Mean  in Statistics: Definition, Types, Formulas, Properties & Examples  

1.0Introduction

In Mathematics and Statistics, the concept of Mean is a fundamental tool for indicating the central tendency of data. For those preparing for JEE, it is essential to be familiar with papers involving different types of means, as these often appear in Statistics, Sequences & Series, Inequalities, and more Probability papers than any other area. When it comes to JEE, the three most important means are:

• Arithmetic Mean (AM)
• Geometric Mean (GM)
• Harmonic Mean (HM)

Each one has its own uses and formulas. The inequality relationship between them (AM ≥ GM ≥ HM) is one of the most common concepts that are tested on the examination.

2.0Definition of Mean

The mean of a group of numbers is the average value that shows the data in a simpler way.

If we have n observations:

$x_1,x_2,x_3,\dots ,x_n$

then the general mean is defined as:

$\text{Mean}=\frac{\text{Sum of all observations}}{\text{Number of observations}}$

Depending on the context, we calculate Arithmetic Mean, Geometric Mean, or Harmonic Mean.

3.0Types of Mean

(a) Arithmetic Mean (AM)

The Arithmetic Mean (AM) is the most commonly used mean.

$AM=\frac{x_1+x_2+\cdots +x_n}{n}$

For two numbers a and b:

$AM=\frac{a+b}{2}$

Example:
The AM of 5, 7, 9 is:

$\frac{5+7+9}{3}=\frac{21}{3}=7$

(b) Geometric Mean (GM)

The Geometric Mean (GM) is useful when values involve ratios or percentages.

$GM=(x_1\cdot x_2\cdot \cdots \cdot x_n{)}^{\frac{1}{n}}$

For two numbers a and b:

$GM=\sqrt{ab}$

Example:
GM of 4 and 16 is:

$\sqrt{4\times 16}=\sqrt{64}=8$

(c) Harmonic Mean (HM)

The Harmonic Mean (HM) is best suited for problems involving rates like speed, work, or efficiency.

$HM=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}}$

For two numbers a and b:

$HM=\frac{2ab}{a+b}$

Example:
HM of 4 and 16 is:

$\frac{2\times 4\times 16}{4+16}=\frac{128}{20}=6.4$

4.0Important Formulas of Mean

Here are the most useful formulas to remember for JEE: