10	10-20	20-30	30-40	40-50	50-60	60-70
No. f Students	2	5	8	15	12	6	3

Solution:

It is clear by the series that the frequency of mode-class 30-40 is highest, Thus, Mode-Class is 30-40. The following formula will be used to calculate the value of mode in this class.

Mode can also be extracted from the following formula :

Example 2 :

 Calculate mode from the following frequency distribution :

Wages (More Than) (in Rs)	30	40	50	60	70	80	90
No. of Workers	519	496	398	208	104	44	6

Solution:

Firstly, cumulative frequency distribution will be converted into simple frequency distribution.

It is clear by inspection that the mode class is 50-60.

Example 3. 

Calculate mode in the following cumulative frequency distribution :

Wages (Less Than) (in Rs)	200	300	400	500	600	700	800	900
No. of Workers	5	18	38	70	90	95	98	100

Solution: 

Firstly the series will be converted into simple frequency distribution.

Example 4. 

Find out the mode from the following data :

Mid-Value	5	10	15	20	25	30	35
Frequency	10	15	28	35	16	7	4