Variance of a Data Set

Introduction to variance of a data set:

Data is nothing but a set of information or collection of facts. We can analyze the data’s with the help of charts and graphs. The method of collecting or preparing the data’s is the data collection.
Variance is the term that  explains how average value of the data set vary from the measured data.

Variance of Data Set:

Following are the steps involved in finding the variance of data set:    

Step 1: First step is to find the Average, that is the arithmetic mean for the given set of numbers.

Mean = Sum of all values / Total number of elements

Step 2: To find the sum of squares for all the given values in the given data set.

Step 3: Divide the result we obtained in Step 2 over the total number of elements in the given data set.

Step 4: Find the difference between the square of mean and the result we obtained from step 3

Step 5: For finding the variance, take the square root of the number that  we obtained as a result from step 4.

Thus, using the above steps, we can easily calculate the variance of the given data set.

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Solved Examples for Finding Variance of a Data Set:

Ex1: Find the variance  of the given data set: { 5, 6, 9, 13, 19 }

Sol:

Step 1: Arithmetic mean             =  (5 +6 + 9+ 13 + 22) / 5

= 55 / 5

=  11

Step 2: Sum of squares               =  52 + 62 + 92 + 132 + 222

=  25 + 36 + 81 + 169 + 484

= 795

Step 3: Sum of squares / Total number of elements

= 795 / 5

= 159

Step 4: Result from Step 3 – square of mean value

= 159 – 112       

= 159 – 121

= 38

Step 5: Variance                        = square root ( 38 )

= 6.16

The variance of the given data set is 6.16

Ex 2: Find the variance  of the given data set: { 2, 3, 4, 5,16 }

Sol:

Step 1: Arithmetic mean             =  (2 +3 + 4+ 5 + 16) / 5

= 30 / 5

=  6

Step 2: Sum of squares               =  22 + 32 + 42 + 52 + 162

=  4 + 9 + 16 + 25 + 256

= 310

Step 3: Sum of squares / Total number of elements

= 310 / 5

= 62

Step 4: Result from Step 3 – square of mean value

= 62 – 62       

= 62 – 36

= 26

Step 5: Variance                        = square root ( 26 )

= 5.1

The variance of the given data set is 5.1

Percentile Definition Statistics

Introduction to percentile definition statistics:

The definition of percentile is dividing the data into 100 equal parts in statistics. We can also refer the percentile as quartiles. The percentile formula of statistics is using the samples. In statistics, the percentile is determined from ordered data. Now we are going to see about percentile definition statistics.

Explanation for Percentile Definition Statistics

Definition of percentile in statistics:

The percentile definition is the position of score that is part of scored values. In other words, it indicates the particular frequency percentage that is percent of scored values. It defines the score position in statistics.

Percentile formula in statistics:

The individual population is calculated by percentile and the statistics use the formula in different forms. They are,

If the ‘x’ values are used in statistics means we can use the percentile formula as `(B + 0.5E)/n` x 100.
If we does not use the ‘x’ values means we can use the percentile formula as` (Number of below x values)/n` x 100.
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More about Percentile Definition Statistics

Example problems for percentile definition statistics:

Problem 1: The scores of student are 22, 24, 25, 27, 30, 31, 32, 33, 33, 34, 36, 37, 39, 40, 42, 44, 45, 47, 50, 52.

Solution:

Total number of scores is 20 and number of below 33 is 7.

Percentile is `(B + 0.5E)/n` x 100 =` (7 + 0.5(2))/20` x 100 = `8/20` x 100 = 40.

Percentile of score 33 is 40.

Problem 2: Student scores are 12, 14, 15, 16, 17, 19, 21, 21, 22, 24, 26, 27, 29, 30, 31, 34, 35, 37, 38, 40. What is the percentile value of 21?

Solution:

Total number of score is 20 and the below 21 is 6.

The calculation for percentile is` (B + 0.5E)/n` x 100.

P = `(6 + 0.5(2))/20` x 100 = `7/20 ` x 100 = 35.

Percentile of score 21 is 35.

Exercise problems for percentile definition statistics:

1. The score of Alex is below 77 is 5 out of 20. What is the percentile value?

Solution: Percentile of Alex is 25.

2. Student’s scores are 2, 3, 4, 4, 6, 7, 9, 10, 12, 13. Determine the percentile of score 4.

Solution: The score 4 is at 30th percentile.

Decimal Place Value System

Introduction:
Decimal place value system is slightly differing from normal place value for numbers. This decimal place value system has a decimal point and numbers on both sides of decimal point. There are different names for the numbers before and after the decimal system. We are going to see how the place value system for decimals.

Explanation to Decimal Place Value System:

General representation of decimal place value system:

Let us see the place value system for the decimal 123456 . 789

Before the decimal point:

123456

6 - ones

5 - tens

4 - hundreds

3 - thousands

2 - ten thousands

1 - hundred thousands

After the decimal point:

789

7 -tenths `1/10`

8 -hundredths `1/100`

9 - thousandths `1/1000`

The above number 123456.789 can be obtained as,

(1 x 100000) + (2 x 10000) + (3 x 1000) +(4 x 100) + (5 x 10) x (6 x 1) + ( 7 x `1/10`) + (8 x `1/100` ) + (9 x `1/1000`)

Example Problems to Decimal Place Value System:
Example: 1

Which of the following is the place value of 5 in decimal 9.5?

a) Ones

b) Tenths

c) Hundreds

d) Decimal point

Solution:

Given, 9.5

5 lie after the decimal point. So, the place value of 5 is tenths.

Answer: b

Example: 2

Which of the following is the place value of 9 for decimal 98.42?

a) Tenths

b) Ones

c) Hundreds

d) Hundredths

Solution:

9 lie before the decimal point.

98

8 – ones

9 – hundreds

Answer: c

Example: 3


How will you write the expanded form the decimal 3.36?

Solution:

Given 3.36

Before the decimal point: 3

3 - Ones

After the decimal point: 36

3 - tenths

6 - hundredths

3.36 = (3 x 1) + (3 x `1/10`) + ( 6 x `1/100`)

Practice Problems to Decimal Place Value System:
Problem: 1

Which of the following is the place value of 8 in decimal 2.08?

a) Ones

b) Tenths

c) Hundredths

d) Hundreds

Answer: c

Problem: 2

Which of the following is the place value of 1 in decimal 0.001?

a) Ones

b) Thousandths

c) Hundredths

d) Hundreds

Answer: b

Algebra Ratio and Proportion

Introduction of algebra ratio and proportion:

Define of algebra ratio:

In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient

Define of algebra proportion:

If the two or more Ratio Proportion encompass all of the quantities in a particular situation.

Source: Wikipedia


Definition of Algebra Proportion:

The ratio `25/5` may be simplified and written as `5/1`.

`25/5`=`5/1`

An equation that states that two ratios are equal is called an algebra proportion. The preceding algebra proportion may also be written in the form 25:5=5:1.

Concept of algebra proportion:

A proportion is an equation that states that two ratios are equal: `m/n`=`p/q`  or  m:n=p:q (provided n?0 and q?0)

Each term of proportion is given a special name according to its position in the proportion.

`m/n`=`p/q`

Where

m is first proportional

n is second proportional

p is third proportional

q is fourth proportional

the pair of terms that form the 1st and 4th  proportional’s are called as the extremes of a proportion; the 2rd  and 3rd proportional of a proportion are called the means of a proportion.

The pair of terms m and p are called as the extremes of the proportion; the pair of terms n and q are referred to as the means of the proportion.

Example for algebra proportional:

Find the first proportional if the remaining terms of a proportion are 1,3 and 9.

Solution:

Let x=first proportional. Then

x/1=3/9

x=`1/3`

x`xx`3=`1/3``xx`3

3x=1

x=`1/3`

The first proportional is `1/3`.

Sometimes we may be able to work with lesser numbers by simplify an arithmetic ratio in the innovative proportion before cross-multiplying.
Algebra is widely used in day to day activities watch out for my forthcoming posts on Degree of Polynomial and What is the Dependent Variable in an Experiment. I am sure they will be helpful.
Definition of Algebra Ratios:

The ration of two numbers a and b(b?0) is the quotient of the numbers. The numbers a and b referred to as the terms of the ratio.

Example problem for algebra ratios:

Example 1:

Solve `25/125`

Solution:

=`1/5`    (numerator and denominator are divided by 25)

=`1/5` 

Answer is `1/5`or 1:5

Example 2:

Solve`(30/15)``xx``(12/2)`

Solution:

=`(30/15)``xx``(12/2)`

= `360/30`(numerator and denominator are divided by 10)

=`36/3`  (numerator and denominator are divided by 3)

=12 or `12/1`

Answer is `12/1`  or 12:1

Absolute Values Algebra Help

Introduction to absolute values algebra help:

Absolute values in algebra help are nothing but we are going to get the help on the absolute values in algebra. In this we are going to solve the problems based on absolute value rules. For example the absolute values of the function or numbers are always positive. So if we solve the absolute value functions we will get the two values for the absolute value functions. We will see some example for absolute values in algebra.

Example Problems for Absolute Values Algebra Help:

Absolute values algebra help example 1:

Solve the variable x where |4x + 2| = |2x + 3|

Solution:

The given absolute value function is |4x + 2| = |2x + 3|

From this we can divide this into (4x + 2) = 2x + 3 ………. (1) – (4x + 2) = 2x + 3 ………… (2)

Equation 1:

(4x + 2) = 2x + 3

Add - 2 on both sides

4x + 2 – 2 = 2x + 3 - 2

4x = 2x + 1

Add – 2x on both sides

4x – 2x = 2x + 1 – 2x

2x = 1

So x = 0.5

Equation 2:

-(4x + 2) = 2x + 3

-4x -2 = 2x + 3

Add + 2 on both sides

-4x - 2 + 2 = 2x + 3 + 2

-4x = 2x + 5

Add – 2x on both sides

-4x – 2x = 2x + 5 – 2x

-6x = 5

So x = - `(5 / 6)`

We will see some more examples for absolute values in algebra. It is better to getting help on absolute values in algebra.

Algebra is widely used in day to day activities watch out for my forthcoming posts on How to Simplify Polynomials and The Binomial Theorem. I am sure they will be helpful.

Absolute Values Algebra Help Example 2:

Solve the variable x where |2x + 1| = |x - 1|

Solution:

The given absolute value function is |2x + 1| = |x - 1|

From this we can divide this into (2x + 1) = x - 1 ………. (1) – (2x + 1) = x – 1 ………… (2)

Equation 1:

(2x + 1) = x – 1

Add - 1 on both sides

2x + 1 – 1 = x – 1 - 1

2x = x - 2

Add – x on both sides

2x – x = x – 2 - x

So x = -2

Equation 2:

-(2x + 1) = x - 1

-2x -1 = x - 1

Add + 1 on both sides

-2x -1 + 1 = x -1 + 1

-2x = x + 0

Add – x on both sides

-2x –x = x - x

-3x = 0

So x = 0

These are some of the examples for absolute value problems in algebra. This is better to getting help on absolute values.

Permutation and Combination Online Study

Introduction for permutation and combination online study:
Permutation:

It is the rescheduling of symbols into clear series. When we set things in order, we say we have made an array. When we alter the order, we declare we have changed the arrangement.

Combination:

It is a non sequence collection of unique sizes. In a permutation the order of occurrence of the objects or the arrangement is essential but in combination the order of occurrence of the objects is not important.

Online:

In general, "online" indicates a state of connectivity, In common usage, "online" often refers to the Internet or the World Wide.For the study of these topic , the following example will be more useful

Web.(Source: wiki)

Formulas for study of  permutation and combinations:

Permutation = nPr =` (n!) / ((n - r)!)`

Combination = nCr =` (_nP_r) /(r!) (or) (n!)/(r!(n-r)!)`

Permutation and Combination Online Study - Examples:

Solve online permutation and combination – Example 1:

What is the number of permutations and combinations: n=5; r=3.

Solution:

Formula for permutation:

Permutation = nPr = `(n!) /((n - r)!)`

Step 1: Find the factorial of 5.

5! = 5 × 4 × 3 × 2 × 1 = 120

Step 2: Find the factorial of 5 - 3.

(n - r)! = (5-3)! = 2! =2

Step 3: Divide 120  by 2

Permutation = `120/2` = 60

Combination = nCr = `(_nP_r) /(r!)`

Step 4: Find the factorial of 3.

3! = 3×2×1 = 6

Step 5: Divide 60  by 6.

Combination =` 60/6 ` = 10

The before example will assist you to get the Permutation and Combination manually.

Solve online permutation and combination – Example 2:

What is the number of permutations and combinations: n=8; r =3.

Solution:

Formula for permutation:

Permutation = nPr = `(n!) /((n - r)!)`

Step 1: Find the factorial of 8.

8! = 8 × 7 × 6 × 5×4×3×2×1 = 40 320

Step 2: Find the factorial of 8-3.

(n - r)! = (8-3)! = 5! = 120

Step 3: Divide 40 320 by 120.

Permutation = `40320/120` = 336

Combination = nCr = `(_nP_r) /(r!)`

Step 4: Find the factorial of 3.

3! = 3×2×1 = 6

Step 5: Divide 336 by 6.

Combination = `336/6` = 56

The before example will assist you to get the Permutation and Combination manually.

Solve online permutation and combination – Example 3:

What is the number of permutations and combinations: n= 15; r =2.

Solution:

Formula for permutation:

Permutation = nPr = `(n!) /((n - r)!)`

Step 1: Find the factorial of 15.

15! = 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5×4×3×2×1 = 1 307 674 368 000

Step 2: Find the factorial of 15-2.

(n - r)! = (15-2)! = 13! = 6 227 020 800

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Step 3: Divide 1 307 674 368 000 by 6 227 020 800.

Permutation = `(1 307 674 368 000)/( 6 227 020 800)`

=  210

Combination = nCr = `(_nP_r) /(r!)`

Step 4: Find the factorial of 2.

2! = 2×1 = 2

Step 5: Divide 210 by 2.

Combination = `210/2 ` = 105

The before example will assist you to get the Permutation and Combination manually.

Permutation and Combination Online Study - Practice Problems:

Practice Problem 1:

What is the number of permutations and combinations: n= 10; r =5.

Answer:

Permutation =30240

Combination = 252

Practice Problem 2:

What is the number of permutations and combinations: n= 50; r =3.

Answer:

Permutation = 117600

Combination = 19600

The Sum of Two Odd Numbers Is

Introduction to the Sum of Two Odd Numbers:

Odd number is the number which cannot divisible by as perfect (without getting remainder). But when we add or sum the two odd numbers, then we get result of the sum of the two odd numbers is equal to even number.

For Example: We know that the number 3 and 5 is the odd number. Now add the odd number 3 and 5, 3 + 5 = 8 where 8 is the even number.

Example Problem – Sum of Two Odd Numbers:

Example 1:

Which of the following form of number is the result when adding the two odd numbers?

Option:

a)     Odd number

b)    Even Number

c)     Zero

d)    Negative Number

Answer: Option b

Explanation:

When we add or sum the two odd numbers, then we get result of the sum of the two odd numbers is equal to even number.

For Example: We know that the number 5 and 7 is the odd number. Now add the odd number 5 and 7, 5 + 7 = 12 where 12 is the even number (12 can divisible 2 as perfect).

Example 2:

Which of the following is the result of the adding the two odd numbers 9 and 11?

Option:

a)     11

b)    9

c)     20

d)    21

Answer: Option c

Explanation:

We know that, when we add or sum the two odd numbers, then we get result of the sum of the two odd numbers is equal to even number.

Here the given option a, b, d are the odd numbers. Option c is the even number, therefore the option c is the correct answer.

Check: 11 +9 = 20

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Practice Problem – Sum of Two Odd Numbers:

Problem 1:

What is the sum of the two odd numbers 45 and 53?

Answer: 98

Problem 2:

What is the sum of the two odd numbers 75 and 51?

Answer: 126

Elementary Math Work

Introduction to elementary math work:
The elementary math work consists of mathematics topics taught in the primary and secondary schools. The most important in elementary mathematics are arithmetic, geometry, algebra, number work and special functions. In secondary school the topics are trigonometry, Calculus etc. Now we are going to see about the elementary math work.

Elementary Math Work:

1. Number work:  The number work consists of the natural, decimal, fractional, rational numbers etc. Let us see one example for the rational number in number work.

Ex  :  Determine 0.55 in the form of a rational number.

Sol :  The expression can be simplify as,

0.55 = 5 tenths + 5 hundredths

= 5 × (1/10) + 5 × (1/100)

= (5/10) + (5/100)

= 55/100

= 11/20.

2. Arithmetic:   In arithmetic, we can see about the housing finance, speed, time, distance, ratio etc. Let us see an example for the arithmetic.

Ex :  Determine the ratio of 3 Kg to 750 g

Sol : The ratio can be calculated as follows,

3 Kg = 3 × 1000 = 3000 g

Thus the required ratio given as

∴Required ratio = 3000 : 750

= 300 : 75

= 60 : 15

= 12 : 3

= 4 : 1

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Other Elementary Math Works:

1. Algebra:  Algebra is the important branch of mathematics consists of many expressions such as variables, constants and coefficients. It also contains some operations as addition, subtraction, multiplication and division.

Ex :   Expand the expression: (2x + 2y + 2z)^2

Sol :  Let us take the equation with (a + b + c)^2

a = 2x, b =2y, c = 2z

(2x + 2y + 2z)^2 = (2x)^2 + (2y)^2 + (2z)^2 + 2(4xy + 4xz + 4yz)

= 4x^2 + 4y^2 + 4z^2 + 8xy + 8xz + 8yz

2. Geometry:  Geometry is about the learning of various size, shape and properties of the geometrical solid figures. The geometry also consists of some constructional properties for all the solid figures.

Ex : The circumference of a circle is given as 10 centimeters. Determine the diameter of the circle?

Sol : The circumference of the circle can be given as,

C = π × d

10 = 3.14 × d

d = 10/3.14

d = 3.18cm.

The diameter of the circle is 3.18 centimeters.

Pyramid Definition Math

Introduction to pyramid definition math:

In this section we will discuss about pyramid definition math. In math the pyramid is comes under the concept of Euclidean geometry. The faces of pyramid are mostly isosceles triangle. The properties of the pyramid and their images are given below. The pyramid has been named as the basis of the base of the pyramid. Let we see about pyramid definition math.

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Definition – Pyramid Definition Math:

Formulas:

Formula for lateral surface area of the pyramid = `(tP)/(2)`

Formula for total surface area of the pyramid = `(tP)/(2) + B`

Formula for volume of the pyramid = (Bh)/(3)

Where, t = lateral height, B = area of base, P = perimeter of base.

Triangular Pyramid

A triangular based pyramid should be a shape that having the base shape as triangular. Simply a triangular based pyramid is a shape of tetrahedron. A tetrahedron is a polyhedron that self-possessed of four numbers of triangular sides. Three of them should meet at every vertex. It also a kind of pyramid, that having flat polygon shaped base as well as triangular faces that linking base at a common point.

Properties of Triangular based pyramid:

This shape having many properties analogous to those of triangles, include in sphere, circumspect, medial tetrahedron as well as exospheres.
The triangular based pyramid should have respective centers namely, in center, circumference, shrieker center, ex centers as well as points like centroid.