Arithmetic Mean in Statistics

Dhristi JEE 2022-24

What is Arithmetic Mean in Statistics?

The arithmetic mean is an easiest and most commonly used measure of a mean, or also referred to as an average. People with the slightest knowledge of math and finance skills can calculate it. Computation through arithmetic simply involves taking the sum of a group of numbers, then dividing that sum by the count of the numbers taken in the series.

One of the widely used measures of central tendency, arithmetic mean is denoted by the symbol x. What makes arithmetic mean a useful measure of central tendency is its tendency to render useful results, even with a huge grouping of numbers.

 

Arithmetic Mean Formula in Statistics

Arithmetic mean is the sum of all observations divided by a number of observations.

The formula to calculate the arithmetic mean is as follows:

Mean = sum of all observations / number of observations

Arithmetic means the formula = X=Σ Xi/in, where i varies from 1 to n. 

 

Example of Arithmetic Mean

For example,  consider the numbers 14, 26, 70, 33, and 47. The result of 14 + 26 + 70 + 33 + 47 is 190. The Arithmetic Mean is 190 divided by 5 or 38.

 

How Does Arithmetic Mean Works In The Field of Finance?

Though not an ideal application in the world of finance, the arithmetic mean maintains its place in some aspects of finance, as well. For example, mean earnings approximate essentially are an arithmetic mean. Suppose that you want to find out the average earnings expectation of the 12 analysts covering a specific stock. Simply, you need to add up all the approximates and divide by 12 to get the arithmetic mean.

 

Similarly, you can also find out a stock’s average closing price during a particular month. Suppose that there are 22 trading days in the month. Simply, consider all the prices, add them, divide by 22 and you will get the arithmetic mean.

 

Limitations of Arithmetic Mean

  1. The Arithmetic mean is not an ideal application when computing the performance of investment portfolios, particularly when it involves compounding, or reinvesting dividends and earnings.

  2. Not ideal to calculate present and future cash flows, which economic analysts use in making their estimates. Using arithmetic means in the situation is sure to mislead numbers.

  3. The arithmetic mean can be misleading when looking at historical returns. In such cases, the geometric mean is most suitable for series that reveal serial correlation, particularly true for investment portfolios.

For example, the arithmetic mean is not feasible, particularly when a single outlier can skew the mean by a huge amount. Suppose you want to estimate the wages of a group of 20 laborers. Fifteen of them get wages between 2000 and 2500 a week. The 20th laborer gets the wages of 4500. That one outlier is not very representative of the group.

 

Arithmetic Mean Solved Examples:

Example 1: Find the arithmetic mean of the first five prime numbers.

Solution: First five prime numbers are 2, 3, 5, 7 and 11. The mean or the average formula is = (2 + 3 + 5 + 7 +11)/5 = 5.6.

 

Example 2 : The Marks Scored by Maria in 5 Subjects are 40, 73, 68, 50, and 54 respectively. Calculate the Mean.

Solution: Marks obtained by Maria in 5 Test Subjects are:

40, 73, 68, 50 and 54

Thus, Mean = Total Marks / Number of subjects

Total Marks = 40 + 73 + 68 + 50 + 54 = 285

Number of Subjects = 5

Mean = 285/5 = 57

 

Example 3: Find out the Arithmetic Mean of the Squares of the First n Natural Numbers.

Solution:

Sum of Squares of the 1st ‘n’ Natural Numbers = n (n+1) (2n+1)/4

Their Arithmetic Mean = sum/n

= n (n+1) (2n+1)/4n

​= (n+1) (2n+1)/4

It is so because the mean of a set of observations is the value that is representative of the whole number as well as occurs most frequently. It considers all the values present in the group and averages them dividing observation into two equal parts.

 

Key Takeaways

  1. Arithmetic mean (average) of any given data set is the sum of a series of numbers divided by the count of that series of numbers.

  2. In the field of finance, the arithmetic mean is generally an inappropriate method to use for calculating an average.

  3. Arithmetic mean is not always ideal, particularly when a single outlier can skew the mean by a big amount

FAQs on Arithmetic Mean in Statistics

1. What is the Best Use of the Arithmetic Mean?

The Arithmetic means enables us to critically categorize the center of the frequency distribution of a quantitative variable by taking into account all of the observations with the same weight allotted to each (as opposed to the weighted arithmetic mean).

2. What are the Methods to Compute the Arithmetic Mean?

 We use methods to compute the Arithmetic Mean for 3 types of series which are as follows:

1. Discrete Data Series

2. Individual Data Series

3. Continuous Data Series

3. What is the Best Use of the Arithmetic Mean?

The Arithmetic mean enables us to critically categorize the centre of the frequency distribution of a quantitative variable by taking into account all of the observations with the same weight allotted to each (as opposed to the weighted arithmetic mean).

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