Numerical Data Table

Introduction of Numerical data table:

Numerical data Table shows the measurement of numerical scale. Statistical methods are using for analyzing the numerical data for numerical data table. Values in the numerical data table is using for charts, histogram and graphs. Numerical data table is associating with collection, organization and interpretation of numerical data. In numerical data table, data should be number.

Description of Numerical Data Table:

Numerical data table is incorporated to statistics. The numerical data table is described with frequency distribution, percent and average. Frequency distribution is used for bar graph, Percent is used for pie chart and Average is used for normal graph.

Ordering data in Numerical Data table:

Ordering the data in the numerical data table will be done by qualitative numerical data and quantitative numerical data.

Ordering qualitative numerical data

It shows the numerical data with some quality parameters.

ordering quantitative data.

It shows the numerical data in terms on quantity.

Qualitative data can be shown in:

Pie charts: Another name of pie chart is circle graph, which is partitioned into sector by sector.

Bar graph: Bar graph consists the rectangular bars, which are representing the values.

Quantitative data can be shown:

Statistical tables: This table contains only  statistical data.

Rank order: This is arranging the data by its rank.

Frequency distribution: Frequency distribution denotes data of  intervals

Understanding numerical data table:

Graphs and number understand the values in the numerical data table. We have already seen the different graphs and number above.Is this topic the prime numbers from 1 to 100 hard for you? Watch out for my coming posts.

Example Problems - Numerical Data Table:

1.compute the average of the numbers, 25, 15, 28, 30, 27and 20 on numerical data table.

Solution:

Mean = 25+15+28+30+27+20 / 6

= 145 / 6

= 24.6             

2.compute the Median of the numbers, 15, 20, 28, 25 and 27 on numerical data table.
Solution:

Median = 15, 20, 28, 25, 27 = 27

3.compute the mode of the numbers, 22, 15, 22, 21, 15 and 20 on numerical data table.

Solution:

Here 15 & 22 repeated twice.

So, the modes are 15, 22.

Delta Mathematical Expression

Introduction for delta mathematical expression:

Delta (uppercase Δ, lowercase δ) is the fourth letter of the Greek alphabet. In the system of Greek numerals it has a value of 4. It was derived from the Phoenician letter Dalet. In the Ancient Greek language, it represented a voiced dental plosive /d/, while in Modern Greek it represents a voiced dental fricative, hence it is pronounced as "th" in "that" or "this". it is Romanized as either "D" or "Dh"

(Source: Wikipedia)

Delta Mathematical Expression - Definition Delta and Symmetric Delta:

Definition for delta:

Delta, Δ, and procedures the rate of change of option value with respect to changes in the fundamental asset's price. Delta is the initial derivative of the value V of the derivative with respect to the fundamental instrument's price S.

`Delta ` = `del`V / `del`S   

Symmetric difference for delta:

In symmetric distinction of two sets is the set of basics which are in one of the sets, but not in together. This procedure is the set-theoretic kin of the exclusive disjunction (XOR operation) in Boolean logic. The symmetric distinction of the sets A and B is often denoted by

A`Delta` B

Delta mathematical expression - slope and application:

Delta from slope formula:

Slope = (y2 – y1) / (x2 – x1)

= Δy / Δx

Where, Δ changes in difference

Delta mathematical expression – Example problems:

Example 1:

Find the slope between two points (4,1) (7,6)

Solution:

Delta from slope = (y2 – y1) / (x2 – x1)

(4,1) (7,6) – (x1,y1) (x2, y2)

= (6 – 1) / (7 – 4)

= (5) / (3)

Δy / Δx = 5 / 3

Example 2:

Find the slope between two points (5,6) (7,8)

Solution:

Delta from slope = (y2 – y1) / (x2 – x1)

(5,6) (7,8) – (x1,y1) (x2, y2)

= (8 – 6) / (7 – 5)

= (2) / (2)

Δy / Δx = 1

Delta Mathematical Expression - Application of Delta in Quadratic Equation:

In the polynomials of quadratic equations, Δ is referred as the discriminant value.

Δ = b2 – 4ac

Where Δ is the discriminant value.

Delta mathematical expression - Example problems:

Example 1:

Find the discriminant value given b = 8, a = 2, c = 6

Solution:

Δ = b2 – 4ac

= 82 – 4 (2)(6)

= 64 – 48

`Delta` = 16

Example 2:

Find the discriminant value given b = 8, a = 2, c = 4

Solution:

Δ = b2 – 4ac

= 82 – 4 (2)(4)

= 64 – 32

`Delta`= 32.

Divide an Angle into Two Equal Angles

Introduction to divide an angle into two equal angles:

Let us assume an angle formed by two rays. The vertex of the angle is the initial points for both the rays. Suppose you need to divide this angle into two equal angles, what is to be done.

One method is to take the help of a protractor to measure the angle. Then you numerically divide that by two, mark a point again using the protractor and join that to the vertex of the angle.

However the better method to divide an angle into two equal angles is by using a compass.

Divide an Angle into Two Equal Angles – Compass Method
The following diagram illustrates the method to divide an angle into two equal angles.



Let OX and OY be the rays making an angle XOY at vertex O.

Set the compass to a convenient radius and strike two arcs from O, one on the ray OX and another on the ray OY. Let the points of intersection of these arcs be A and B respectively. (Fig i)

Now set the compass again for a convenient radius, draw an arc from point A. Without changing the compass setting, draw a similar arc from point B.  Let these arcs intersect at P. Draw a ray connecting the vertex O and the point P. The ray OP divides angle XOY into two  equal angles.  (Fig ii)

The ray OP is called the internal angle bisector of the angle XOY.

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Divide an Angle into Two Equal Angles – Proof for the Method

Join AP and BP (Fig iii)

Considering the triangles  OAO and OBP,

OA = OB  (By construction)

AP = BP  (By construction)

Side OP is common.

Hence the triangles  OAO and OBP are congruent as per SSS postulate.

Therefore , angle AOP = angle BOP (corresponding parts of congruent triangles)

Thus it is proved that the angle XOY is divided into two  equal angles.

Binomial Distribution Normal

Introduction to binomial distribution:
In a specified series of n independent experiments, the discrete probability distribution for the number of successes is called as normal binomial distribution. Bernoulli research is also an experiment of success/failure in normal binomial distribution.Bernoulli experiment is also known as  Bernoulli trial. In normal binomial distribution when n=1, it  is called as Bernoulli distribution.

Mean and Median

Consider the random variable K having the parameters p and n follows the normal binomial distribution. We can write it as K ~ B(n, p).

Probability mass function gives the probability of receiving successes(k) in n trials.

f(k;n,p)=Pr(K=k)=(n k) p^k (1-p)^n-k. for k = 0, 1, 2, ..., n,

where f(k;n,p)=f(n-k;n,1-p)

Mean and variance: If X ~ B(n, p) X  is a binomially distributed variable, then we can say that the expected value of X is E[X]=np. and the variance is Var[X]=np(1-p)

Mode and Median

Normally the mode and the median of a normal binomial B(n,p) distribution is said to be identical to (n + 1)p. However when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes namely the (n + 1)p and (n + 1)p - 1. However the small special results have been established in the given ratio:

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Mode, median and mean gets coincide for np is the given integer.
Median m should lie in the known interval ?np? = m = ?np?.
For any median m the value cannot recline too far away from the given mean: |m-np| = min {ln2, max {p, 1-p}}.
When p = ½ and n is odd, any m in the given interval ½(n-1) = m = ½(n+1) is said to be a median for the binomial distribution.
If p = ½ and n is the even, then the m = n/2 is an unique median for the given Var(X)=E(X^2)-(E(X))^2. For using the formula need the expected value of X 2:

Geometry Solving Ratios

Introduction to geometry solving ratios

Usually we come across with the ratio of two numbers, if two numbers are x and y then their ratio are given by x : y. Same thing can be extended to geometry of same kind like, between the lengths, areas, perimeters, surface areas, volume etc. These ratios in geometry will help us to know the enlargement or the shrinking of the sizes or shapes of a geometrical figure.

Now let us see few problems of this kind in the following examples.

Example Problems on Geometry Solving Ratios:
Ex 1: The ratio of two supplementary angles are in the ratio 3 : 7. Find the measure of each angle.

Soln: Let the angles be given by 3x : 7x.

Since their sum is supplementary, then 3x + 7x = 180.

Therefore, 10x = 180

`=>` x = 18.

Therefore the angles are 3(18) and 7(18). Hence, they are 540 and 1260.

Ex 2: If the lengths of the sides of a triangle are in the ratio 3 : 4 : 5. Find the area of the triangle if its perimeter is 144cm.

Soln: Given: Perimeter = 144 cm.

The ratios are 3x + 4x + 5x = 144

`=>` 12x = 144

x = 12.

Therefore, the sides of the triangle are given by 3(12), 4(12), 5(12). That is, 36cm, 48cm, 60cm.

To find the area, let us use the Heron’s formula.

Area = `sqrt[s(s-a)(s-b)(s-c)]` . The values of a, b, c can be taken as 36, 48, 60 respectively.

Here S = `[a+b+c]/2`

=`[36 + 48 + 60]/2`

= `144/2` = 72.

Therefore, the area is given by:

A = `sqrt(72(72-36)(72-48)(72-60))`

= 864 cm2.


Between, if you have problem on these topics Nets for Solid Figures, please browse expert math related websites for more help on Types of Sequences.

More Example Problem on Geometry Solving Ratios:

Ex 3: The total surface area of solid cylinder is 616 cm2. if the ratio between its curved surface area and its total surface area is 1 : 2, find volume of the cylinder. [Take `pi = 22/7` ]

Soln: Given: Total surface area = 2`pi` r(r + h)

= 616 cm2 -----------------(1)

Also, the ratios of curved surface area : total surface area = 1 : 2.

`=>` 2`pi` rh : 2`pi` r(r+h) = 1 : 2

`h/(r+h) = 1/2`

2h = h + r

`=>` h = r.

Therefore from (1), we get, 2`pi` r(r+r) = 616

4 `pi` r2 = 616

r2 = 49

r = h = 7.

Therefore, the volume of the cylinder = `pir^2h` = `22/7 xx 7^2 xx 7` = 1078 cm3.                                                

Perimeter of a Right Angle Triangle

Introduction about right angle triangle:

A right triangle or right-angled triangle is a triangle in which one angle is a right angle (that is, a 90 degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The length of boundary is called perimeter triangle. We can calculate the perimeter of right angle triangle by adding the all three sides. In this article we shall discus how to calculate the perimeter of right triangle.

Formula to Find the Perimeter of the Right Triangle:

Perimeter of triangle (P) = a + b + c units

a, b and c are the names of three sides.

Example Problems to Find the Perimeter of the Right Angle Triangle:

1.      Find the perimeter of right angle triangle whose sides are 3 cm, 4 cm and 5 cm.

Solution:

Step 1:  Given:

Side a = 3 cm

Side b = 4 cm

Side c = 5 cm

Step 2:       Perimeter of right angle triangle (P) = a + b + c units

= 3+ 4 + 5

Step 3:        Perimeter of right angle triangle (P) = 12 cm

2.      Find the perimeter of right angle triangle whose sides are 5 cm, 12 cm and 13 cm.

Solution:

Step 1:  Given:

Side a = 5 cm

Side b = 12 cm

Side c = 13 cm

Step 2:     Perimeter of right angle triangle (P) = a + b + c units

= 5 + 12 + 13

Step 3:    Perimeter of right angle triangle (P) = 30 cm

3.      Find the perimeter of right angle triangle whose sides are 9 cm, 12 cm and 15 cm.

Solution:

Step 1:  Given:

Side a = 9 cm

Side b = 12 cm

Side c = 15 cm

Step 2:    Perimeter of right angle triangle (P) = a + b + c units

= 9 + 12 + 15

Step 3:  Perimeter of right angle triangle (P) = 36 cm

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Practice problems to find the perimeter of the right angle triangle:

4.      Find the perimeter of right angle triangle whose sides are 6 cm, 8 cm and 10 cm.

Answer: perimeter of right angle triangle (P) = 24 cm

5.      Find the perimeter of right angle triangle whose sides are 4 cm, 2 cm and 4.47 cm.

Answer: perimeter of right angle triangle (P) = 10.47 cm

6.      Find the perimeter of right angle triangle whose sides are 5 cm, 10 cm and 11.18 cm.

Answer: perimeter of right angle triangle (P) = 26.18 cm

Odd Number Series

Introduction to odd number series:

Odd Number definition: A number ‘x’ which is not divided exactly by 2 then ‘x’ is odd number. For example 3 is odd number which is not divided exactly by 2. If it is divided by 2 then ‘x’ is even number. In another way we can find the odd number easily. Whether the last digit of number ends with 1, 3, 5, 7 and 9 then we can say that as odd number.

For example:  87 are odd or even?

The number end with 7 so according to our definition 87 is odd.

We write the odd number series from 1 to 100 as

1, 3, 5, 7 ... up to 99.
1 + 3 + 5 + 7 + … + 99
To find the nth term of number series formula is

tn = a + (n – 1)d

To find the sum of an number series formula is

Sn = n/2 [ 2a + (n – 1)d ]

Or

Sn = n/2 [ a + l ]

Here ‘n’ is number of term

‘a’ is first term of series

‘d’ is difference between two consecutive terms

‘l’ is Last term of the series

Example: 1 + 3 + 5 + … find 13th term of this series?

Solution:

a = 1;

d = 2nd term – first term = 3 – 1 = 2

n = 13

tn = a + (n – 1)d

= 1 + (13 – 1)2

= 1 + (12)2

= 1 + 24

= 25

t13 = 25

Example Problem on Odd Number Series

Find Sum of first three odd numbers?
Solution:

Let us take first three odd number is 1, 3, and 5.

Sum of three odd number = 1 + 3 + 5

= 9

List out odd numbers greater than 2 and smaller than 20.
Solution:

Which is not divisible by 2 then the number is called odd number.

Odd numbers = 3, 5, 7, 9, 11, 13, 15, 17 and 19.

8 odd numbers available in between the greater than 2 and smaller than 20.

Prove that sum of an even number and odd number as odd number?
Proof:

Let us take ‘2x’ is Even number and ‘2y + 1’ Odd number.

Sum of these even and odd numbers:

2x + (2y +1) = 2(x + y) + 1

Let N = x + y and we can write the sum as

2x + (2y +1) = 2N +1

Let,

x =1 then 2x = 2; y = 2 then 2y + 1 = 5;

2x + 2y + 1 = 2(1 + 2) +1

= 6 + 1

= 7 this is odd number.

Therefore “sum of an even and odd number is odd number”

Sum of three consecutive odd numbers are 21 then find their odd numbers?
Solution:

Let us take 2x + 1 is one odd number

Then consecutive three odd numbers = 2x + 1, 2x + 3, 2x + 5

Sum of these odd numbers = 21

2x + 1+ 2x + 3+ 2x + 5 = 21

6x + 9 = 21

6x = 12

x = 12 / 6

x = 2

Then    2x + 1 = 2 (2) + 1 = 5

2x + 3 = 2 (2) + 3 = 7

2x + 5 = 2 (2) + 5 = 9

The odd numbers are 5, 7, and 9.

3, 7, 11, … find the sum upto 15th term?
Solution:

a = 3; d = 7 – 3 = 4; n = 15

Formula is,

Sn = n/2 [ 2a + (n – 1)d ]

= 15/2 [2*3 + (15 – 1)*4]

= 15/2 [6 + 56]

= 15/2 [62]

S15 = 465

Between, if you have problem on these topics Even Number, please browse expert math related websites for more help on Convert Fraction to Percent.

Exercise on Odd Number Series

Practice problem on odd number series

The sum of two odd consecutive numbers is 16. Find their odd numbers?
Sum of two odd numbers is _______?
The sum of four odd numbers is 40. The three odd numbers is 9, 11 and 13 find the 4th odd number?
1 + 7 + 13 +…. Find the 20th term?
1 + 5 + 9 +….  Find sum up to 18th term?
Answer Key of odd number series

7, 9
Odd
7
115
630