Uses of Arithmetic Mean

Introduction

Mathematics often revolves around data sets. A data set is a collection/ group of numbers/data. These data are obtained through mathematical calculations, measurements, surveys, or other data collection techniques.

For example, a data set can contain the roll number of all the students in a class, a list of all the items available in a grocery store, etc. These lists are fairly small, but a larger data set can exist, like the total number of species on the planet and their traits, etc.

To deal with larger data sets easily, we use the methods of mean, median, and mode. They help us understand data sets that contain many numbers.

1. Mean

• The literal meaning of mean is average. Though average and mean are two different terms, they have the same property of a data set. By using the properties of addition and division, we can determine the mean of a data set.
• Mean is the sum total of all the terms divided by the total number of terms.

Mean= Sum total of all terms/total number of terms

• Example: Consider a data set of 5 different numbers {2,4,6,8,10}, the mean of this sequence will be:
  Mean= (2+4+6+8+10)/5
  30/5=6

2. Median

• The Median is defined as the middle of the data set. The number divides the sequence into two equal groups/sets. In median, the date set is supposed to contain numbers in an order (descending or ascending)
• Example:
  Consider the data set of 3 different terms {1,2,3}; here, the median is 2

3. Mode

• Mode is technically the value in a data set that occurs more often or frequently.
• Example: Consider the data set of 6 numbers: (1,2,2,3,3,3}, the mode is 3 as it occurs more frequently.

Types of Mean

1. Arithmetic Mean

• When three terms, say, a, b, and c, are in Arithmetic Progression, then the middle term(b) is said to be the arithmetic mean (A.M.) of the other two terms (i.e. a, and b)
  i.e., If a, b, and c are in AP, then b= (a+c)/2; here, b is called the arithmetic mean of a and b

• If we extend the same to n such terms, say a₁, a₂, a₃...a_n are n numbers, then the arithmetic mean(A) of these terms is:
  A=1/n *(a₁, +a₂+ a₃+...+a_n); this is the same as adding all the numbers and dividing them by the total number of observations.

2. Geometric Mean

• When three terms say a, b and c are in Geometric Progression, then the middle term(b) is said to be the Geometric Mean (G.M.) of the other two terms (i.e., and b)
  i.e., If a, b, and c are in GP, then b²=ac or b=√ac; here, b is called the geometric mean of a and b

• If we extend the same to n such non-zero positive terms, say a₁, a₂, a₃...a_n are n numbers, then the geometric mean(G) of these terms is:
  G=(a₁*a₂* a₃*...*a_n)^1/n

3. Harmonic Mean

• The Harmonic Mean of two terms, say a and b, is said to be H if: a, H, b are in HP.
  H=(2*a*b)/(a+b)

• Similarly, if we extend the same to n such terms, say a₁, a₂, a₃...a_n are n numbers, then the harmonic mean(H) of these terms is:
  H=n/ [(1/ a₁) +(1/ a₂) +...+1/ a_n)]

Drawbacks of Arithmetic Mean

• It cannot be determined either by graphical location or by inspection.
• Arithmetic mean is affected by the extreme values.
• The arithmetic mean is unsuitable whenever the data items in a data set are missing.

Arithmetic Mean in Real Life

Using Arithmetic Mean, we can determine the following things:

• Average weather conditions of an area to determine the climate.
• Average goals taken by a football player.
• Average marks scored by the students in a particular class.
• Average working age population of a country.

Examples

1. Find the Arithmetic Mean of the sequence: 2,5,7

Solution:

Given: n=3

AM=1/n *(a₁, +a₂+ a₃+...+a_n)

1/3*(2+5+7) =14/3

2. Find the Geometric Mean of the sequence: 3, 12, 14

Solution:

Given: n=3

GM= (a₁, +a₂+ a₃+...+a_n)^1/n

(3*12*48)^1/3= (3³*2³*2³)^1/3

12

3. Find the Arithmetic Mean of the sequence: 12, 48, 123, 15

Solution:

Given: n=4

AM=1/n *(a₁, +a₂+ a₃+...+a_n)

1/4*(12+48+123+15) =1/4*(198)

49.