Arithmetic Mean

JEE Crash Course 2022

What is an Arithmetic Mean?

The mean or average of a set of data or a collection of data is known as the arithmetic mean. The arithmetic mean is evaluated by adding the given collection of numbers and dividing the sum by the count of numbers in the collection. The arithmetic mean is used in surveys and experimental studies. It comes under the statistics part of mathematics. In mathematics, there are different types of means such as arithmetic mean, arithmetic harmonic mean, geometric mean, and geometric harmonic mean. The term arithmetic mean is just used to differentiate it from the other means.

 

The arithmetic mean is mostly the same as the average, but due to some cases such as the distribution having open-end classes, the distribution is highly skewed, we cannot agree to that.

 

Arithmetic Formula

The simplest and the best measure of central tendency is the arithmetic mean. It is the ratio of the sum of the items to the number of items. The selection of the average depends on the distribution of data and the purpose for which it is used.

 

If a1, a2……., an are the values of variable a, then the mean of a is calculated as below.

 

Arithmetic mean = \[\frac {(a_1+a_2+.....+ a_n)} {n} \]

 

Where 𝑛 is the total number of variables in the set.

 

= X1+X2+X3...XN

 

\[\bar{NX}\] =X1+X2+X3...XNN

 

Where,

 

X̅ = The Mean

 

X1 = The First Value

 

X2 = The Second Value


X3 = The Third Value

 

XN = The Last Value

 

N = The Number of Value

 

Arithmetic Mean for Ungrouped Data

The simplest and the most used central tendency measure is the arithmetic mean. It is the ratio of the sum of variables to the number of variables present in a given set. Let’s understand this with the help of an example.

1. Calculate the arithmetic mean from the following data.

20,30,40,70,80

Solution:

Sum of terms = 20 + 30 + 40 + 70 + 80 =  240

 

Number of terms = 5

 

Arithmetic mean = \[\frac {240} {5} \] = 48

 

Therefore the arithmetic mean of the above five numbers is 48.

 

Arithmetic Mean Between Two Numbers

How to find the arithmetic mean between 2 numbers?

 

Consider any two numbers, say x and y.

 

And m be the arithmetic mean of these numbers x and y.

 

The sequence will be x, m, y in arithmetic progression.

 

m - x = y - m

 

M = \[\frac {(X+Y)} {2} \] = \[\frac {\text {Sum of the numbers}} {\text {No.of.Terms}} \]

 

This is the answer to the question.

 

How to Find the Arithmetic Mean of a Series

The arithmetic mean can be found out for three different series namely:

  • Individual series

  • Discrete series

  • Continuous series


Arithmetic Mean in Individual Series

An individual data series is a set of numbers that are arranged in ascending order. The arithmetic mean of individual series is calculated as below:

 

If x1, x2, x3, ……, xn are the 𝑛 items, then

 

Arithmetic mean = \[\frac {(a_1+a_2+.....+ a_n)} {n} \]   = \[\frac {\Sigma X} {\Sigma N} \]

 

Arithmetic Mean in Discrete Series

Discrete series is a series where the variables have corresponding frequencies, but the variables are without class intervals. The arithmetic mean of discrete series is calculated as below

If x1, x2, x3, ……, xn are the 𝑛 items and f1, f2, f3,....,fn are the corresponding frequencies, then

 

Arithmetic Mean =\[\frac {(f_1x_1+f_2x_2+f_3x_3+.....+ f_nx_n)} {N} \] = \[\frac {\Sigma f_X} {N} \]

 

Where,

 

N = Σf

 

Arithmetic Mean in Continuous Series

Continuous series is a series where the variables are in the class interval, and each has a corresponding frequency. The formula for calculating the arithmetic mean of continuous series is the same as the discrete series; you only need to find out ‘x’. In this case, the mid-value is considered as x.

 

Mid Value = \[\frac {\text {(lower limit + Upper limit)}} {2} \]

 

Weighted Arithmetic Mean

In weighted arithmetic mean, different weights are given to the individual values based upon which the mean is calculated. This mean is calculated as below:

 

If  x1, x2, x3, ……, xn are the 𝑛 items and w1, w2, w3,.....,wn are the corresponding weights, then

 

Weighted arithmetic mean = \[\frac {(W_1X_1+W_2X_2+W_3X_3+.....+ W_nX_n)} {\Sigma W} \] = \[\frac {\Sigma W_X} {\Sigma W} \]

 

Where, 

 

w = number of weights

 

Arithmetic Mean and Geometric Mean

The geometric mean of two numbers is calculated by finding out the square root of the product of two numbers.

 

If the two numbers are x and y

 

Geometric Mean = √xy

 

Relationship between A.M and G.M

 

A.M > G.M

 

Solved Problems

Example 1

Calculate the arithmetic mean from the following data

x

1

2

3

4

y

5

10

15

20

 

Solution:

x

1

2

3

4

f

5

10

15

20

fx

5

20

45

80

 

Arithmetic mean = \[\frac {\Sigma f(x)} {\Sigma f} \]

 

= \[\frac {150} {50} \]

 

= 3

FAQs (Frequently Asked Questions)

1. What are the Characteristics of Arithmetic Mean?

The characteristics of the arithmetic mean are:

  • The arithmetic mean is very easy to compute because it’s calculation is very easy.

  • The arithmetic mean is very simple, and a common man can easily understand its concepts without wasting much time.

  • One of the important features of arithmetic mean is that it is not at all affected by any changes or fluctuations.

  • The arithmetic mean is based on all the values in the data set and gives an actual result on the set of data available.

All these characteristics make arithmetic mean one of the most used and efficient measures of central tendency. It is better than mode and median when used in a survey or an operational study.

2. Discuss the Merits and Demerits of the Arithmetic Mean.

The arithmetic mean has merits as well as demerits.


Merits:

  • It provides a perfect and systematic description of data available that can be used for different purposes, such as surveys and operational studies.

  • The description interpreted gives a complete picture of the data provided.

  • It helps in analyzing data and is considered as the basis of statistical analysis.

  • Different data from various categories can be compared using the arithmetic mean.

Demerits:

  • Extreme values affect the arithmetic mean.

  • It is not capable of calculating the average of ratios and percentages properly.

  • It is not suggested for calculating the average of a highly skewed collection of data.

  • If one of the items gets missing, then it’s hard to find results.

3. What is an arithmetic mean vs mean?

Average, also called the arithmetic mean, is the sum of all the values divided by the number of values. Whereas, mean is the average in the given data. In statistics, the mean is equal to the total number of observations divided by the number of observations.

4. Is studying this arithmetic mean chapter necessary when compared to other topics in mathematics?

Understanding and studying arithmetic in-depth requires not only the importance of mathematical ideas but also helps in recognizing how these ideas actually relate to one another. Your understanding continues to grow with experience and as a result of opportunities around you to embrace new ideas and find new connections among the already familiar ones.

5. How to concentrate more while solving this arithmetic mean chapter in math?

This is a common problem every student will face before their examinations. They need to select a place where they might feel comfortable and peaceful to sit and study for hours. Usually, a silent place will help you store many things in your memory better and longer and will help you do the calculations perfectly. Try writing down the concepts, formulas, and theories separately in a notebook and revise them whenever you get the time and even before entering the exam hall.

6. Where can we find useful study materials and necessary resources or guides for the arithmetic mean with solutions?

Best handpicked and expert-curated online resources can be downloaded from the Vedantu app or website. Each and every topic is exclusively covered by professionals here, which will be more accurate and reliable. Also, the current mathematics study material released for the new academic session 2021-2022 is best for self-study in this pandemic period, as most of the students are bound to stay at home and learn by themselves. Viewing these materials online is one of the easiest methods one can follow because they will have access to the material whenever they need it.

7. Why do we use the arithmetic mean?

The arithmetic mean is appropriate when all values in the data sample have the same units of measure, e.g. all numbers are heights, or dollars, or miles, etc. When calculating the arithmetic mean, the values can be positive, negative, or zero. Arithmetic mean is the most popular measure of the central tendency. It is easy to calculate and understand the arithmetic mean calculations and procedures. And it is based on all the values of the series.

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