What is Mean?

Mean refers to the mathematical average calculated for a set of two or more values. There are primarily two ways of calcualting it: arithmetic mean, where all the numbers are added and then divided by the number of items and and geometric mean, where we multiply the numbers together and then take the Nth root and subtract it with one.

Mean Formula

The formula of arithmetic mean is calculated by adding all the available periodic returns and divide the result by the number of periods.

Arithmetic mean = (r₁ + r₂ + …. + r_n) / n

where Ri = return in the i^th year and n = Number of periods

The formula of the geometric mean 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.

Geometric mean = [(1 + r₁) * (1 + r₂) * …. * (1 + r_n)] ^1/n – 1

Calculation of 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 r₁, r₂, ….., 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, the arithmetic average 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 Geometric Mean

Step 1: First of all, determine the various periodic returns which are denoted by r₁, r₂, ….., 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 average 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

You can download this Mean Formula Excel Template here – Mean Formula Excel Template

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

Return of 1^st year, r₁

Return of 1^st year, r₁ = [(Closing stock price / Opening stock price) – 1] * 100%
= [($110.15 / $100.00) – 1] * 100%
= 10.15%

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

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Return of 2^nd year, r₂= [($117.35 / $110.15) – 1] * 100%

= 6.54%

Return of 3^rd year, r₃= [($125.50 / $117.35) – 1] * 100%

= 6.95%

Return of 4^th year, r₄= [($130.10 / $125.50) – 1] * 100%

= 3.67%

Return of 5^th year, r₅= [($140.00 / $130.10) – 1] * 100%

= 7.61%

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

Arithmetic mean = (r₁ + r₂ + r₃ + r₄ + r₅) / n
= (10.15% + 6.54% + 6.95% + 3.67% + 7.61%) / 5

Arithmetic Average of Returns will be –

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

Geometric mean = [(1 + r₁) * (1 + r₂) * (1 + r₃) * (1 + r₄) * (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 Average of Returns will be –

Therefore, arithmetic 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 is basically a statistical indicator used to estimate a company’s stock performance over a certain period, which can be days, months or years.