MEASURES OF CENTRAL TENDENCY BCOM 1ST YEAR: Median, Mean and Mode. Features of Good Average. All previous year questions.
Question: What do you mean by Arithmetic Mean? Differentiate between Simple and Weighted Arithmetic Mean. When is the usage of median is considered more appropriate than mean?
Answer:
ARITHMETIC MEAN: Arithmetic Mean or the mean is the number which is obtained by dividing the values of all the items of the series and dividing the total by number of items.
Arithmetic Mean is denoted by x̄. It can be represented by the following formula:
Mean= x̄ =
OR
Mean= Sum of value of items/ Total number of items.
Arithmetic Mean can be of two types:
Simple Arithmetic Mean: In simple arithmetic mean, all items of a series are given equal importance.
Weighted Arithmetic Mean: Under Weighted Arithmetic Mean, different items of the series are assignment weights according to their relative importance.
DIFFERENCE BETWEEN SIMPLE AND WEIGHTED ARITHMETIC MEAN
| BASIS OF DIFFERENCE | SIMPLE ARITHMETIC MEAN | WEIGHTED ARITHMETIC MEAN |
| MEANING | Simple average is the average of a set of values calculated with each value being assigned equal importance or weightage. | Weighted average is the average of a set of values calculated by giving weightage to the relative importance of each value. |
| FORMULA NUMERATOR | In simple average calculation, the numerator of the formula is the sum total of all the values in the set. | In weighted average calculation, the numerator of the formula is the sum total of – the values in the set multiplied by the weightage assigned to each value. |
| FORMULA DENOMINATOR | In simple average calculation, the denominator of the formula is the total number of values in the set. | In weighted average calculation, the denominator of the formula is the sum total of all the weights assigned to the values in the set. |
| WEIGHTS ASSIGNED | In simple average calculation, weights are not assigned to each value. | In weighted average calculation, weights are assigned to each value in relation to their specific importance/relevance. |
| USEFUL WHEN | Simple average calculation is useful in simpler data analysis when all values are equally important. It is more relevant in simple mathematical analysis. | Weighted average calculation finds more relevance in accounting and financial calculations such as – weighted average cost of inventory, weighted average cost of capital. |
| INDICATION OF | Simple average is an indication of arithmetical mean or centre point of the set of values. | 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. |
| EASE OF CALCULATION | Simple average is easier to calculate. | Weighted average is more complex to calculate than simple average. |
| ACCURACY | Simple average is a less accurate method of average calculation especially in more complex sets of data. | Weighted average considers the relative importance of all values and thus is a more accurate representation of the average of a set. |
CASES WHEN USAGE OF MEDIAN IS CONSIDERED APPROPRIATE THAN MEAN
Median is a positional measure. It is the middle value of the series. 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.
Median should be preferred over Mean in the following cases:
- In case of data containing the extreme values, median should be applied rather than the mean. It is so because median is not affected by the value of extreme items in the series and mean get affected a lot by the extreme items. This happens because in case of mean we take the value of item but in median we only consider the number of items not the values of the item.
2. Median should be used where the distribution of data values is skewed.
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.
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.
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.