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The important measures of central tendency are arithmetic mean,median and mode. These are most commonly used in the measures of central tendency.

i. Arithmetic mean : Arithmetic mean is the most popular and important means among mathematical means, which is generally used by the common man in routine life. The arithmetic mean of a series is the value which is obtained by dividing the sum of all the values of the series by the number of items present in it.

According to W.I.King : “The arithmetic average may be defined as the sum or aggregate of a series of items divided by their number.”

According to H.Secrist : “ Arithmetic mean is the amount secured by dividing the sum of values of the items in a series by their number.”

Thus, it is clear that the arithmetic mean is found in the sum of all the values of a general Category, divided by the number of values.

For example: if the monthly income of 5 families is Rs 2000, 3000, 4000, 5000 and Rupees 6000, then for finding out the arithmetic mean or average income of the families, all the incomes of these households are added together, which is Rs. 20000 and then total income will be divided by the total number of items which is 5, The average monthly income will be Rs. 4000, that is the arithmetic mean. 

Arithmetic mean is of two types :

1. Simple Arithmetic Mean 

2. Weighted Arithmetic Mean

Merits of Arithmetic Mean : 

Following are the merits of arithmetic mean-

• Easy to compute and understand : It is the simplest average to understand and easiest to compute. A layman can also understand it easily. 
• Based on all items of the series : It takes into consideration every item in the series in computation. Thus, it is a good representative value. 
• Definity : It is defined by a rigid mathematical formula with the result that everyone who computes the average gets the same answer. 
• Stability : In comparison to other averages, mean is quite stable. It does not vary too much when repeated samples are taken from one and the same population, at least not as much as some other kinds of statistical descriptions. 
• Suitable for algebraic treatment : Being determined by a rigid formula, it lends itself to subsequent algebraic treatment better than the median or mode. 
• No need for arranging data : It is not necessary to arrange the values in an array form.

Following are the demerits of arithmetic mean :

• Effect of extreme value : Since the value of arithmetic mean depends upon each and every item of the series, therefore, extreme items, i.e. very small and very large items affect the average figure disproportionately. 
• Unrealistic : Sometimes it may represent such figure which seems to be unrealistic. 
• Graphical representation is not possible : It cannot be calculated by graphical method. 
• Calculation difficulties : In comparison to positional averages, calculation of arithmetic mean is more difficult because (i) It cannot be located by mere inspection, while some other averages can be located by mere inspection, (ii) It cannot be determined even if one of the values is not known because it takes into consideration every item in the series in computation, (iii) It is not suitable for qualitative facts. 
• Fallacious conclusions : Sometimes it gives fallacious and inconsistent conclusions. 
• Not suitable in the study of rate, ratio and percentage : It is not suitable for the study of rate, ratio and percentage.

ii. Mode : An important measure of the central tendency is ‘mode’. The value which is the most frequently seen in the series, is called ‘mode’. It means that the value having the highest frequency is called ‘mode’. 

For example, if most men wear ‘7 number shoe’ then the ‘7’ size is ‘mode’.

From the above definitions, it is clear that mode is the value which occurs most often in the series. The mode is expressed by letter Z of the English language.

Properties 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. The average size of the items of daily use such as stitched clothes etc., is located by ‘mode’. 
• Best Representative : The value of the ‘mode’ in a series is the one whose recurrence is most often. So this is the best representative of the series. Its value is also taken from the values of the series. 
• Minimum impact of extreme values : Another important feature of the mode is that it is not affected by the extreme values of the class. Extreme values have a great effect on arithmetic mean. 
• Determination by graphical method : Another advantage of mode is that it can also be determined by graphical method. It can be calculated with the help of a rectangular diagram. 
• It is possible to find the mode of qualitative facts : The mode of all those qualitative facts can be found which can be classified and graded. 
• Unaffected by deviations : There is no effect of class deviations on mode. 
• The calculation of all the frequencies is not required : Only the preceding and succeeding frequencies of the mode item is sufficient.

Demerits of Mode: 

Following are the flaws of mode :

• Uncertain and ambiguous : It is frequently uncertain and ambiguous. It is difficult to determine the value of the mode when each observation occurs the same number of time, also many times, a series has more than one modes.
• Lack of algebraic treatment : It is not suitable for further algebraic treatment. 

For example: from the modes of two sets of data we cannot calculate the overall mode of the combined data. 

• Complexity in the computation process : If mode could be calculated by inspection, it could be simple, else when computed by grouping or interpolation it can be very difficult for laymen. 
• Illusory mean : In many situations mode does not represent the series correctly. In such a situation, this mean creates the illusion. 
• Less importance to extreme values : It does not give importance to extreme values in the series i.e. it rejects all exceptional instances and is, therefore, not useful in those cases where weights are to be given to extreme values. 
• The change in class-magnitude also changes the mode : One drawback of mode is that when class-magnitude changes, its value also changes.

iii. Median : Median is that variable value of a data-item series which divides the ordered series into two equal parts in such a way that all the values in one part are greater than the median and all the values of the other part are lesser than the median value.

Following are the four advantages of median:

1. It is especially useful in case of open-end classes since only the position and not the values of items must be known. The median is also recommended if the distribution has unequal classes, since it is easier to compute than the mean. 

2. It is not influenced by the magnitude of extreme deviation from it. 

3. It is the most appropriate average in dealing with qualitative data, i.e. where ranks are given or there are other types of items that are not counted or measured but are scored. 

4. Perhaps the greatest advantage of median is , however, the fact that the median actually does indicate what many people incorrectly believe the arithmetic mean indicates. 

5. The median indicates the value of the middle item in the distribution. This is a clear-cut meaning and makes the median a measure that can be easily explained.

Following are the disadvantages of median:

• Lack of Representation : The median does not represent the average of a group, in which there is considerable difference in the values of different items. 
• No algebraic treatment : It is not capable of algebraic treatment. 
• For example: median cannot be used for determining the combined median of two or more groups as it is possible, in case of mean.
• Sorting class problem : To find the median, it is necessary to arrange the data class in ascending or descending order. This work takes time. 
• Unrealistic : When the median is somewhere between two values, then this is only a possible value, not real. 
• Equal significance of all positions : In its calculation all the positions are given equal importance, which is faulty. 
• Ignores marginal values : Marginal values have no effect an median. If some values are of greater importance or weight, then the use of median is inappropriate.

Correct Answer is: (a) 16

Correct Answer  is: (b) Median

Owing to the fact that calculation process of median is simple and easy, it is very useful from a practical viewpoint. Median is used for distribution of wealth and property. Median has great utility in analysis of social problems. Median is extremely useful in measurement of attribute aspects, i.e., health, poverty, intelligence, etc. The use of median is suitable where values do not have to be weighted. The dividing values(especially quadrants) are widely used in measuring divergence and skew.

Following are the four flaws of median. 

1. The median does not represent the average of a group, in which there is considerable difference in the values of different items. 

2. It is not capable of algebraic treatment. 

For example, median cannot be used for determining the combined median of two or more groups, as is possible in case of mean. 

3. When the median is somewhere between two values, then this is only possible value, not real. 

4. To find the median, it is necessary to arrange the data class in ascending or descending order. This work takes time.

Partition Values : Median divides a data item series into two eaqual series. A series can be divided into four, five, eight, ten and hundred equal parts, using the principle of median. Therefore, the values which divide the series into multiple parts are called partition value. Partition values divide the series into the median, quadrant, pentant, octant, decatant and centatant, respectively, or 2, 4, 5, 8, 10 and 100 parts respectively.

Correct Answer is: (a) Median

Correct Answer is: (a) 2

Correct Answer is: (a) 12.5

Correct Answer is: (a) Median

Correct Answer is: (b) 4

M = Value of \([\frac {N + 1}{2}]\) th item

Following are the four advantages of median : 

1. It is easy to calculate. 

2. It is less affected by extreme values. 

3. Since it is one of the values of the series itself it is a real value.

4. They can be calculated by graphical method.

Correct Answer is: (c) Quartile

Z = 3M – 2 \(\overline X\)

60 = 3M – 2 x 75

60 = 3M – 150

3M = 60 + 150 

3M = 210

M =210/3 

M = 70

Correct Answer is: (d) Median

In continuous series the median is of N/2^th item’s value not a value of \([\frac{N + 1}{2}]\) th item. Because the value of median have to be similar in ascending and descending order. If we considered is N/2 located on centre point than the value of median would comes out similar and it is suitable to determine the median by cumulative frequency curve because the centre point of the curve is located on N/2.

When the facts are of qualitative type, then the use of median is the most suitable.

Correct Answer is: (b) 19.33

It is the clear and completely defined mean.

The extreme values are ignored in it.

Median means is suitable for open-ended class- intervals.

When the item-values are written in order of 4,3,2,1 etc. from large to small, then it is called descending order.

Median is that variable value of a data-item series which divides the ordered series into two equal parts in such a way that all the values in one part are greater than the median and all the value of the other part are lesser the median value.

Q₁ is low quartile.

Median is that variable value of a data-item series which divides the ordered series into two equal parts in such a way that all the values in one part are greater than the median and all the value of the other part are lesser the median value.

It is called second quartile or median.

When the item-values are written in order of 1, 2, 3, 4 etc. from small to large, then it is called ascending order.

1, 2, 3, 4, 5, 7, 8 .

The values which divide a series into many parts are called the partition values. Series can be divided into four, five, eight, ten and hundred equal parts.

Median is used where the values have are not to be weighted.

Correct Answer is: (b) Median