Relationship between Arithmetic, Geometric and Harmonic Mean

The observations based upon any test which happened, can be any experiment for reading the changes in value, and 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, geometric and harmonic 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, Geometric and Harmonic 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. While the inverse of the arithmetic mean is the harmonic mean for the given sample of data.

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, for two items, say p and q,  

Arithmetic Mean=p+q2, Geometric Mean=pq and Harmonic Mean=(1p+1q 2)-1=2pqp+q

Now, from the above computations, we can easily find the relationship between the three means for any experiment. 

Thus, we can evaluate the relationship between these as, 

AMHM=p+q2.2pqp+q=pq

Now, we note that, this is equivalent to GM2

THus, we can say that, AMHM=GM2

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,geometric and harmonic mean. 

Thus, 

Arithmetic mean=2+82=5 and Geometric mean=2*812=16=4

Harmonic Mean=12+182=2*2*82+8=3210=3.2

Thus, both the friends evaluated the mean and found the deviations. Also, they were eager to find the relation between these means which they computed.

Thus, AMHM=GM2, hence substituting the values, we have, 53.2=16=42

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. Mean gives us an overall view of the data without looking at individual observations.Arithmetic mean ,geometric mean and harmonic mean are related to each other as discussed above and the relation is AMHM=GM2.