Why is arithmetic mean the most popular measure of the central tendency?
Arithmetic is a branch of mathematics that consists of the study of numbers, which focuses on the properties of the traditional operations on such numbers. These operations include addition, subtraction, multiplication, division, exponentiation, and extraction of roots. Out of these, addition(+), subtraction(-), multiplication(×), and division(÷) are the four basic operations, which are bound to be used in day-to-day life, whether by a school student, a person owning a business, or any working professional. Every individual needs to calculate stuff frequently almost everyday. Such is the prominence and inevitability of these operations, that they cannot be separated from our daily lives. In fact, at least a basic understanding of arithmetic is fundamental in the study of algebra, geometry, and analysis. In other words, arithmetic is the foundation upon which all other advanced mathematics are built, the aspect of math that people are most familiar with.
- Addition: 2 + 2 = 4
- Subtraction: 5 − 3 = 2
- Multiplication: 3 × 3 = 9
- Division: 9 ÷ 3 = 3
The above examples are relatively simpler for the sake of understanding. It is to be noted that the calculations in real life can be greater in magnitude than those illustrated above, depending on the situation. Regardless of it all, such exercise is to be still referred to as an arithmetic operation.
To clearly understand the topic of arithmetic mean, one needs to be familiar with the concept of arithmetic progression.
Such a sequence in which the difference between any two consecutive terms is constant is called an arithmetic progression or arithmetic sequence. The difference between the consecutive terms is known as the common difference and is denoted by d.
One example of such a series could be 2, 5, 8, 11, 14, 17. We clearly observe that the difference between consecutive terms in the series is constant. Here, d = 5 – 2 = 17 – 14 = 3. Thus, the given series is an arithmetic progression.
The sum of all of the numbers in a list divided by the number of terms in that list gives the arithmetic mean of that list. In the arithmetic progression, we know that if the three numbers are in AP, that means if a, b and c are in AP, then basically the first two terms a and b will have the difference which will be equal to the next two terms b and c.
So we can say, b – a = c – b. Rearranging the terms,
2b = a + c
⇒ b =
So we can say that this term b is the average of the other two terms a and c. This average in the arithmetic progression is called the arithmetic mean.
Hence, Arithmetic mean = A = S/N.
Arithmetic mean of the series: 2, 4, 6, 8, 10, 12.
Here, S = 2 + 4 + 6 + 8 + 10 + 12 = 42
N = Number of terms = 6
Hence, A.M. of the given series = S/ N = 42/ 6 = 7.
This was the case of calculating the arithmetic mean in the case of an individual series. It implies that all these items appear only once in the given series. But in a lot of cases, the items may repeat themselves in a series. They need to grouped together, and assigned a number as to how many times they appear in such a series. Such series may be referred to as discrete series or continuous series.
Question 1. Find the mean of the first five prime numbers.
We know that the first five prime numbers are 2, 3, 5, 7 and 11.
Arithmetic mean = Sum of items/ number of given items
= sum of first five prime numbers/ number of prime numbers
= (2 + 3 + 5 + 7 + 11)/ 5
= 28/ 5
Thus, the mean of the first five prime numbers is 5.6
Question 2. Find the mean of the first six multiples of 4.
The first six multiples of 4 are 4, 8, 12, 16, 20 and 24.
Mean = Sum of the first six multiples of 4/ number of multiples
= (4 + 8 + 12 + 16 + 20 + 24)/ 6
= 84/ 6
Thus, the mean of the first six multiples of 4 is 14.
Question 3. Find the arithmetic mean of the first 7 natural numbers.
The first 7 natural numbers are 1, 2, 3, 4, 5, 6 and 7.
Mean = sum of first 7 natural numbers/ number of natural numbers
= (1 + 2 + 3 + 4 + 5 + 6 + 7)/ 7
= 28/ 7
Thus, the mean of the first 7 natural numbers is 4.
Question 4. If the mean of 9, 8, 10, x, 12 is 15, find the value of x.
As per the formula of arithmetic mean, AM of the given numbers = (9 + 8 + 10 + x + 12)/ 5
= (39 + x)/ 5
It is given that the mean = 15
⇒ 39 + x = 15 × 5
⇒ 39 + x = 75
⇒ 39 – 39 + x = 75 – 39
⇒ x = 36
Question 5. List out some demerits of the arithmetic mean.
- Difficult to Compute: Arithmetic mean can only be computed with the application of formulas. It can neither be determined by inspection or by graphical location.
- Only Quantitative Aspects: Arithmetic mean can not be computed for qualitative data. In other words, AM is based on quantitative aspect of the given data.
- Affected by Extreme Observations: The smallest and largest observations in a given data set distort arithmetic mean. Hence it may not represent accurate picture of the data set on hand.
- Non- coincidental: Arithmetic mean sometimes does not coincide with any item in the given data set. This also shows how it might not be the most accurate representation of the data.