The expression of the central value of a data set is an average of a list of data. It is defined mathematically as the ratio of the sum of all the data to the number of units in the list.

In actual life, the average formula has numerous uses. If we need to get the average age of men or women in a group or the average male weight in India, we may do it by adding all of the values and dividing them by the number of values.

This article will discuss each of these and discuss the **properties of AM, GM and HM**.

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**The formula for the calculation of the average of a given collection of numbers is given below:-**

Average = Sum of Units/ Number of Units

The mean is commonly referred to as the average. The mean differs from the median and the mode in that it is a calculated measure of the central tendency from the data. As a result, depending on the data type, there are many methods for calculating the mean.

The arithmetic, geometric, and harmonic mean are three types of mean computations that you may encounter. There are additional methods and many more measurements of central tendency, but these three are perhaps the most used (e.g. the so-called Pythagorean means).

**What does Arithmetic Mean?**

By far, the most common average is the arithmetic mean. It is the most straightforward to compute and comprehend. Many people refer to it as The Average. Most individuals are only familiar with and have used the arithmetic mean; yet, it can be erroneous and misleading when employed wrongly.

The arithmetic mean is calculated by dividing the sum of the values by the total number of values, denoted by N.

- Arithmetic Mean = (A1 + A2 + … + An) / n

When all values in a data sample have the same units of measurement, such as heights, dollars, kilometres, and so on, the arithmetic mean is acceptable.

The values used to calculate the arithmetic mean can be positive, negative, or zero.

The arithmetic mean is most useful in the following situations:

- The information is not distorted (no extreme outliers)
- Individual data points are independent of one another (see the section below for examples of where data are interrelated, e.g., financial analysis)

**What is the Geometric Mean?**

The geometric mean is a method for determining the central tendency of a group of numbers by calculating the nth root of the product of n integers.

It is not the same as the arithmetic mean, calculated by adding the observations and then dividing the sum by the number of observations.

However, in the geometric mean, we determine the product of all observations and then the nth root of the product, where n is the number of observations.

The geometric mean is calculated by taking the N-th root of the product of all values, where N is the number of values.

- Geometric mean = n-root(A1 * A2 *… * An)

**Geometric Means Application**

Geometric mean has several advantages over arithmetic mean and is utilized in a wide range of applications.

- It is employed in stock indexes since many value-line indexes used by financial departments use G.M. to determine the annual return on investment portfolios.
- The geometric mean is used in finance to calculate average growth rates, also known as compounded annual growth rates (CAGR).
- Geometric Mean is also utilized in biological investigations such as cell division and bacterial growth rate, among other things.

For example, if the data set only contains two values, the geometric mean is the square root of the product of the two values.

The cube-root is used for three values, and so on.

Assume you invested $1000 and received a 10% return the first year, a 20% return the second year, and a 30% return the third year.

You have $1000 * 1.1 * 1.2 * 1.3 = $1716.00 after three years.

If you take the arithmetic mean, it’s 10 + 20 + 30 = 20% annual return, so after three years you’d have $1000 * 1.4 * 1.4 * 1.4 = $2744.

As we can see, arithmetic means overestimates earnings by roughly $1028, which is incorrect because we used an additive operation on a multiplicative process.

When evaluating the success of an investment or portfolio, investors frequently favour geometric mean over arithmetic mean.

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**What is Harmonic Mean?**

The harmonic mean is a sort of average derived by dividing the number of values in a data series by the sum of the reciprocals (1/x i) of each value in the data series.

One of the three Pythagorean means is a harmonic mean (the other two are arithmetic mean and geometric mean).

Among the Pythagorean means, the harmonic mean always has the lowest value.

Harmonic Mean of n number of values = n / [(1/A1)+(1/A2)+(1/A3)+…+(1/An)]

The harmonic mean formula is used in a variety of applications, some of which are listed below.

- Given particular conditions used in calculating the average
- generating Fibonacci sequences
- In finance, it is used to calculate average multiples.

The weighted harmonic mean is a subset of harmonic mean in which all weights are equal to one. It is analogous to the simple harmonic mean. If the set of weights w1, w2, w3,…, wn is related to the sample space A1, A2, A3,…., An, then the weighted harmonic mean is defined as

WHM = n / [ (f1/A1) + (f2/A2) + (f3/A3) + (f4/A4)….+ (fn/An) ]

**Relation Between AM/GM/HM**

GM² = AM HM is the relationship between AM, GM, and HM.

As a result, the square of the geometric mean equals the product of the arithmetic and harmonic means.

Let us also look at why the GM for a particular data set is always less than the arithmetic mean.

Let AM and GM stand for Arithmetic Mean and Geometric Mean, respectively.

GM=√ab and AM=(a+b)/2.

Let us now subtract the two equations.

AM-GM = (a+b)/2 ab = (a+b2√ab)/2 = (√a−√b)2/2 ≥ 0

AM−GM ≥ 0

This implies that AM ≥ GM.

**Conclusion**

Now that you know all about AM and GM, you can ace the knowledge tests given online and make sure that all the knowledge you have is put to good use.