## What is the Real Meaning of Average in Statistics?

Average is a single value which is calculated to represent the whole data. It may be calculated for Sample or Population data.

It is a single value that represents whole data. It is a value somewhere in the center, where most of the items of the series cluster. Such values are called Measures of central tendency. A choice of the proper average is the job of an expert who is calculating the average.

### Types of Average in Statistics

The following averages are usually used:

• Arithmetic mean
• Geometric mean
• Harmonic mean
• Median
• Mode

### What is Arithmetic Mean in Statistics?

Arithmetic mean is the sum of values, divided by the total number of values. it is also called X bar. It is obtained by adding up all observation and dividing by total number of observations.

\frac{Ex}n

\frac{Efx}{Ef}

### Properties of Arithmetic Mean as a Measure of Central Tendency

It is most widely used measure of location. Its properties are:

• A set of numerical data has one and only one mean, so it is unique.
• All the values are included in computing arithmetic mean
• The sum of squared deviation from mean is minimum.

### Demerits of Arithmetic mean

• It cannot average ratios and percentage properly.
• it can not be computed if any item is missing.
• it is highly affected by extremely large values.

### Geometric Mean Examples with Solutions

The Geometric mean of a set of data of n positive number is the nth root of their product. It is obtained by multiplying all the values and then extracting the relavent root of the product.

### Properties of Geometric Mean

• The Geomatric mean in the terms of A.M and H.M is:
• G.M = \sqrt{A.M\;\times H.M}
• The geometric mean is always less then arithmetic mean.
• It is considered best tool for constructing index number.
• Geometric mean is used for calculation of average percentage increase or decrease.

### Demerits of Geometric Mean

• its calculation is rather difficult.
• it is not easy to understand.
• it cannot be calculated if any item is zero or negative.

### What is Harmonic mean?

Harmonic mean(H) is defined as the number of values, divided by sum of reciprocal of each value. Or Harmonic mean of series is the reciprocal of the arithmetic mean of the reciprocal of the values.

#### Properties of Harmonic mean

• It is used to measure average speed, average price , average of profit and loss.
• Sometimes Harmonic mean is used as an alternative to weighted arithmetic mean.

#### Demerits of Harmonic mean

• its calculation is rather difficult.
• it gives high weightage to small values.
• it is usually a value that does not exist in given data.

### Define Median

The median is the middle value of set of Sara that are arranged in ascending or descending order of magnitude. If sample size n is an odd number then median is middle value. If sample size n is an even number, the median is average of two middle values. Median divides the data into two equal parts. It can also be called X_tilde.

#### Properties of Median

• Median is unique, there is only one median in a set of data.
• It is not affected by extremely large values or small values.
• The median is used for an Open_ended distribution.

#### Demerits of Median

• it is not capable for further mathematical treatments.
• it cannot give correct value when multiplied by number if items.

### Define Mode

The mode is the value in a set of Sara that appears maximum number of times. In the other words mode is the value that appears with the highest frequency in a set of data.

#### Properties of Mode

• The mode is very easy to find and thus is used to guide to the typical values in the sample.
• It can be found for quantitative data as well as qualitative data.
• The mode is not affected by occurrence of any extreme value.

#### Demerits of Mode

• it has no significance when number of items is not large.
• it is not based on all observations.
• it is not capable for further mathematical treatments.

#### EMPIRICAL RELATION BETWEEN MEAN ,MEDIAN , MODE

• Mean – median = \frac12 (median – mode)
• Mean – mode= 3 (mean – median)
• Median – mode= 2(Mean- Median)

OR

• Mean =\frac12 (3 median – mode)
• Median = \frac13 (2 mean + mode)
• Mode = 3 median – 2 Mean

### Relationship between A.M ,G.M ,H.M:

if all observations are same then:

A.M=G.M=H.M

## what is Random Variable?

The word Random Variable is quite commonly used in our daily life. Table of Random numbers have desired properties no matter how chosen from the table by rows, columns, diagonal, or irregularly. The first such table was published by L.H.C. tippet in 1927. The Table of Random numbers contains digits 1,2,3…9. Most modern methods of selecting a sample are based on the theory of random selection.
Discrete Random Variable and Continuous Random Variable are the types of Random Variable.



### According to a simple definition of Random number

“A random number is a numerical quantity whose value depends on chance.”

### According to a proper definition of Random variable

“A Random variable is a set of values assigned to all possible outcomes of a random experiment.” A random variable can also be written as r.v. If we write A, B, C…F on the six faces of a die these letters are not Random variables but if we right some numerical values like 1,2,3,4,5,6 on six faces of the die, Then we have a set of values called Random variable. A distribution that gives probability to each value of the random variable is called a probability distribution. random variable also called a chance variable

### Examples of Random Variable

• The number of errors per page in a balance sheet
• The height (in cm) of players of a basketball team
• A countable number of values

### Types of Random Variables

There are two types of Random variable

• Discrete random variable
• Continuous Random variable

## Define Discrete Random Variable?

A random variable X that can assume finite or countably infinite or only some selected values in a given interval is called a discrete random variable. Its probability is denoted by p(x). Discrete probability function provides a probability for each value of the discrete random variable.

#### Examples of discrete random variable

• The number of bacteria in 1cc of water
• The number of fatal accidents
• Number of tails obtained in the toss of four coins
• Number of houses in a certain town

## Define Continuous Random Variable?

A random variable X that can assume an unlimited number of variables in a given interval is called a Continuous Random variable. The probability density function provides probabilities for each value of a continuous random variable. It can be a formula or equation.

### Examples of a continuous random variable

• The price of a car
• Weight of a person
• Length of a bridge
• The height of a person
• The amount of rainfall

### Properties of Random numbers

• Random number is used to obtain the number of items in a population.
• By using a random number table, even digits 0,2,4,8 will stand for head and odd digits 1,3,5 7,9 will stand for the tail.

## What is Sequence?

Sequences are also called progression. Sequences are used to represent ordered lists of members. As the member of a sequence is indefinite order so correspondence can be established by matching them one by one with the numbers 1, 2, 3, 4,…

### Example of Sequence

for example, if the sequence is 1, 4, 7, 10, … . nth member then such a correspondence can be set up as follows

thus a sequence is a function whose domain is a subsequence of the set of natural numbers. A sequence is a special kind of function from a subset of n to R or C. Sometimes the domain of a sequence is taken to be a subset of the set { 0, 1, 2, 3,…}, i.e. the set of nonnegative integers. If all the members of a sequence are real then it is called a real sequence.

### Notation for Sequence

Sequences are usually denoted with letters a, b, c, etc., and n is used rather than x as a variable. If the natural number n be a part of the domain of a sequence a, the reciprocal element in its range is denoted by an. A special notation an is adopted for a(n) and the symbol { a_n } or a_1 , a_2, a_3 …, a_n ,… is used to represent the sequence.

The elements in the range of a sequrence{ a_n } are called its terms that is a_1 is the first term, a_2 is the second term, and a_n is the nth term or general term of the sequence.

for example, the term of the sequence {n+(-1)^n} can be written by assigning to n, the values 1, 2, 3, … if we denote the sequence by { b_n }, then

b_n = n+ (-1)^n and we have

b_1 = 1+ (-1)^1 = 1-1 =0

b_2 = 2+ (-1)^2 = 2+1 =3

b_3 = 3+ (-1)^3 = 3-1 =2

b_4 = 4+ (-1)^4 =4+1 =5etc

##### Note for Sequence:

If the domain of the sequence is a finite set, then it is called the finite sequence otherwise an infinite sequence.

An infinite sequence has no last term.

### Types of sequences

• Arithmetic sequence
• Geometric sequence
• Harmonic sequence
• Fibonacci sequence

### Define Arithmetic sequence

A sequence { a_n } is an arithmetic sequence if a_na_{n-1} is the same for all n belongs to N and n>1.

In other words, the difference of two consecutive terms of a sequence is the same then it is called an arithmetic sequence. The difference is called the common difference. The arithmetic sequence is also called the arithmetic progression. The d is the notation that is used for denoting the difference between two terms.

#### Exampleof Arithmetic sequence

5, 10, 15, 20, 25,….

Here the common difference is 5

If in the arithmetic progression the first term is a_1 and the common difference is d then the nth term of this sequence is given

a_n = a_1 +(n-1)d this is also called the general term.

### What is Geometric sequence?

A sequence { a_n } is said to be a geometric sequence if a_n \ a_{n-1} is the same non-zero number for all n belongs to N and n>1.

In other words, if the successive terms of a sequence have the same ratio between them then it is called a geometric sequence. the other name of a geometric sequence is a geometric progression. The term a_n \ a_{n-1} is the common ratio and r is the notation of this term. It is clearly defined that r is the ratio of any term of the G.P to its predecessor. There is no term of a geometric sequence that is zero.

#### Example Geometric sequence

1, 3, 9, 27,…

the nth term of a geometric sequence is given as

a_n = a_1r^{n-1}

### What is Harmonic sequence?

A sequence is called the harmonic sequence if the reciprocals of its term are in an arithmetic sequence. It is also known as harmonic progression. Since the reciprocal of 0 is not defined so zero is not the term of harmonic sequence

#### ExampleHarmonic sequence

1,\frac13, \frac15 , \frac17

This is a harmonic sequence and their reciprocal 1 3 5 7 are in a geometric sequence.

If a_1 is the first term of harmonic sequence and d is the difference then the nth term is given as

1\ a_1 +(n-1)d

### Define Fibonacci sequence?

In any sequence, we are adding the two-term to get the next term this process is continued until we get our required term this type of sequence is called the Fibonacci sequence.

#### Example Define Fibonacci sequence

3, 1, 4, 5, 9,14,…

In this sequence, we can see that

a_1=3 and a_2= 1

a_3 = a_2+ a_1 = 3+ 1 =4

a_4 = a_3 + a_2 = 4 + 1 =5 and so on.

So the Fibonacci Sequence for the nth term is given as

a_n = a_{n-2}+ a_{n-1}, n>2

There are some other types of sequences that are we discuss here as An integer sequence is a sequence in which elements are integers. A polynomial sequence is a sequence in which elements are polynomial. A binary sequence is a sequence in which elements have one of two distinct values like that a term coin which has only two values head or tail.

### Increasing and Decreasing of Fibonacci

A sequence is monotonically increasing if each term in the sequence is greater or equal to the previous term.

#### Exampleof Fibonacci

1, 4, 7, 9,

A sequence is monotonically decreasing if each term of the sequence is less than or equal to the previous term

10, 6, 6, 4