Arithmetic Mean is denoted by x\u0304. It can be represented by the following formula:<\/p>\n\n\n\n
Mean= x\u0304 = <\/p>\n\n\n\n OR<\/p>\n\n\n\n Mean= Sum of value of items\/ Total number of items.<\/p>\n\n\n\n Arithmetic Mean can be of two types:<\/p>\n\n\n\n Simple Arithmetic Mean: <\/strong>In simple arithmetic mean, all items of a series are given equal importance.<\/p>\n\n\n\n Weighted Arithmetic Mean: <\/strong>Under Weighted Arithmetic Mean, different items of the series are assignment weights according to their relative importance.<\/p>\n\n\n\n Median is a positional measure. It is the middle value of the series.\u00a0 It is the value that divides the series into two equal parts. Thus, the values less than and more than median will be equal. It is denoted by M.<\/p>\n\n\n\n Median should be preferred over Mean in the following cases:<\/p>\n\n\n\n 2. Median should be used where the distribution of data values is skewed.<\/p>\n\n\n\n 3. Median should be used where the data is to presented using the graph. This is so because median can be calculated by using Less than ogives and More than ogives graph but arithmetic mean cannot be calculated by using the graphic method.<\/p>\n\n\n\n 4. Median is preferred over mean to those areas where direct quantitative measurement is not possible. It is not possible to measure intelligence directly. But it is possible to arrange the group of persons in ascending or descending order of intelligence to locate who is the most intelligent person.<\/p>\n\n\n\n 5. In case of open end series, median is preferred over mean. There is no need in median to convert the open end series into close end series but in case of mean, the open end series need conversion. <\/p>\n\n\n\n\n\n\n\n It is true than Median does not get affected by the extreme values but mean gets affected by the extreme values. This is because in case of mean the values of the items are considered but in case of median only the numbers of values are taken into consideration and not the values.<\/p>\n\n\n\n Example: Suppose X= 1, 2, 4, 6, 8, 9, 10<\/p>\n\n\n\n Mean = (1+2+4+6+8+9+10)\/ 7= 40\/7= 5.714<\/p>\n\n\n\n Median = (N+1)\/2= (7+1)\/2= 4th<\/sup> item = 6<\/p>\n\n\n\n Now, lets introduce an extreme item in the given series<\/p>\n\n\n\n X= 1,2,3,4,6,8,9, 10,42<\/p>\n\n\n\n Now Mean is= (1+2+3+4+6+8+9+10+42)\/9= 85\/9= 9.44<\/p>\n\n\n\n Now Median is= (9+1)\/2= 5th<\/sup> item= 6<\/p>\n\n\n\n When the extreme items are introduced, the mean becomes 9.44 directly from 5.714 but the median remains same i.e. 6. This shows that Median is not affected by the extreme values. <\/p>\n\n\n\n\n\n\n\n Answer: <\/strong><\/p>\n\n\n\n Average is a statistical measure representing a group of individual values in simple and comprehensive manner.<\/p>\n\n\n\n ACCORDING TO CLARK<\/p>\n\n\n\n “Average is an attempt to find one single figure to describe whole of figures”.<\/p>\n\n\n\n ACCORDING TO LEABO According to Ya Lun Chou<\/p>\n\n\n\n Average is a typical value in the sense that it is sometimes employed to represent all the individual values in a series or of a variable.<\/p>\n\n\n\n According to Prof. Yule and Kendall, an average should possess the following properties <\/p>\n\n\n\n Rigidly Defined. <\/strong>It means, everyone finds the same results. If an average is left to the observer and if it is not definite value then it cannot be representative of the series. If the investigator is biased, the value of the average would be definite and stable. In brief, we can say that the average should lead to one and only one interpretation by different persons. It means the definition should not be left to the will of the investigator. If it is left to the observer, then its value would not be definite and fixed. To solve this problem average should be defined by algebraic formula, which can universally be recognized. <\/p>\n\n\n\n Based on All Items:<\/strong> An average will not be representative if some of the items of the series are excluded from the group. There are some averages which do not take into account all the values of a group. Such averages cannot be satisfactory averages. A good measure should take into account all observations of a group.<\/p>\n\n\n\n Easy to Calculate and Follow<\/strong>: In case the calculation of an average involves too much mathematical processes, it will not be easily understood and its use will be confined only to a limited number of persons and hence, this average cannot be a popular average. It should be easy to understand so that its meaning can be made clear even to a layman. The important quality of good average is that it should not be too mathematical and its calculation should not be too difficult. The mere knowledge of plus, minus, division and multiplication should be required. <\/p>\n\n\n\n Not Affected by Variations of Sampling: <\/strong>The difference in the values of the averages for different samples is called fluctuations of sampling. If in a specific field, two independent sample studies are made, the average should not differ much. But when two independent enquiries are made, there is bound to be a difference in the average values. The average values in some cases would be more whereas in case of others it may be less. If the fluctuation of sampling’ is less, then that average will be regarded better than those where the difference is more.<\/p>\n\n\n\n Capable of Further Algebraic Treatment<\/strong>: If an average does not enjoy this quality, then its use is bound to be very limited. In that case, it may not be possible to calculate combined average of two or more than two from their individual averages. Besides, it will not be possible to study the average relationship of different parts of a variable if it is expressed as the sum of two or more variables. That means a good average should be amenable to further algebraic treatment. <\/p>\n\n\n\n Affected by Extreme Items<\/strong>: In average, each and every item should affect the value of the average. No item should affect the average unduly. If one or two very small very large items unduly affect the average, in that case average cannot be typical of the complete group. That way extreme items may distort the average and adversely affect it.<\/p>\n\n\n\n Not affected by skewness: <\/strong>A good average is the one which is not affected by skewness in the distribution. Contrary to this, if it is affected by skewness, it cannot become a true representative.<\/p>\n\n\n\n Average can be found by Graphic Method: <\/strong>A good average is one which can be found by arithmetic as as graphic method.<\/p>\n\n\n\n![\"MEASURES](\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/1.png\")
DIFFERENCE BETWEEN SIMPLE AND WEIGHTED ARITHMETIC MEAN<\/h4>\n\n\n\n
BASIS OF DIFFERENCE<\/strong><\/td> SIMPLE ARITHMETIC MEAN<\/strong><\/td> WEIGHTED ARITHMETIC MEAN<\/strong><\/td><\/tr> MEANING<\/strong><\/td> Simple average is the average of a set of values calculated with each value being assigned equal importance or weightage.<\/td> Weighted average is the average of a set of values calculated by giving weightage to the relative importance of each value.<\/td><\/tr> FORMULA NUMERATOR<\/strong><\/td> In simple average calculation, the numerator of the formula is the sum total of all the values in the set.<\/td> In weighted average calculation, the numerator of the formula is the sum total of \u2013 the values in the set multiplied by the weightage assigned to each value.<\/td><\/tr> FORMULA DENOMINATOR<\/strong><\/td> In simple average calculation, the denominator of the formula is the total number of values in the set.<\/td> In weighted average calculation, the denominator of the formula is the sum total of all the weights assigned to the values in the set.<\/td><\/tr> WEIGHTS ASSIGNED<\/strong><\/td> In simple average calculation, weights are not assigned to each value. <\/td> In weighted average calculation, weights are assigned to each value in relation to their specific importance\/relevance. <\/td><\/tr> USEFUL WHEN<\/strong><\/td> Simple average calculation is useful in simpler data analysis when all values are equally important. It is more relevant in simple mathematical analysis.<\/td> Weighted average calculation finds more relevance in accounting and financial calculations such as \u2013 weighted average cost of inventory, weighted average cost of capital.<\/td><\/tr> INDICATION OF<\/strong><\/td> Simple average is an indication of arithmetical mean or centre point of the set of values.<\/td> Weighted average on the other hand does not necessarily indicate this. It would be more tilted towards the values which have been assigned a greater weight in the set.<\/td><\/tr> EASE OF CALCULATION<\/strong><\/td> Simple average is easier to calculate.<\/td> Weighted average is more complex to calculate than simple average.<\/td><\/tr> ACCURACY<\/strong><\/td> Simple average is a less accurate method of average calculation especially in more complex sets of data.<\/td> Weighted average considers the relative importance of all values and thus is a more accurate representation of the average of a set.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n CASES WHEN USAGE OF MEDIAN IS CONSIDERED APPROPRIATE THAN MEAN<\/h4>\n\n\n\n
MEASURES OF CENTRAL TENDENCY BCOM 1ST YEAR<\/h2>\n\n\n\n
QUESTION: <\/strong>“Median is not affected by extreme values unlike arithmetic mean.” Discuss.<\/h3>\n\n\n\n
MEASURES OF CENTRAL TENDENCY BCOM 1ST YEAR<\/h2>\n\n\n\n
QUESTION: What is statistical Average? State the properties of good average?<\/h3>\n\n\n\n
“The average is sometimes described as a number which is typical of the whole group.”<\/p>\n\n\n\nPROPERTIES OF A GOOD AVERAGE<\/h4>\n\n\n\n
![\"MEASURES](\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/2-1024x576.jpg\")