Arithmetic Mean is the average of a set of data or a collection of numbers. In Mathematics there are different types of means such as arithmetic mean, arithmetic harmonic mean, geometric mean, and geometric harmonic mean. 

Mean is called Arithmetic Mean to differentiate it from other types of the mean. The word Arithmetic Mean means good average. Sometimes Mean cannot be used in cases like distribution with open-end classes, highly skewed distribution, and an average taken for ratios and percentages. 

The faculties at Matrix Academy, the best IIT-JEE academy in Sikar have explained Arithmetic Mean as the simplest measure of central tendency and it is the ratio of the sum of items to the number of items. It is considered as the best measure of central tendency. 

The choice of the average depends upon the distribution of the data and the purpose for which it is used. If a1, a2,……, an are the values of variable “a”. 

Different Ways of Calculating Arithmetic Mean

Arithmetic Mean of Ungrouped Data

The most used measure of central tendency is Arithmetic Mean. It is the ratio of aggregate values to the total number of values. 

Here is an example by Matrix Academy to explain how we can find the arithmetic mean of ungrouped data- 

Question– Calculate the arithmetic mean of the ages of 10 players: 20, 30, 40, 78, 65, 98, 77, 79, 63, 64.

Solution:

Sum of terms = 20 + 30 + 40 + 78 + 65 + 98 + 77 + 79 + 63 + 64 = 614

Number of terms = 10

Arithmetic Mean = 614/10 = 61.4

Therefore, the arithmetic mean of the age of 10 players is 61.4.

Arithmetic Mean of Two Numbers

Let’s consider any two numbers, say a and b. And C be the arithmetic mean between two numbers.

The sequence will be a, C, b in M.C.

C – a= b – C

C = (b + a)/2 = (Sum of the numbers)/(number of terms)

Arithmetic Mean of a Series 

There are three different types of series to calculate Arithmetic Mean as listed below- 

• Individual Series 
• Discrete Series 
• Continuous Series 

Individual Series 

The arithmetic mean for individual series is calculated by summing the values of the observations and dividing the result by the number of observations.

x – arithmetic mean for the sample

n – number of observations

x_i – the ith observation of the characteristic

Discrete Series 

Discrete Series is a series created by dividing all observations into groups. Each group represents a specific value of the characteristic. It is only possible when the values are discrete, i.e. they come from a countable set: age in years, points in a test. For each characteristic value identified, the number of its occurrences in the observation test should be provided.

The formula used to calculate the arithmetic mean for a discrete data series:

x – arithmetic mean for the sample

n – number of observations

x_i – ith value of the characteristic

n_i – number of observations for the ith value of the characteristic

Continuous Series 

A continuous data series divides all values into groups but somewhat differently than the discrete data series. Each grouping of observations represents not a specific value but a range of values.

The formula is- 

x – arithmetic mean for the statistical sample

n – number of observations

n_i – number of observations for the ith interval

x_i – the center of the ith interval

Weighted Arithmetic Mean

Weighted Mean is an average computed by giving different weights to some of the individual values. 

The weighted mean for a given set of non-negative data is- 

X= x1, x2, …….., xn with non-negative weights, W= w1, w2, ….., wn. 

Frequently Asked Questions in JEE exam-

Ques 1- What are the Characteristics of Arithmetic Mean? 

Answer- The characteristics of Arithmetic Mean are- 

1. To get a general idea about the whole group.
2. To represent the whole group and hence summarizes the whole data.
3. For summarizing data.
4. For decision-making.

Ques 2- What are the merits and demerits of Arithmetic Mean?

Answer- Advantages of Arithmetic Mean: 

• It is simple to understand. 
• Can be easily calculated.
• Capable of further algebraic treatment, the result is stable.

Disadvantages:

• Arithmetic Mean need not coincide with any of the observed values,
• Not good for ratios and percentages and sometimes give absurd answers. 

Ques 3- Is arithmetic mean the same as average?

The arithmetic mean is a good average. But, it cannot be used when:

• The distribution has open-end classes.
• The distribution is highly skewed.
• Averages are taken for ratios and percentages.

The arithmetic mean is the best measure of central tendency that shows the average value of a character in a given statistical sample. It is the most popular measure, vitally important, and applied in virtually every area of life.

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