## Formula to Calculate Mean (Arithmetic & Geometric)

The mean formula is calculated by the sum of all the values denoted by the summation of X given in the data divided by the number of values in the data set denoted by N. The mean is also used to determine a company’s performance on the basis of its earnings over a number of years among others. However, it is primarily used in the case of the portfolio for determining its average returns over a certain period of time.

The term “mean” in the financial parlance refers to the mathematical average calculated for a set of two or more periodic returns. However, there is more than one way for the calculation of the mean for a certain given set of numbers which includes the methods of the arithmetic mean and geometric mean. The equation for a mean of returns based on the arithmetic mean is calculated by adding all the available periodic returns and divide the result by the number of periods.

**Arithmetic mean = (r**

_{1}+ r_{2}+ …. + r_{n}) / nwhere Ri = return in the i^{th} year and n = Number of periods

The formula for a mean of returns based on the geometric mean is computed by initially adding one to each of the available periodic returns, then multiplying them and raising the result to the power of the reciprocal of the number of periods and then deduct one from it.

Mathematically, Geometric Mean represented as,

**Geometric mean = [(1 + r**

_{1}) * (1 + r_{2}) * …. * (1 + r_{n})]^{1/n}– 1### Calculation of Arithmetic & Geometric Mean (Step by Step)

#### Steps to Calculate Arithmetic Mean

**Step 1:**Firstly, determine the returns for various periods based on the value of the portfolio or investment at various points in time. The returns are denoted by r_{1}, r_{2}, ….., r_{n}corresponding to 1^{st}year, 2^{nd}year,…., n^{th}year.**Step 2:**Next, determine the number of periods, and it is denoted by n.**Step 3:**Finally, for the arithmetic mean of returns is calculated by adding all the periodic returns and divide the result by the number of periods as shown above.

#### Steps to Calculate G**eometric Mean**

**Step 1:**First of all, determine the various periodic returns which are denoted by r_{1}, r_{2}, ….., r_{n}corresponding to 1^{st}year, 2^{nd}year,…., n^{th}year.**Step 2:**Next, determine the number of periods, and it is denoted by n.**Step 3:**Finally, for the geometric mean of returns is calculated by initially adding one to each of the available periodic returns, then multiplying them and raising the result to the power of the reciprocal of the number of periods and then deduct one from it as shown above.

### Examples

**Let us take an example of company stock with the following stock price at the end of each of the financial year.**

Calculate the arithmetic mean and geometric mean of the annual returns based on the given information.

**Return of 1 ^{st} year, r_{1}**

- Return of 1
^{st}year,**r**= [(Closing stock price / Opening stock price) – 1] * 100%_{1} - = [($110.15 / $100.00) – 1] * 100%
- = 10.15%

Similarly, we have calculated the returns for all the year as follows,

Return of 2^{nd} year, **r _{2 }**= [($117.35 / $110.15) – 1] * 100%

= 6.54%

Return of 3^{rd} year, **r _{3 }**= [($125.50 / $117.35) – 1] * 100%

= 6.95%

Return of 4^{th} year, **r _{4 }**= [($130.10 / $125.50) – 1] * 100%

= 3.67%

Return of 5^{th} year, **r _{5 }**= [($140.00 / $130.10) – 1] * 100%

= 7.61%

Therefore, the calculation of arithmetic mean equation is done as follows,

- Arithmetic mean = (r
_{1}+ r_{2}+ r_{3}+ r_{4}+ r_{5}) / n - = (10.15% + 6.54% + 6.95% + 3.67% + 7.61%) / 5

**Arithmetic Mean of Returns will be –**

Arithmetic Mean = 6.98%

Now, the calculation of geometric mean equation is done as follows,

- Geometric mean = [(1 + r
_{1}) * (1 + r_{2}) * (1 + r_{3}) * (1 + r_{4}) * (1 + r_{n})]^{1/n}– 1 - = [(1 + 10.15%) * (1 + 6.54%) * (1 + 6.95%) * (1 + 3.67%) * (1 + 7.61%)]
^{1/5}– 1

**Geometric Mean of Returns will be –**

Geometric Mean = 6.96%

Therefore, arithmetic mean and the geometric mean of the returns are 6.98% and 6.96% respectively.

### Relevance and Uses

From the perspective of an analyst, an investor or any other financial user, it is very important to understand the concept of mean which basically a statistical indicator used to estimate a company’s stock performance over a certain period which can be days, months or years.

### Mean Formula In Excel (with excel template)

Now let us take the example of stock prices of Apple Inc. for 20 days to illustrate the concept of mean in the excel template below.

The calculation of Arithmetic Mean is as follows,

**Arithmetic Mean will be-**

The Geometric Mean is as follows,

**Geometric Mean will be-**

The table provides the detailed calculation of the arithmetic mean and geometric mean.

### Recommended Articles

This has been a guide to Mean Formula. Here we learn how to calculate arithmetic & geometric mean using its formulas for the annual returns of the company. You can learn more about our articles from the following –

- What is the Weighted Mean Formula?
- Sample Standard Deviation
- Relative Standard Deviation
- Harmonic Mean Formula
- Variance vs Standard Deviation – Compare
- Calculate Weighted Average in Excel
- Calculate Moving Average in Excel
- Mean vs Median – Compare

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