Statistical Data

Quantitative : Discrete, Continuous

Referred to as numeric,  Quantitative data have two types: Discrete and Continuos. Continuous are measurement and discrete are counts, as a general rule.

A count that can't be made precise more is discrete data. It involves integers typically. It is a discrete data which is the number of children (or pets, or adults), for instance, the reason is whole are counting by yourself, entities indvisible: 1.3 pets or 2.5 kids, you can't have it.

On the other hand, continuos data, to finer and finer levels it could reduce and could be divided. For examlple, at progressively more precise - beyond, milimiters, centimeters and meters - so height are continuos data.

Qualitative 

Qualitative data is characterizes or approximates by data but properties, attributes, characteristics, etc are not measured,  of phenomenon or a thin. Whereas quantitative data defines is where qualitative data describes.

Measures of dispersion

Standard Deviation

Within a set of data, a measure of the spread of scores are the standard deviation defines.

Formulas for the standard deviation

Sample standard deviation formula is:

where,

= sample mean
n = number of scores in sample.
s = sample standard deviation
= sum of...

Measures of central tendency

Mean:
                      -- The mean is found when the numbers of data entries dividing and adding up all of the given data. It is also known as the average.
                Example:
                             - the grades were as follows,  the class of grade 110 math had a mathematics test recently:
 
                                       78
                                       66
                                       82                                  464 / 6 = 77.3
                                       89
                                       75                        Hence, the mean average of the class is 77.3.
                                    + 74
                                      464
     Median:
                      -- The middle number is the median.  First you arrange the numbers in order from lowest. Firstly, from the lowest, in order you arrange the numbers
                to highest, by crossing off the numbers until you reach the
                middle then you find the middle number.
              Example:
                               - to find the median, use the above data:
                                                66  74  75  78  82  89\
                            - There is no middle number, as you can see we have two numbers  What should we do?
                    find the average and we take the two middle numbers , ( or mean ).  It is simple;
                                                             75 + 78 = 153
                                                              153 / 2 = 76.5
                             Hence,  76.5 is the middle number.
 
         Mode:
                        -- this is the number that most often occurs.
                 Example:
                                 - the following data below, find the mode:
 
                                    78  56  68  92  84  76  74  56  68  66  78  72  66
                                    65  53  61  62  78  84  61  90  87  77  62  88  81
                                                   78 is the Mode.

Sets and Venn Diagram

At the left shows overlap of two sets A and B of the Venn diagram. U is the universal set.  In the center region labeled where the circles overlap is where the values that belong to both set A and set B are located. "Intersection" of the two sets is the region are called.

 

Videos of Shading regions Venn Diagrams

(Can only be played at google chrome)

Geometric Progression

Geometric Progression, Series & Sums

Introduction

A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e.,

where	r	common ratio
	a1	first term
	a2	second term
	a3	third term
	an-1	the term before the n th term
	an	the n th term

The geometric sequence is sometimes called the geometric progression or GP, for short.
For example, the sequence 1, 3, 9, 27, 81 is a geometric sequence. Note that after the first term, the next term is obtained by multiplying the preceding element by 3.
The geometric sequence has its sequence formation:
To find the nth term of a geometric sequence we use the formula:

where	r	common ratio
	a1	first term
	an-1	the term before the n th term
	n	number of terms

Arithmetic Progression

An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant.
For example, the sequence 1, 2, 3, 4, ... is an arithmetic progression with common difference 1.
Second example: the sequence 3, 5, 7, 9, 11,... is an arithmetic progression
with common difference 2.
Third example: the sequence 20, 10, 0, -10, -20, -30, ... is an arithmetic progression
with common difference -10.

Notation

We denote by d the common difference.
By a_n we denote the n-th term of an arithmetic progression.
By S_n we denote the sum of the first n elements of an arithmetic series.
Arithmetic series means the sum of the elements of an arithmetic progression.

Properties

a₁ + a_n = a₂ + a_n-1 = ... = a_k + a_n-k+1

and

a_n = ½(a_n-1 + a_n+1)

Sample: let 1, 11, 21, 31, 41, 51... be an arithmetic progression.
51 + 1 = 41 + 11 = 31 + 21
and
11 = (21 + 1)/2
21 = (31 + 11)/2...

---

If the initial term of an arithmetic progression is a₁ and the common difference of successive members is d, then the n-th term of the sequence is given by

a_n = a₁ + (n - 1)d, n = 1, 2, ...

---

The sum S of the first n numbers of an arithmetic progression is given by the formula:

S = ½(a₁ + a_n)n

where a₁ is the first term and a_n the last one. or

S = ½(2a₁ + d(n-1))n

Number Pattern

(can only be played in google chrome)

Pattern or a certain sequence that are followed by a list of numbers.


For Example:  tarts at 1 and jumps 3 every time... 1, 4, 7, 10, 13, 16,

Another Example: doubles each time ...  2, 4, 8, 16, 32

Linear Inequalities

Linear inequalities in general dimensions

In Rⁿ linear inequalities are the expressions that may be written in the form

${\displaystyle f({\bar {x}})<b}$ or ${\displaystyle f({\bar {x}})\leq b,}$

where f is a linear form (also called a linear functional), ${\displaystyle {\bar {x}}=(x_{1},x_{2},\ldots ,x_{n})}$ and b a constant real number.
More concretely, this may be written out as

${\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}<b}$

or

${\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\leq b.}$

Here ${\displaystyle x_{1},x_{2},...,x_{n}}$ are called the unknowns, and ${\displaystyle a_{1},a_{2},...,a_{n}}$ are called the coefficients.
Alternatively, these may be written as

${\displaystyle g(x)<0\,}$ or ${\displaystyle g(x)\leq 0,}$

where g is an affine function.^[4]
That is

${\displaystyle a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}<0}$

or

${\displaystyle a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\leq 0.}$

Note that any inequality containing a "greater than" or a "greater than or equal" sign can be rewritten with a "less than" or "less than or equal" sign, so there is no need to define linear inequalities using those signs.

Systems of linear inequalities

A system of linear inequalities is a set of linear inequalities in the same variables:

${\displaystyle {\begin{alignedat}{7}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\cdots +\;&&a_{1n}x_{n}&&\;\leq \;&&&b_{1}\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\cdots +\;&&a_{2n}x_{n}&&\;\leq \;&&&b_{2}\\\vdots \;\;\;&&&&\vdots \;\;\;&&&&\vdots \;\;\;&&&&&\;\vdots \\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\cdots +\;&&a_{mn}x_{n}&&\;\leq \;&&&b_{m}\\\end{alignedat}}}$

Here ${\displaystyle x_{1},\ x_{2},...,x_{n}}$ are the unknowns, ${\displaystyle a_{11},\ a_{12},...,\ a_{mn}}$ are the coefficients of the system, and ${\displaystyle b_{1},\ b_{2},...,b_{m}}$ are the constant terms.
This can be concisely written as the matrix inequality

${\displaystyle Ax\leq b,}$

where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants.
In the above systems both strict and non-strict inequalities may be used.

• Not all systems of linear inequalities have solutions.

Linear programming (LP) (also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization).
More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such a point exists.
Linear programs are problems that can be expressed in canonical form as

${\displaystyle {\begin{aligned}&{\text{maximize}}&&\mathbf {c} ^{\mathrm {T} }\mathbf {x} \\&{\text{subject to}}&&A\mathbf {x} \leq \mathbf {b} \\&{\text{and}}&&\mathbf {x} \geq \mathbf {0} \end{aligned}}}$

• where x represents the vector of variables (to be determined), c and b are vectors of (known) coefficients, A is a (known) matrix of coefficients, and ${\displaystyle (\cdot )^{\mathrm {T} }}$ is the matrix transpose. The expression to be maximized or minimized is called the objective function (c^Tx in this case). The inequalities Ax ≤ b and x ≥ 0 are the constraints which specify a convex polytope over which the objective function is to be optimized. In this context, two vectors are comparable when they have the same dimensions. If every entry in the first is less-than or equal-to the corresponding entry in the second then we can say the first vector is less-than or equal-to the second vector.

Linear programming can be applied to various fields of study. It is widely used in business and economics, and is also utilized for some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design.

LOGARITHM

WHEN WE ARE GIVEN the base 2, for example, and exponent 3, then we can evaluate 2³.

2³ = 8.

Inversely, if we are given the base 2 and its power 8 --

2^? = 8

-- then what is the exponent that will produce 8?

That exponent is called a logarithm.  We call the exponent 3 the logarithm of 8 with base 2.  We write

3 = log₂8.

We write the base 2 as a subscript.
3 is the exponent to which 2 must be raised to produce 8.
A logarithm is an exponent.

Since

10⁴ = 10,000

then

log₁₀10,000 = 4.

"The logarithm of 10,000 with base 10 is 4."

4 is the exponent to which 10 must be raised to produce 10,000.

"10⁴ = 10,000" is called the exponential form.

"log₁₀10,000 = 4" is called the logarithmic form.

Here is the definition:

log_bx = n   means   bⁿ = x.

That  base  with that  exponent  produces x.

Example 1.   Write in exponential form:   log₂32 = 5.

 Answer.   2⁵ = 32.

Example 2.   Write in logarithmic form:  4−2  =

1 16

.

Answer.   log4

1 16

= −2.

Problem 1.   Which numbers have negative logarithms?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

Proper fractions.

Example 3.   Evaluate  log₈1.

 Answer.   8 to what exponent produces 1?  8⁰ = 1.

log₈1 = 0.

We can observe that, in any base, the logarithm of 1 is 0.

log_b1 = 0

Example 4.    Evaluate  log₅5.

 Answer.   5 with what exponent will produce 5?   5¹ = 5.  Therefore,

log₅5 = 1.

In any base, the logarithm of the base itself is 1.

log_bb = 1

Example 5 .   log₂2^m = ?

 Answer.   2 raised to what exponent will produce 2^m ?   m, obviously.

log₂2^m = m.

The following is an important formal rule, valid for any base b:

log_bb^x = x

This rule embodies the very meaning of a logarithm.  x -- on the right -- is the exponent to which the base b must be raised  to produce b^x.

The rule also shows that the exponential function b^x is the inverse of the function log_bx.  We will see this in the following Topic.

Example 6 .   Evaluate  log3

1 9

.

Answer.

1 9

is equal to 3 with what exponent?

1 9

= 3−2.

log3

1 9

=

log33−2  =  −2.

Compare the previous rule.

Example 7.   log₂ .25 = ?

 Answer.   .25 = ¼ = 2⁻².  Therefore,

log₂ .25 = log₂2⁻² = −2.

Laws of Indices

When working with indices, there are three rules that should be used:

When m and n is positive integers,

1.	am × an = am + n
2.	am ÷ an = am – n   or	am an	= am – n   (m ≥ n)
3.	(am)n = am × n

This is the definition of aⁿ logical consequences, these three are the results, but formal proof are really needed.With particular examples as below, you can 'verify' them, but this are not the proof.

27× 23	=	(2 × 2 × 2 × 2 × 2 × 2 × 2) × (2 × 2 × 2)
	=	2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
	=	210	(here m = 7, n = 3 and m + n = 10)

or,

27 ÷ 23	=	2 × 2 × 2 × 2 × 2 × 2 × 2 2 × 2 × 2
	=	2 × 2 × 2 × 2
	=	24	(again m = 7, n = 3 and m – n = 4)

Also,

(27)3	=	27 × 27 × 27
	=	221	(using rule 1)   (again m = 7, n = 3 and m × n = 21)

Below, the first rule proof is given below:

Proof

am × an	=	a × a × ... × a m of these	×	a × a × ... × a n of these
	=	a × a × ... × a × a × a × ... × a (m+n) of these
	=	am+n

In a similar way, for all positive integers m and n, the second and third rules can be shown.
Using rule 2 we can see an important result:

	xn xn	= xn – n = x0
but	xn xn	= 1,	so

x⁰ = 1

This is true for any non-zero value of x, so, for example, 3⁰ = 1, 27⁰ = 1 and 1001⁰ = 1.