Arithmetic Mean in Statistics

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What is Arithmetic Mean in Statistics?

The arithmetic mean is an easiest and most commonly used measure of a mean, or also referred to an average. People with slightest knowledge of math and finance skills can calculate it. Computation through the arithmetic mean 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 ˉx. What makes arithmetic mean a useful measure of central tendency is its tendency to render useful results, even with huge grouping of numbers.


Arithmetic Mean Formula in Statistics

Formula to calculate arithmetic mean is as follows:

Mean = sum of all observations / number of observations


Example of Arithmetic Mean

For example, take 14, 26, 70, 33, and 47. The sum is 190. The Arithmetic Mean is 190 divided by 5 or 38.


How 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 approximates 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 of  dividends and earnings.

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


Solved Examples of Arithmetic Mean

Example 1: 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 2: 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 which is the representative of the whole number as well 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.

FAQ (Frequently Asked Questions)

Q1: What are the Methods to Compute the Arithmetic Mean?

Answer: 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

Q2: What are the Several Other Types of Mean?

Answer: People around the world also use various other types of means, such as the weighted mean, geometric mean and harmonic mean. These types of mean set into motion in certain situations like finance and investing. Another example is the trimmed mean, which is used while calculating CPE and CPI.

Q3: What Do We Understand by Geometric Mean?

Answer: The average that implies the central tendency or essential value of a set or series of numbers by using the product of their values is known as the Geometric mean. The geometric mean is opposed to the arithmetic mean that we also refer to as mean or average.

Q4: What is the Best Use of the Arithmetic Mean?

Answer: 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).