Sample Mean Formula

Introduction to Mean:

Mean is one of the more common term in statistics and it's easy to compute. The average is calculated by arranging the values from the set in a particular way and computing a single number as being the average of the set. The mean may different with median, mode or range. For example, mean income is twisted upwards by a small number of people with very large incomes, so that the majority has an income lower than the mean.


Mean Formula

In general sample mean (A.M) or average of n observations x1, x2, …, xn is defined to be the number x such that the sum of the deviations of the observations from x is 0. That is, the arithmetic mean x of n observations x1, x2, …, xn is given by the equation

(x1 ? x) +(x2 ? x) + ... +(xn ? x) = 0

Hence sample mean formula = x1+x2+x3+…….xn / n

sample mean formula= sum of elements / number of elements


Sample Mean Example

Following steps used to calculate the sample mean value

Step1: find the sum of the numbers

Step2: Calculate the total numbers

Step3: Using the formula finding the mean



Example:

The salaries of its nine employees:

The CEO makes $100,000 per year,

Two managers make $50,000 per year,

Four factory workers make $15,000 each, and

Two trainees make $9,000 per year.

So add 100000+50,000 + 50000+15,000 + 15000+15,000 + 15000 + 9,000 + 9000(all the values in the set of data) which gives 278, 000. Then divide that total by 9 (value in the set of data).

That gives you the mean, which is $30,889.

Not a bad average salary. But be careful when using this number. After all, only three of the nine workers at xyz Co. make that much money.


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Practice Problem:

Using the mean formula find the mean value

The salaries of its ten employees:

The CEO makes $100,000 per year,

Two managers make $50,000 per year,

Four factory workers make $15,000 each, and

Three trainees make $9,000 per year.

Answer: 28700

how to do math proportion

Introduction to math proportion:

Algebra is that separation of mathematics in which calculation are made by using any arbitrary characters to stand for the quantities or things considered. Which are used to correspond to numbers is often given the name literal numbers or simply literals. Since the literal numbers are used to represent numbers,

In our daily life, by learning ratio and proportion many a times we compare two quantities of the same type. Thus, in convinced situations, comparison by division makes better sense than comparison by taking the difference. The comparison by division is the Ratio. We denote ratio-using symbol ‘:’. If two ratios are equal, we state that they are in proportion and use the symbol ‘:’ or ‘=’ to equate the two ratios.


Math proportion – Definition and types:

Definition of proportion:

If two ratios are not equal, then we state that they are not in proportion. In a statement of learning proportion, the four quantities involved when taken in order are known as respective terms. First and fourth terms are known as extreme terms. Second and third terms are known as middle terms.

a: b = c : d

There are two types of propositions,

1. Simple proposition:

A propositions consisting of just one subject and one predicate is called a simple proposition.

Example: The following are simple proposition

1. Ram is blind.

2. The flower is not red.

2.  Compound proposition

A proposition consisting of two or more simple propositions in the form of a single sentence is called a compound proposition.

Example: The following are compound propositions,

Quadrilateral ABCD is a square and each side of this quadrilateral is 4cm long.


Math proportion – Example problems:


Problem 1:

Are the ratios 60g: 30g and 32 kg: 48 kg in proportion?

Solution:

60 g: 30 g =60/30 = 2:1

= 2:1

32 kg: 56 kg = 4 / 7 = 4:7

= 4: 7           So, 60: 30 = 32: 56.

Therefore, the ratios 60 g: 30 g and 32 kg: 56 kg are in proportion,

i.e. 60 : 30 :: 32 : 56.

The middle terms in this are 30, 32 and the extreme terms are 60, 56.

Problem 2:

If 4: 5 = 8: X is a proportion, find the missing term.

Solution:

Product of extremes = 4 × x

Product of means = 5 × 8 = 40.

Since it is a proportion, 4 × x = 40

Divide both sides by 4

(4 × x) / 4 = 40 / 4

x =40 / 4 = 10.

X = 10

Problem 3:
The income and Savings of a family are in the ratio 5: 2. If the income of the family is Rs.5, 800. Find how much is being saved.

Solution:

Let the savings be Rs. x.

The proportion is 5: 2 = 5800: x

(Income: Saving) = (Income: Saving)

5x = 2 × 5800

5x / 5 = (2 × 5800) / 5

x = 11600 / 5

x = 2320

The Savings = Rs.2320.

grade 5 math probability

Introduction to grade 5 math probability:

Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory that is used extensively in such areas of study as mathematics, statistics, finance, gambling, and science. In this article we shall discuss about grade 5 probability problems.(Source: wikipedia) I like to share this Probability of Independent Events with you all through my article.


Grade 5 math probability example problems

Here we are going to discuss grade 5 probability problems with detailed solutions.

Problem:

An unbiased die is rolled once at a time find the probability of shown the number 6 in dice.

Solution:

A dice is rolled once at a time, the possible chance of sample space is S= {1, 2, 3, 4, 5, 6} and, therefore, the sample space n(S)=6..

Consider E1 = event of rolling a dice getting the number 6

E1= {6} and, therefore, n (E1) =1.

Therefore probability (getting number 6) = P (E1) = n (E1) / n(S) = `1 / 6`

Example:

An unbiased die is rolled once at a time. Find the probability of getting a number greater than 2.

Solution:

A dice is rolled once at a time, the possible chance of sample space is S= {1, 2, 3, 4, 5, 6} and, therefore, the sample space n(S)=6..

Let E1 = event of rolling a dice getting number greater than 2 Then,

E1= {3, 4, 5, 6} and, therefore, n (E1) =4.

Therefore Probability of getting a number rolling dice greater than 2 = P (E1) = n(E1) / n(S) = `4 / 6` = `2 / 3` .

Example:

An unbiased die is rolled once at a time. Find the probability of getting a number less than 3.

Solution:

A dice is rolled once at a time, the possible chance of sample space is S= {1, 2, 3, 4, 5, 6} and, therefore, the sample space n(S)=6..

Let E1 = event of rolling a dice getting number less than 3 Then,

E1= {2 , 1} and, therefore, n (E1) =2.

Therefore Probability of getting a number rolling dice less than 3 = P (E1) = n(E1) / n(S) = `2 / 6` = `1 / 3` .

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Grade 5 math probability example problems


Problem:

An unbiased die is rolled once at a time find the probability of shown the number 1 in dice.

Answer:

`1/6.`

Problem:

An unbiased die is rolled once at a time. Find the probability of getting a number greater than 4.

Answer:

`1/3`

Y Intercept Slope Form

Introduction on y intercept slope form:

If the line passes through the points A (x1, y1) and B (x2, y2) and then its slope is,

Slope (m) = y2-y1/x2-x1.

The slope is defined as the relation between the perpendicular line distance and flat distance between any two points. The slope is denoted by letter ‘m’.

m=y/x where y and x distances between the two points

The point slope equation of a line is,

Y=mx+b

Where x and y is the points, m is the slope and b is the Y-intercept

The Y intercepts are taking place when horizontal values are zero.

Example Problem on Y Intercept Slope Form:

Example 1

Given point (5, 2) and the slope is 4. Find the equation of this line and the Y intercept.

Solution:

Point (5, 2)? (x1, y1)

Slope m= 4.

Equation of the slope point form is

y-y1 = m (x-x1)

y-2=4(x-5)

y-2=4x-20 (using distribute property)

y-2+2=4x-20+2 (Add 2 on both side)

y=4x-18.

Slope m=4. Y intercepts b=-18.

The line of the equation is y=4x-18

Example 2:

Given point (7, 3) and (-2, 5). Find the equation of this line and the Y intercept.

Solution:

Points (7, 3) ? (x1, y1)

(-2, 5) --- > (X2, y2)

To find the equation of line is,

y-y1 = m (x-x1)

Step 1: Take points (7, 3) (-2, 5)

Slope m = y2-y1/ x2-x1

m=5-3/-2-7 ? 2/-9

Step 2: Now take slope m=-2/9 and point (-2, 5)

y-y1 = m (x-x1)

y-5=-2/9(x-(-2))

Step 3: Simplification

y-5=-2/9x+4/9 (Using distributive property)

y-5+5=-2/9x+0.44+5 (Add 5 on both side)

y=-2/9x+5.44.

The equation is in slope intercept form

Slope m=-2/9 and Y intercept b=5.44

The line of equation is y=-2/9x+5.44

some more Examples on Y Intercept Slope Form:

Example3

Find the Y intercepts of the function f(x) =3.5x+25.

Solution:

To find the Y intercept, plug x=0

F (0) = 3.5(0)+25

=25

This means the Y intercepts taking place at (0, 25)

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Example 4:

Find the Y intercepts of the function f(x) =1.2x+3.5

Solution:

To find the Y intercept, plug x=0

F(0)=1.2(0)+3.5

=3.5

This means the Y intercepts taking place at (0, 3.5)

Practice Root Mean Square

Introduction for root mean square practice:

Generally root mean square is the main topic for statistics and mathematics subject. It is known as well as quadratic mean. The root mean square is a learning topic of the statistical measure of the degree of the variable quantity; it is helpful for while positive and negative integers are required for mean value. Here in this article we are going to explain about root mean square, and solving root mean square example and practice problems.

Practice Root Mean Square:

The root mean square values are the set of values.  It is the average mean value of the squares of the unique (original) values. Root mean square abbreviated from is RMS or rms.

The square root of arithmetic mean is square values of the casual variables. In other terms, we can explain that the root mean square is a statistics evaluate of the degree of the different quantity. It can be evaluate for the serious of discrete ideals or for a constant varying function.

Root mean square formula:

Root mean square =` sqrt (((x1^2) + (x2^2) + (x3^2)+ .....+ (xn^2))/ N)`

Where as, x = individual value

N = number of total values.

This formula is very significant for practice problems solving in root mean square.

Between, if you have problem on these topics Parallelogram Geometry, please browse expert math related websites for more help on formula for a cube.

Practice Problems from Root Mean Square:

Practice problem 1: Solve the root mean square value of (-6, 10, 15, 21, -12, 16).

Solution:

Step 1: to count the total number of values (N),

Here N = 6.

Step 2: Square all the values,

36, 100, 225, 441, 144, 256

Step 3: Take the average of all the square values,

`(36+100+225+441+144+256)/6`

= `1202/6`

= `200.33`

Step 4: To take the square root of the average values,

Rms = `sqrt (200.33)`

= 14.1539

Answer is ~ 14.15.

Practice problem 2: To solving the following numbers and find the root mean square values (-15, 12, 50, 8, -32).

Solution:

Here the total numbers are, N = 5.

Square all the values (225, 144, 2500, 64, and 1024)

Take the average of the square values.

`(225 +14 4 + 2500 + 64 + 1024) / 5`

= `3957 /5.`

= `791.4`

Take the square root of average values.

Rms = `sqrt (791.4)`

= `28.131`

Answer is = ~28.1.

Home work practice problems of root mean square:

Problem 1: To find the Rms value of (-6, 12, 24, 36, 18, -12) the above values.

Answer is: 20.49

Problem 2: To Solve the values and fined the Rms value of (-12, 24, -10, 20, -8, 16)

Answer is: 16.02

Those all above explanations and example problems make clear and know how to solving the roots mean square practice problems.

Definition of Slope in Math

Introduction for definition of slope in math:

Definition:

The slope of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line. Given two points (x1,y1) and (x2,y2) on a line, the slope m of the line is

Slope (m) = `(y_2 - y_1)/(x_2 - x_1)`

Source: Wikipedia

The definition of slope in math example problems and practice problems are given below.

Example Problems for Definition of Slope in Math:

Example problem 1:

Determine the slope of the line, 5x - y = -13

Solution:

The slope intercept form of a line is y = mx + b, where m is the slope of the line.

Given equation is in the form of ax + bx + c = 0. To find the slope of the line, we have to convert the equation from general form to slope intercept form.

The given equation can be converted from general form to slope intercept form as follows,

5x - y = -13

Subtract 5x on both sides,

5x - y - 5x = -13 - 5x

-y = -13 - 5x

Divide by (-1) on both sides.

`-y/(-1) ` = -`(13)/(-1)` - `(5x)/(-1)`

y = 13 + 5x

y = 5x + 13

Now the equation is in the form of y = mx + b, so the slope of the given line is 5

Example problem 2:

Determine the slope of a line, which contains the points A (4, -3), B (-2, 8).

Solution:

The slope of a line which contains two points (x1, y1) and (x2, y2) is given by,

Here, x1 = 4, x2 = -2, y1 = -3, y2 = 8.

Slope of the line, m = ` (y_2 - y_1)/(x_2 - x_1)`

m = `(8 + 3)/( -2 - 4)`

m = `-11/6`

Slope of the line (m) = `-11/6`

So, the slope of a line, which contains the points A (4, -3), B (-2, 8) is `-11/6`

Practice Problems for Definition of Slope in Math:

Practice problem 1:

Determine the slope of the line, 8x - y = -60

Answer: Slope (m) = 8

Practice problem 2:

Determine the slope of the given points: (12, -4) and (2, 8)

Answer: Slope (m) =  `-6/5`

Corollary of Theorem

Corollary of theorem - Introduction:

In math a corollary normally follows a theorem. The exercise of the expression corollary, slightly than proposal or theorem, is basically subjective. Proposal B is corollaries of proposal A if B can readily be assume from A, but the meaning of gladly varies depending on the author and context. The importance of the corollary is frequently measured secondary to that of the primary theorem; B is suspect to be termed a corollary if its math consequences are as important as those of A. Occasionally a corollary has a proof that explicate the derivation; on occasion the derivation is considered to be self-evident.

Corollary of some Theorems:

Corollary of Theorem

Let Y ~ B (n, `pi` ), where `pi` is called the population proportion, n is the sample size and Y is the number of “success”s in the sample. Then the sample proportion is p = `Y/n` . Given these,

`lim_(x->oo)P(|y/n - pi| <= epsi)= lim_(x->oo)P(|p-pi|<=epsi) =1`

This states that as the sample size increases the sample proportion converges in probability to the population proportion, that is, p->`pi` .

Theorem

Strong Law of Large Numbers or almost sure convergence

Let X1, X2,.. Xn be a random sample from a population (of X’s) with mean E(X) = mX and a finite variance, Var(X) = `sigma^2_x < oo` . Let `bar(X) = 1/n sum_(i=1)^n X_i` . Then, for any positive real number e,

`P(lim_(n->oo) |bar(X)- mu_x| <= epsi =1 or p(lim_(n->oo)) | bar(X) - mu_x|>=epsi) = 0`

Angle Sum Theorem:

The sum of the angle measures in any triangle is 180°

PROVE IT!

x + x + y = 180° (Triangle)

y + y + y = 180° (Line)

y = x (Alternate Interior Angles)

Corollary to the Angle Sum Theorem:

If ∆ABC is any triangle, then an exterior angle of ABC has the same measure as the sum of the measures of the two nonadjacent interior angles.

PROVE IT!

m `angle` 1 + m `angle` 2 = m `angle` 3

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Altitude Similarity Theorem

The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.

`Delta` ABC ~ `Delta` ACD ~ `Delta ` CBD

Corollary 1

The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.

`(AD)/(CD) = (CD)/(DB) CD = sqrt(AD(DB)) `

Corollary 2

The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse.

`(AD)/(AC) = (AC)/(AB') , (BD)/(CB) = (CB)/(AB)`