Arithmetic Mean vs Geometric Mean

The observations based upon any test which happened, it can be any experiment for reading the changes in value, can be noted to vary between a range. The value for each experiment may not be identical. These values may be noted to be within a range of numbers. Thus, the range may not be useful for all the scenarios. Few observations work on range, but not all. 

In the statistical domain, the observation can be any set of values regardless of the experiment. Few scenarios can be height of people, marks of students, sales value per month, and many more. Therefore, it becomes abruptly difficult to get all the values and note them. Missing out values can make a serious issue. Hence, the concept leads to the origin of a new variable denoting this unique value such that it represents the overall observation. 

The arithmetic and geometric mean were introduced to be a value which can represent the overall data for the taken observation. Supporting the experiment, one can easily find the value representing the observed values as a whole. 

Arithmetic Mean vs Geometric Mean

Assume that a sample experiment takes place such that the observed values are in a given range. Suppose, a total of m readings were noted and analysed. Now, the readings can have different values, wherein few can be repeated. Now, the term denotes the overall experiment as a whole. Thus, we can find the mean for the whole lot as one to represent. 

The experiment had m readings, and the values can be unique or repeating depending on the type of experiment we had. Suppose, the different values are m1, m2, m3…. and so on. 

Now, the mean will represent the overall data from the experiment carried out. 

Now, to evaluate the arithmetic mean, we find the average, we initially observe the values we have from the experiment. These different values can be added together to get a single value. This summation of the observation is taken into consideration for finding out the mean to represent as a whole. Now, this value is divided by the total number of observed values to get the average value for the experiment. This value represents the whole lot uniquely and this is known as the mean for any given data. The arithmetic mean represents the mean for the given arithmetic observations. 

Similarly, to evaluate the geometric mean, we find the multiplication of the different observations and thus the total observations are used to root this multiplication. This way, the geometric mean for the given experiment can be evaluated. 

Thus, one can say that, 

Arithmetic Mean = m1+m2+m3+….. / m and Geometric mean=(m1.m2.m3…)1 / m

Where, 

The different observations are m1, m2, m3….

And m represents the number of values we noted from carrying out the experiment.

For different cases and scenarios, the formula can be wisely used. Thus, based on dependency, arithmetic mean is used for independent values whereas, dependent values are used in geometric mean. Mostly, the mean should be accurate and more feasible, thus, the geometric mean is more accurate than the arithmetic mean for a larger set of values. Geometric mean is used for positive numbers only. With simple calculation, the arithmetic mean is more easy to use and is larger than the geometric mean. Depending on the application, the method to evaluate the mean is decided. Arithmetic mean is used on a daily basis while financially geometric mean is widely used.

Example

There are two friends having cakes. The first one with 2 cakes and the second one with 8 cakes. Both the friends decided to split the cake equally so that both have the same number of cakes. They tried to do this using the mean. Thus, they computed arithmetic and geometric mean. 

Thus, 

Arithmetic mean=2+8 / 2=5 and Geometric mean=2*81 / 2=16=4

Thus, both the friends evaluated the mean and found the deviations. 

Conclusion

The arithmetic mean and geometric mean of different observations for any set of tests or experiments can be used to represent the whole as a one valued observation. This value can be part of the experimental observations or a unique value for the experiment. Depending on the number and value of the observations, the mean can have different values.