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Correct Answer is: (b) 20

Correct Answer is: (c) Mode

Another important measure of determining central tendency is the mode. The value that comes most often in the value series is called mode. It, means that the value for which the frequency is highest is called mode. 

For example: if the shops of size number ‘7’ are worn by most people, then the size ‘7’ will be mode.

Important definitions of mode : 

• According to Dr. Bowley : “The value of the classified quantity in a statistical group (wages, height, or any other measurable quantity) where the registered number is highest, is called the mode, or more density place, or the most important value. 
• According to Prof. Cannon : “The mode is that value which has been repeated most frequently in the category.” From the above definitions, it is clear that the mode is the value that occurs most often in the series. The mode is expressed hy the ‘Z’ or ‘M0’ of the English language.

Following are the merits of mode :

• Simple and popular : This is a simple and popular mean. In some circumstances it is calculated only by inspection. This mean is very popular in daily life. Daily usage items, e.g. average size in stitched clothes, etc. implies the mode. 
• Best representation : The value of the mode is the one whose recurrence is the highest. So it is the best representative of the series. Its value is also from among the values of the series. 
• Minimal impact of extreme values : Another important feature of mode is that it is not affected by the extreme values of the series. Extreme values have a great effect on the arithmetic mean. 
• The calculation of all the frequencies is not necessary : To calculate it, there is no need to know the frequency of all the values in the series. Only the preceding and succeding frequencies of the mode class serves the purpose.

Following are the demerits of mode :

• Uncertainty and ambiguity : Its biggest flaw is its uncertainty and ambiguity. If the frequency of all the values in the series is equal, then it cannot be calculated. Also, many times the series has more than one mode. All this reflects the uncertainty of this mean.
• Lack of algebraic interpretation : This flaw is also found in mode like the median. Its algebraic manipulation is not possible. Due to this defect, the use of this mean is very rare in many statistical methods. 
• Complexity in the calculation process : If the mode is determined by inspection method then it remains simpler, otherwise it becomes very difficult for the layman to compute it through grouping and interpolation process.

Utility of Mode : Mode is used by common people in routine life to a great extent. Mode is a useful mean in business sphere, meteorology, life-sciences, consumer preference, etc. Especially the greatest concentration point of values, such as average collar size, monthly expenses of students, number of average telephone calls per day, the number of words in an average page of a book, average number of children per couple, etc. can be determined using mode. Mode is thought to be more indicative and leading in business forecasting or prognostications. Rainfall, wind-speed, heat related forecasts are made using mode. In this way, mode is very useful in practical life.

\((\overline X - Z ) = 3 (\overline X - M)\)

Z = 3M – 2 \(\overline X\)

Z= (3 × 41.6 ) – (2 × 38.2)

= 124.8 – 76.4

= 48.4

The mutual relationship between these three means depends on whether the series is symmetrical or asymmetrical.

Symmetrical Series : In these series the value of arithmetic mean, median and mode are equal.

Asymmetric Series : In these series the values of these three means are different and generally \((\overline X - Z)\) is equal to 3 \((\overline X - M)\) .

In this state if the value of any two means is known then the probable value of the third can be determined by the following formula :

\((\overline X - Z)\) = 3  \((\overline X - M)\)

Z = 3 M 2 \(\overline X\)

The mutual relationship between these three averages depends on whether the series is symmetrical or asymmetrical.

Symmetrical Series : In these series the value of arithmetic mean, median and mode are equal.

Asymmetric Series : In these series the values of these three average are different and generally \((\overline X - Z)\) is equal to 3 \((\overline X - M).\) 

In this situation, if the value of the two means is known, then the probable value of the third can be determined by the following formula-

\((\overline X - Z) = 3 (\overline X - M)\)

Z= 3 M-2 \(\overline X\) 

Example : 

If the arithmetic mean of the distribution is 20.2 and median is 31.6 then what will be the most likely value of the mode?

\((\overline X - Z) = 3 (\overline X - M)\)

Z = 3M – 2 \(\overline X\)

Z= (3 × 31.6 ) – (2 × 20.2)

= 94.8 – 40.4

= 54.4

Density test is used in mode when the frequency of the data series is random or irregular because, in such a situation, the maximum frequency is not detected. Density testing is used when there are irregularities found in the decrease- increase or concentration in frequencies, etc.

Following are the four flaws of mode : 

1. Its algebraic investigation is not possible. 

2. It is not always possible to calculate mode. Sometimes there is more than one mode in the series. 

3. Where the extreme terms are to be given importance, its calculation is not suitable. 

4. Changing the class-magnitude also changes its value.

Correct Answer is: (d) Mode

Following are the uses of mode : 

1. The mode is used by common people in routine life to a great extent. 

2. The mode is a useful mean in meteorology, life-sciences, consumer incomes, etc. 

3. The mode is thought to be more indicative and leading in business forecasting or prognostication. 

4. Rainfall, wind-speed, heat-related forecasts are made using mode.

First of all, a table is made in grouping method, in which there are 6 columns for additional frequencies apart from variable values (x).

After grouping the frequencies in the above manner, each column’s greatest frequency is circled with a pencil and by marking the variable values of these greatest frequencies, calculation are done through analytical table. The variable value having the greatest number of marks in front of it is the mode value.

Following are the methods of determining mode-

1. By inspection 

2. By grouping

Following are the two properties of mode: 

1. It is not affected by extreme values 

2. Its calculation is possible by graphical method.

Two methods of determining the mode are there in a discrete series.

1. Converting individual series into discrete series. 

2. Converting into continuous series. 

3. Estimation of Mode with the help of Arithmetic mean and median.

Three Methods

Following are the two flaws of mode: 

1. Changing the class-magnitude also changes its value. 

2. Where extreme values should be determined, its calculation is not suitable.

Following are the two utilities of mode : 

1. For finding the maximum frequency point of values. 

2. Mode is considered more useful in business forecasts or predictions.

Following are the four characteristics of mode : 

1. This is a simple and intelligible mean. 

2. It is not affected by extreme values. 

3. This is the most probable value of the distribution. 

4. Its calculation is possible by the graphical method.

Mode a place-related mean.

Difference between mode and median : 

1. The mode is the value which has the highest frequency, while the median series is placed in ascending or descending order, there is a middle term which divides the series into two equal parts. On one side lie lesser values than the median, and on the other side lie higher values than the median. 

2. The calculation of mode can be done by inspection, but the median cannot be calculated merely by inspection.

Fashion or custom which is prevalent.

From the French language word ‘La Mode’.

It is clear in series that the maximum frequency is 25 and its value is 46. So 46 is mode.

This is regular series. Thus, the mode can be found out by inspection, (highest frequency = 28) 

So Mode ( Z) = 14

Correct Answer is: (c) 5

Correct Answer is: (c) Mode

Correct Answer is: (a) Mode

Correct Answer is: (b) Mode

It is clear by inspection that the highest frequency is 17 and the corresponding item value is 3. Thus Mode = 3.

Correct Answer is: (b) Mode

Answer: Mode.

Where the information of the most popular value of distribution is to be made, the mode is the most suitable mean. In the business area, it is becoming very popular. This is the best mean to get professional forecasting, know about fashion, ideal wage determination, and the ideal time period for production.

Correct Answer is: (b) Mode

Z = 1₁ + f₂/f₀ + f₂ x i

Z = 3M - 2\(\overline X\)

= 3 x 21 - 2 x 20

= 63 - 40

Z = 23

Z= 3M -2 \(\overline X\) = ( 3 × 15.73) – (2 × 15.6)

Z= 47.19 – 31.20

Z = 15.99