Standard Deviation

Standard Deviation is defined as "the measure of the variability". If the spread is more then the standard deviation will also be more. Standard deviation can also be stated as the "square root of the variance". Standard deviation has its application in computing the the annual returns of the banks

Arithmetic Mean:

A mathematical representation of the series of the number which is totalled as the sum of all the number in series and it is divided by the total number of values in the set.

Variability :

When the  standard deviation is raised to the power of two , the value which we get is the variance i.e Variance is equal to the square of the standard deviation. It is the measure of the degree of the spread among certain set of the values. Variance is also stated in the other way as follows"Variance is the measure of tendancy of individual values that vary from the arithmetic mean value

Formula to determine the Standard deviation :

[Formula of Standard deviation]

Where  ,     Σ = sum of

xi = Individual scores

m = arithmetic mean of all scores

n = Size of the number of scores

Variance :

Variance = σ2

Example

Example 1: To find the standard deviation of  2 , 4 , 6 , 8 , 10

Step-1 :Calculate the mean and the deviation

xi       m        xi-m         (xi-m)2

2        6        -4              16

4         6        -2              4

6        6          0              0

8        6          2              4

10       6         4             16

Step-2 :Find the sum of  (x-m)2

16 + 4 + 0 + 4 + 16 = 40

Step-3 :n = 5 , the total number of values , find n-1

5-1 = 4

Step-4 :Now find standard deviation

√40 / √4 = 3.162

3.162 is the standard deviation

To find the variance just square the standard deviation value

Step-5 : Variance = (3.162)2 = 9.998244

Example 2

Example 2 :To find the standard deviation of 3 , 6 , 9 , 12 , 15

Step-1 : Calculate the mean deviation

xi          m          xi-m           (xi-m)2

3           9           -6               36

6           9           -3                 9

9           9            0                 0

12          9            3                 9

15          9           6                36

Step-2 :

calculate the sum of (x-m)2

36+9+0+9+36 = 90

Step-3 :

n=5 ,the total number of values , find n-1

5-1 = 4

Step-4 :

Now let us find the standard deviation

√90/√4 = 4.743

The standard deviation is 4.743

Step-5 :

Variance = (4.743)2

= 22.496

Non Terminating Decimals

In this page we are going to discuss about Non terminating decimals.Below we will get introduction to Non terminating decimals. A non terminating decimal is one of the type of decimals which are divided into two types. They are,

Terminating Decimals
Non terminating Decimals

Non terminating Decimals is the continuous decimal number. It has been going on till the value infinitive. It is the very big value sometimes these non terminating decimals are rounded to its nearest values. But the Terminating decimal number is the fixed value of the decimal value.

Calculating Non Terminating Decimals

Below you can see the steps of calculating non terminating decimals:

Divide 5/3

Step 1: This is the type of an improper fraction. because here numerator takes the large value than the denominator.

Step 2: One time 3 is 3. So the remainder is 2 and the quotient value is 1

Step 3: Now 2 is not divided by 3

Step 4: So we can add zero to the remainder 2 and put a point the quotient value 1

Step 5: Now the value is 20. Again we can do the same procedure

Step 6: Now 6 times 3 is 18 again the remainder is 2 and the quotient value is 6

Step 7: Again we can add zero to 2 then the value is 20

Step 8: Again 6 times 3 is 18. Again the remainder is 2 and the quotient value is 6

Step 9: Now the decimal value is 1.66 …….its going on

Step 10: This decimal part 1.66……is known as the non terminating decimal.

Step 11: It can be rounded as 1.67

Step 12: Because it’s decimal part value is greater than or equal to 5. So the value of the last decimal value is consider as one and that value is added to the previous decimal value.

Step 13: Otherwise the last decimal part value is less than five. So the value of the last decimal value is simply leave. And the previous decimal value is considered as the last decimal digit value.

Pictorial Representation of this problem is shown below:

Non- terminating decimals

Practice problems

Here are some practice problems on non terminating Decimals:

Identify which of the fractions gives the non terminating decimals:

(4)/(2)
(2)/(3)
(6)/(3)

Answer:

Terminating decimals (4)/(2) =2 and (6)/(3) =2

Non terminating decimal (2)/(3) = 0.6666.....

vertices of a pyramid

A vertex of a polyhedron is the point of intersection of three or more faces of the structure.

A pyramid is a polyhedron with a polygonal base connected to a single point called the apex.

The number of vertices of a pyramid varies with the different types of pyramids.


How to determine the number of vertices

In a pyramid, the side faces (originating from the apex) meet at the base.

pyramid

So basically, for the square pyramid shown above, the four vertices of the square base become four vertices of the pyramid.

In addition, we have the apex, where the slant surfaces meet.

Hence, in a pyramid with a n-sided polygon as base, we have (n+1) vertices.

Some examples of vertices of a pyramid.

Now let us try to find the number of vertices of some common pyramids.

triangular pyramid

This is a triangular pyramid, i.e., the base is a triangle (3-sided polygon).

Hence, the number of vertices would be (3+1) = 4


rectangular pyramid

This is a rectangular pyramid, i.e., the base is a rectangle (4-sided polygon).

Hence, the number of vertices would be (4+1) = 5


pentagonal pyramid

This is a pentagonal pyramid, i.e., the base is a pentagon (5-sided polygon).

Hence, the number of vertices would be (5+1) = 6


Thus, it is quite simple to find the number of vertices of a pyramid given we know the type of polygon it has as its base. The square pyramid (with a square as its base) is the most common type of pyramid found in various monuments all over the world.

Construction of a Regular Pentagon

In this part we are going to see in detail about the introduction to construction of a regular pentagon. It’s easy to construct a regular polygon by simply using the compass and the ruler. And so is the construction of the regular pentagon. Most of the pentagons constructed in this method are done by inscribing it into the circle.

Construction of a regular pentagon Method 1:

Draw a circle of diameter AB centered at O.
Draw a line CD perpendicular to AB. Now mark the mid points of OA and OB as P and Q respectively. Draw circles with center as P and Q and radius OP and OQ respectively.
Draw a line from C to P. This line intersects the circle with center Q at T.
With C as center and CT as radius draw a circle. This circle intersects the main circle at E and F which are the successive pentagon vertices.
With length EF find the other vertices of the pentagon.

Since the pentagon has all its sides equal it will be easy to find the other sides if find any one of the vertices.

Construction of a regular pentagon Method 2:

• Draw a circle with center O and Diameter XX’.

• Then draw a line AQ perpendicular to the line XX’.

• Now Mark a point P such that it bisects the line OX’.


• With P as center and length AP draw an arc such that it cuts the line OX. Mark the point G.

• Then with AG as radius and A as center draw an arc such that it cuts the circle at B and C



• With AB or AC as radius find the points D and E.

• Join the points ABCDE to get pentagon.

Study Calculating Median

Study Median:

Median is the middle value of the element in their ascending order. If the size of the elements is even, then the average value of two middle numbers is the median. In the sequence of elements, divide equally the sequence of elements are on one side, half on the other, then the average value of middle elements is the median.

Study Calculating Median Formulas:

If ‘n’ is the total number of elements and ‘n’ is the ‘odd’, then the formula for finding the middle position after the arrangement of elements in ascending order is: (n+1)/2.

If ‘n’ is the total number of elements and ‘n’ is the ‘even’, then you have to find the two middle positions after the arrangement of elements in ascending order. Formulas for finding the two middle positions are: (n/2) and (n/2)+1. After finding the two middle elements, find the average that is the median.

Study Calculating median - Example Problems:

Study Calculating median Problem 1:

Calculate the median of 5, 4, 1, 8, and 7.

Solution:

Count the elements in the sequence has given.

n = 5(odd)

Arrange the elements in ascending order.

1, 4, 5, 7, 8

Formula for finding the middle position is (n+1)/2

(5+1)/2 = 6/2 = 3

Median = 3.
Study Calculating Median Problem 2:

Calculate the median of 8, 6, 1, 2, 3, 4, 7, and 5

Solution:

Count the elements in the sequence has given.

n = 8(even)

Arrange the numbers in ascending order.

1, 2, 3, 4, 5, 6, 7, 8

Find the two middle positions using n/2 and (n/2)+1

Positions are n/2= 8/2 = 4 and 5

4th and 5th position numbers are 4 and 5

The average is (4+5)/2 = 4.5

Median = 4.5

Study Calculating Median Problem 3:

Calculate the median of 2, 4, 6, 3, 1, 6, 3, 7, 4, and 6

Solution:

Count the elements in the sequence has given.

n = 10(even)

Arrange the numbers in ascending order.

1, 2, 3, 3, 4, 4, 6, 6, 6, 7

Calculate the two middle positions using n/2 and (n/2)+1

Positions are n/2= 10/2 = 5 and 6

4th and 5th position numbers are 4 and 4

The average is (4+4)/2 = 4

Median = 4

Study median practice Problems:

Study median practice problem 1:

Calculate the median of 4, 3, 1, 2, and 6.

Answer: 3
Study median practice Problem 2:

Calculate the median of 7, 9, 1, 4, 3, 6, 2, and 8

Answer: 5
Study median practice Problem 3:

Calculate the median of 5, 4, 6, 2, 3, 1, and 9

Answer: 5

Composition and Invertible Function

An invertible function for ƒ is a function from B to A, with the property that a round trip (a composition) from A to B to A returns each element of the first set to itself. A function ƒ that has an inverse is called invertible; the inverse function is then uniquely determined by ƒ and is denoted by ƒ−1. Function composition is the applications of one function to the results of another. For instance, the functions f: X → Y and g: YZcomprised by computing the output of g when it has an argument of f(x) instead of x.  (Source. Wikipedia) .

Examples for invertible function:

To define the invertible of a function f i.e. f−1 (read as ‘f inverse’), the function f must be one-to-one and onto.

Example 1:

Let A = {1, 2, 3}, B = {a, b, c, d}. Consider a function f = {(1, a), (2, b),(3, c)}. Here the image set or the range is {a, b, c} which is not equal to the co domain {a, b, c, d}. Therefore, it is not onto.


For the inverse function f−1 the co-domain of f becomes domain of f −1.

i.e. If f : A → B then f−1 : B → A . According to the definition of domain, each element of the domain must have image in the co-domain. In f−1, the element‘d’ has no image in A. Therefore f −1 is not a function.

Example 2:

f = {(1, a), (2, a), (3, b)} where A = {1, 2, 3}, B = {a, b}

Here the two different elements ‘1’ and ‘2’ have the same image ‘a’.

Therefore the function is not one-to-one.

The range = {a, b} = B. ∴ The function is onto.

f(1) = a

f (2) = a

f(3) = b

Here all the elements in A has unique image

f −1 (a) = 1

f −1 (a) = 2

f −1 (b) = 3


The element ‘a’ has the two images 1 and 2. It violates the principle of the function that each element has a unique image. This is because the function is not one-to-one.

Thus, ‘f −1 exists if and only if f is one-to-one and onto’.

Examples for composition function:

Example :

Let A = {1, 2}, B = {3, 4} and C = {5, 6} and f : A → B and g : B → C such that f(1) = 3, f(2) = 4, g(3) = 5, g(4) = 6. Find gof.

Solution:

gof is a composition function from A → C.

Identify the images of elements of an under the composition function gof.

(gof) (1) = g(f(1)) = g(3) = 5

(gof) (2) = g(f(2)) = g(4) = 6

i.e. image of 1 is 5 and image of 2 is 6 under gof

∴ gof = {(1, 5), (2, 6)}

How to Calculate Median

To find the median in a set of data, intial step is rearrange the set of data in ascending order or descending order. The mid-term value is the median for thae set of data.Half the numbers in the record are fewer, and half the numbers are greater. Establish the Median, put the numbers are known in value arrange and locate the middle number. But there are two middle numbers (as happens when there is an even amount of numbers) then average those two numbers.


Single middle number:

Example 1:

Establish the median for the following listing of value:

8, 13, 4, 7, 5

Solution:

Find the Median of: 8, 13, 4, 7, and 5 (Odd amount of numbers)

Line up your numbers: 4, 5, 7, 8, and 13 (smallest to largest)

The Median is: 7 (The number in the middle)

Example 2:

Establish the median for the following listing of value:

3, 8, 3, 4, 3, 6, 4, 1, 3

Solution:

Find the Median of: 3, 8, 3, 4, 3, 6, 4, 1, and 3 (Odd amount of numbers)

Line up your numbers: 1, 3, 3, 3, 3, 4, 4, 6, and 8 (smallest to largest)

The Median is: 3 (The number in the middle)


Double middle numbers:

Example 1:

Establish the median for the following listing of value:

8, 3, 4, 7, 2, and 6

Solution:

Find the Median of: 8, 3, 4, 7, 2, and 6 (Even amount of numbers)

Line up your numbers: 2, 3, 4, 6, 7, and 8 (smallest to largest)

Add the 2 middles numbers and divide by 2:

= (4 + 6) / 2

= 10 / 2

= 5

The Median is 5.

Example 2:

Establish the median for the following listing of value:

8, 8, 8, 9, 9, 9, 11 and 10

Solution:

Find the Median of: 8, 8, 8, 9, 9, 9, 11 and 10 (Even amount of numbers)

Line up your numbers: 8, 8, 8, 9, 9, 9, 10 and 11 (smallest to largest)

Add the 2 middles numbers and divide by 2:

= (9 + 9) / 2

= 18 / 2

= 9

The Median is 9


Example 3:

Establish the median for the following listing of value:

7, 8, 6, 9, 8, 7, 12 and 11

Solution:

Find the Median of: 7, 8, 6, 9, 8, 7, 12 and 11 (Even amount of numbers)

Line up your numbers: 6, 7, 7, 8, 8, 9, 11 and 12 (smallest to largest)

Add the 2 middles numbers and divide by 2:

= (8 + 8) / 2

= 16 / 2

= 8

The Median is 8