ARITHMETIC MEAN
Arithmetic Mean is the number which is obtained by adding the values of all the items of a series and dividing the total by the number of items. It is also known as Average. It is an important and most commonly used measure of central tendency.
ACCORDING TO HORACE SACRIST
“The arithmetic mean is the amount secured by dividing the sum of value of items in a series by their numbers.”
Thus mean is calculated by adding values of all the items and dividing their total by the number of items. In case of Discrete series and Continuous series, the values of the frequencies are also taken into account.
TYPES OF MEAN OR AVERAGE
There are two types of Mean or Average:
SIMPLE ARITHMETIC MEAN: In case of simple arithmetic mean, all the values are given equal importance. It means all the values are taken into consideration while calculating simple mean.
WEIGHTED ARITHMETIC MEAN: Weighted Arithmetic Mean is calculated by assigning different weights to the items according to their importance in the particular field to which it belongs.
STEPS TO CALCULATE WEIGHTED ARITHMETIC MEAN
- Denote the variable as ‘X’
- Denote weights as ‘W’
- Multiply the ‘X’ by ‘W’ and the result will be XW
- Make total of W and XW, the result will be ∑W and ∑XW
- Apply the formula = ∑XW/ ∑W
PROPERTIES OF MEAN OR AVERAGE
The important mathematical properties of mean are:
PROPERTY 1:
The sum of the deviations of the items from the actual mean is always zero (0).
It can be illustrated as:
X | |
2 3 4 5 6 | 2-4=-2 3-4=-1 4-4=0 5-4=1 6-4=2 |
∑X=20 | 0 |
PROPERTY 2:
Sum of the squares of the deviations from mean is always the least.
Example:
X | (X- MEAN) | ∑(X- MEAN) ^{2} |
10 12 15 20 23 | -6 -4 -1 4 7 | 36 16 1 16 49 |
∑X= 80 | 0 | 118 |
PROPERTY 3:
If each item is replaced by the mean the sum of the item of series will be the same as the sum of those means. This proves that the product of the mean with the number of items is always equal to sum of all the items.
PROPERTY 4:
If the means of different series along with the number of items is given then combined arithmetic mean can be calculated.
PROPERTY 5:
If the items in a series are increased, decreased, multiplied or divided by the some constant the mean is also increased, decreased, multiplied or divided by the same constant.
X | X+3 | X-3 | X*3 | X/3 |
4 16 10 18 | 7 19 13 21 | 1 13 7 15 | 12 48 30 54 | 1.33 5.33 3.34 6.00 |
∑X=12 | ∑X=60 | ∑X=36 | ∑X=144 | ∑X=16.00 |
MEAN =12 | MEAN =15 | MEAN =9 | MEAN =36 | MEAN =4 |
PROPERTY 6:
The Arithmetic Mean is not affected by change in origin. This means that for any value of the assumed mean, the value of arithmetic mean remains the same.
PROPERTY 7:
The standard error of arithmetic mean is always less than that of the other measures of central tendency.
MERITS OF MEAN
RIGIDLY DEFINED: The value of mean is rigidly defined as the calculations are based on the mathematical properties and as such there is no scope for misinterpretation.
EASY TO UNDERSTAND AND SIMPLE TO CALCULATE: It is easily understandable and simple to calculate which makes it most commonly used measure of central tendency.
BASED ON ALL OBSERVATIONS: Arithmetic mean takes into consideration all the values of the series. That is why; the value of mean truly represents the series.
BASIS FOR OTHER STATISTICAL METHODS: Arithmetic Mean is used as a base for other statistical measures like measures of dispersion, measures of skewness, correlation, regression etc because it has rigid mathematical properties.
ALGEBRAIC TREATMENT: Arithmetic mean is capable of further algebraic treatment.
LEAST AFFECTED BY SAMPLING FLUCTUATIONS: Arithmetic mean is most stable average as it is least affected by sampling fluctuations.
BEST SUITABLE FOR COMPARISON: Arithmetic Mean has rigid mathematical properties and it the stable average. That is why it is used to compare the various issues from one place to another.
DEMERITS OF MEAN
NOT SUITABLE FOR QUALITATIVE DATA: Arithmetic Mean is not calculated if the data is not expressed in quantitative terms. There is no way to calculate the mean of qualitative aspects like intelligence, poverty, honesty etc.
NOT LOCATED BY INSPECTION: Mean is not located by inspection. By taking a glance of the data, one cannot tell about the value of arithmetic mean.
CANNOT BE LOCATED GRAPHICALLY: Mean cannot be determined graphically. Whereas median and mode can be determined graphically.
MAY NOT BELONG TO SERIES: Sometime the value of mean does not belong to the series. In this case, the value is known as Fictitious average.
CANNOT BE CALCULATED FOR OPEN END SERIES: Arithmetic Mean cannot be calculated if we have open end class intervals. The series need to be converted before calculating mean.
AFFECTED BY EXTREME VALUES: The most serious demerit of mean is that it is too much affected by the extreme values in the data. Any large or small value in the series can change the value of the arithmetic mean.
GIVES ABSURD RESULTS SOMETIMES: Mean sometimes gives absurd results. Likewise the value 42.6 is not reliable as the frequency cannot be taken in fractions.
Also Study: | Also Study: | Also Study: | Also Study: | Also Study: |
Measures of central tendency | Arithmetic Mean | Median | Mode | Statistics |
Functions of statistics | Scope of statistics | Limitations of Statistics | Distrust of statistics |
Nice answer of mean
Thank you!
These are very helpful notes .
Thank you