Arithmetic Mean Formula

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Arithmetic Mean Formula in Statistics

Arithmetic Mean also known as Mean or Average is basically the sum total of all of the numbers in a list divided by the number of objects in that list.

For example, the mean of the numbers 0, 1, 5, 8, 11 is 4 since 1 + 5 + 8 + 11 = 25 and 25 divided by 5 is 5.

Moreover, while working with an average, there is one central formula which is widely used to answer questions pertinent to an average. However, the formula can also be manipulated in different ways, enabling test writers to create various iterations on mean problems.


Calculation of Arithmetic Mean

Given below is the formal mathematical formula used to find the arithmetic mean.

Where,

A = arithmetic mean (or average)

n = the number of items or terms being averaged

x1 = value of every single item in the list of numbers being averaged

\[A = \frac{1}{n} \times \sum_{i=1}^{n} x_{i}\]

In addition, below is the formula for the arithmetic mean, stated in a more understandable form.

\[A = \frac{S}{N}\]

Where,

A = Arithmetic mean (or average).

n = The number of items or terms being averaged.

S = The sum total of the numbers in the set of interest being averaged.

Note: One common error most students make is that they automatically divide the sum of numbers by 2. However, dividing the sum of numbers by 2 is only appropriate when the list of numbers includes two terms. That being said, when there are more than two terms for averaging, dividing by two will give the wrong answer.


Usage and Importance of Arithmetic Mean

The Arithmetic mean is simple, and most people with even a little knowledge about math and finance can calculate it. Average or arithmetic mean is also a useful measure of central tendency, as it is disposed to provide meaningful outcomes, even with huge groupings of numbers.


Weighted Arithmetic Mean Formula

Weighted Mean is an average calculated by providing different weights to some of the individual values. If all the weights are equivalent, then the weighted mean will be similar to the arithmetic mean. It depicts the average of the given data. The Weighted mean is somewhat the same as the sample mean or the arithmetic mean.


Following is the mathematical formula to calculate the weighted arithmetic mean:

\[W = \frac{\sum_{i=1}^{n} w_{i} X_{i}}{\sum_{i=1}^{n}w_{i}}\]

Where,

W = weighted average mean.

n = Number of terms or items in a series to be averaged.

Wi = weights pertaining to to x values.

Xi = data values to be averaged.


Arithmetic Mean Formula For Grouped Data

In order to compute the mean of grouped data, the 1st step is to identify the midpoint of each class interval or the class marks. These midpoints should then be multiplied by the frequencies of the correspondent classes. The value of the mean will then be equal to the sum of the products divided by the number of values. Following is the formal mathematical formula used to find the arithmetic mean for grouped data.

The arithmetic mean for grouped data formula = x̅ = A + ∑f

Where,

A = Assumed mean of the assigned data.

∑f = Summation of the frequencies assigned in the grouped data

∑fd = Summation of the frequencies and deviation of an assigned mean data

d = Deviation of a mean data.

x̅ = Average or arithmetic mean.


Geometric Mean

The geometric mean is an average that represents the central tendency or typical set of number values by using the product of their values. Following is the formula used to find the geometric mean:

\[(\prod_{i=1}^{n} x_{i})^{\frac{1}{n}} = \sqrt[n]{x_{1}x_{2}x_{3} . . . x_{n}}\]

Where,

∏ = Geometric mean average.

n =  Total number of values.

Xi = data values to be averaged.


Example of the Arithmetic vs. Geometric Mean

Let's say that a stock's returns over five years are -10%, -1%, 6%, 20%, and 6%. The arithmetic mean would simply sum up the value of return and divide by five which comes out to be 4.2% per year average return.

On the other hand, geometric mean follows a more mathematical approach and would instead be computed as (0.9 x 0.99 x 1.06 x 1.2 x 1.06)1/5 -1 = 3.74% per year average return.

Remember that the arithmetic means will always be greater than the geometric mean, and is also a more appropriate calculation in this case.


Fun Facts on Arithmetic Mean

  • In reference to finance, the arithmetic mean is not generally a suitable method for computing an average, particularly when a single outlier can skew the mean by a huge amount.

  • Geometric mean and harmonic mean are other averages used more commonly in finance.

FAQ (Frequently Asked Questions)

Q1. What is Meant by the Arithmetic Mean?

Answer: The arithmetic mean is the easiest and extensively used measure of a mean, or what we say average. It simply requires taking the sum of a group of numbers, then dividing that sum total by the count of the numbers used in the series.


For example, take the numbers 9, 21, 46, and 68. The sum is 144. The arithmetic mean is 144 divided by 4 (144/4), or 36.

Q2. How Many Types of Mean are There?

Answer: There are several other types of means that are being widely used such as the geometric mean and harmonic mean. These types of means come into play in certain situations like investing, stock trading analysis and finance. Another example includes the trimmed mean, used when computing economic data such as CPI and CPE.

Q3. How the Arithmetic Mean Works?

Answer: The arithmetic mean surprisingly reserves its position in finance as well. For example, mean earnings approximates essentially are arithmetic mean. Say you want to find out the average earnings expectation of the 21 analysts covering a specific stock. Simply add up all the approximates and divide by 21 to obtain the arithmetic mean.


The same holds true when we seek to find out a stock’s average closing price during a specific month. Say there are 22 trading days in the month. Simply consider all the prices, add them up, and divide by 22 to obtain the average or arithmetic mean.