Arithmetic Mean: This is a value calculated by adding all of the elements together and then dividing the sum by the number of items. It’s also known as average. It is the most extensively used and popular measure for representing all data as a single value.

Arithmetic mean may be either:

1. Simple arithmetic mean, or

2. Weighted arithmetic mean.

The arithmetic mean, which is defined as the sum of all observations divided by the number of observations, is one of the measures of central tendency.

Properties of Arithmetic Mean:

1)  It is rigidly defined.

2)  It is based on all the observations.

3)  It is easy to comprehend.

4)  It is simple to calculate.

5) The presence of extreme observations has the least impact on it.

6) The sum of deviations of the items from the arithmetic mean is always zero.

7) The Sum of the squared deviations of the items from A.M. is minimum, which is less than the sum of the squared deviations of the items from any other values.

8) If each item in the series is replaced by the mean, then the sum of these substitutions will be equal to the sum of the individual items.) It is amenable to mathematical treatment or properties. 

Draw backs of arithmetic mean:

1)  It is very much affected by sampling fluctuation.

2) Arithmetic mean cannot be advocated to open en classification.

Merits:

1. It is straightforward to calculate and comprehend. It is for this reason that it is the most widely used central tendency measure.

2. Every item has an impact because it is included in the calculation.

3. The result remains the same since the mathematical formula is rigid.

4. When repeated samples are gathered from the same population, fluctuations are minimal for this measure of central tendency.

5. Unlike other measures like as mode and median, it can be subjected to algebraic treatment.

6. A.M. has an advantage in that it is a calculated quantity that is not depending on the order of terms in a series.

7. Due to its strict definition, it is mostly used to compare issues.

Demerits or Limitations:

1. It cannot be located graphically.

2. A single component can have a significant impact on the outcome. If there are three terms, for example, 4, 7, and 10, X is 7. The new X is 4+7+10+95/4 = 116/4 = 29 when we add a new term 95. When compared to the size of the X- in the first three terms, this is a significant change.

3. Only if the frequency is regularly distributed will it be useful. If the skewness is greater, the results will be ineffectual.

4. In the case of open end class intervals, we must assume the intervals’ boundaries, and a small fluctuation in X is possible. This is not the case with median and mode, as the open end intervals are not used in their calculations.

5. Because data cannot be stated numerically, qualitative forms such as Cleverness and Riches cannot provide X.

6. Unlike the mode and median, X cannot be found by inspection.

7. It can sometimes come to absurd or impossible conclusions, for example, if three courses have 60, 50, and 12 pupils, the average number of students is 60+50+42/3 = 50.67, which is impossible because students cannot be in fractions.

Conclusion

The Arithmetic Mean (AM), often known as average in statistics, is the ratio of the sum of all observations to the total number of observations. Outside of statistics, the arithmetic mean can be used to inform or model concepts. The arithmetic mean can be conceived of as a gravitational centre in a physical sense. The average distance the data points are from the mean of a data set is referred to as standard deviation. In the physical paradigm, the square of standard deviation (i.e. variance) is comparable to the moment of inertia.