Selecting a sample

In the previous post we have discussed about Sampling errors and In today's session we are going to discuss about Selecting a sample.
Selecting a sample:
In this session, we will talk about selecting a sample from a group of samples for grade VI.
In terms of probability, sampling is a method that aids in random selection process.
The important condition is that all the events to be chosen must have same probability.
For example, the random numbers chosen by a computer program is a random selection process as all the numbers can be chosen with equal probability.
Sample size: sample size is the total sum of sizes of all its cases.
Before we continue, let us understand some basic notations,
N=total number of cases in the sampling space.
n=number of cases in a given sample
f=sampling fraction=n/N
Simple random sampling:
We have to select n units from N.
N is the total cases in sampling space.
All the n must have the equal probability to be chosen.
nCN=total number of subsets of n from N.
Let us make this concept clearer through an example.
Let us take an example that we have to do a survey of a company by selecting its past clients. We want 100 clients to be a part of the survey.
Company provides a list of 1000 employees whose records are in the database of the company.
Sampling fraction=n/N=100/1000=0.10
Hence sampling fraction is 10%.
We have several ways to select the 100 clients. An easy way is to put all the 1000 clients in a group, subdivide the group and randomly select any subgroup. This is a mechanical way to select but it is not as efficient as the quality of samples would depend on how we have subdivided the group and how randomly we choose the subgroups. (know more about  Sampling, here)
The computerized method is more efficient as compared to the mechanical method.
Principally, our main motive is to understand the selection of sample out of many. To know more about cbse syllabus.

The main point here to keep in mind is that the samples must have equal probability to be chosen.
If the selecting probability depends on the previous selection, then it is said to be conditional probability.
In the next topic, we are going to discuss Sampling errors.

Sampling errors

In sampling errors we take a number of samples of the given mode. The samples are the models on which an analyst is working to find the errors from the number of terms which are under examination. We use the numbers of samples and then find the mean to reduce the sampling errors of the data model. In this way we define sampling errors.
For example, population of the state (increasing or decreasing); now if we want the mean height of 5th standard youngsters then we measure all the heights of sample of 5th graders in the state.
The best parameter to estimate the population is sample mean. But, there is difference between the mean sample or observed sample and the true population mean. So that, the sampling method is good or bad, if the rate of sampling is bad then likely should be some errors occur in the sample static.

The sampling errors are not easy to reduce. For example, any state wants to contact people to know how many people are homeless, the number of senior citizens, etc. The state government wants to find the parameter information but this is not easy. Then there is an error due to imperfect data collection. To reduce the sampling errors we take a number of samples of the model and then find the mean value of these samples. There are some other errors like non sampling, standard error. (know more about icse syllabus 2013, here)

The non sampling error is caused by human error by which a statistical analysis is done. These errors are not limited and not eliminated. Standard error helps us to measure the sample accuracy of the sample model. The representative sample is an unbiased indication of the data model. This is the sampling errors for grade VI. In the next session we will discuss about Selecting a sample.

Constructing sample spaces

Constructing sample spaces:
Sample space is generally denoted by S, Ω or U. Fundamentally it’s the set of all random trials; like in tossing a coin the sample space is (head, tail). If we toss two coins then sample space for this random trial is (head, head), (head, tail), (tail, head), (tail, tail). In the trial of tossing a single sided die sample space s is (1, 2, 3, 4, 5, and 6). It’s not necessary that one trial should have only one sample space; like in a trial of drawing a card from a standard deck of 52 playing cards one probability of outcome is could be the rank (ace through king) and the another sample space could be suits i.e. (club, diamond, hearts or spades). The subsets of sample space are called event. Problems would be critical after increasing the number of sample spaces. So it would be problematic when there are infinite sample spaces means there are infinite events.
Sample space construction:
Sample space is the set of all possible outcomes and it’s necessary to consider all possibilities. It may be a difficult task and for this purpose counting principle can be used. If there is more than one event it’s important to determine all possibilities that exist. It can be stated that:
“If there are A ways of an event to occur and B ways for occurrence of second event then there are A.B ways for both to occur”. This is the concept of the counting principle.
An example is given below which can help to understand sample space better. (know more about cbse sample papers, here)
Example: one jar contains 1 red, 3 green, 2 blue or 4 yellow balls. Then what would be the probability of each outcome if a single ball is chosen from the jar.
Solution: Sample space for this random trial is
S = (red, green, blue, yellow)
Probability can be calculated as:
Probability:  P (red ball) = 1/10
P (green ball) = 3/10
P (blue ball) = 2/10 = 1/5
P (yellow ball) = 4/10 = 2/5
In the next session we will discuss about Sampling errors. 

Estimation of Solutions in Grade VI

In this session we are going to discuss Estimation of Solutions. Before we proceed for any project we need to find the estimated budget for the project to be attained. This is done by rounding the values to nearest tens, hundreds or thousands.  Suppose we go to buy some fruits and other purchase items. We make a rough budget of the articles to be purchased and take Rs 200 in my purse. This is called estimating solutions. Now when we buy the fruits, the seller tells the following costs:  Apples of Rs 84.00, Bananas of Rs 35 and grapes of Rs 18.
  The shopkeeper gives us the bill of Rs 84.00
                                                          Rs 35.00
                                                    +   Rs 18.00
                                                        _____________
                                                             137.00
We simply round off the figure to nearest tens and pay Rs 140 to him. Another way to make the payment will be to round off the above figure in nearest hundred and pay Rs 150.00 estimation of solutions is done in order to make any project a success, may it be a window shopping, budgeting for the parties big or small about how many expected guest will be attending the party or even organizing big projects.
In Grade VI We follow the following methods for estimating solutions:
To round off the value to nearest tens, we see the ones place digit, if it is less than 5, we make ones place 0. If it is greater than 5 we add 1 to tens place and make ones place 0. E.g.:
      47 round off to tens becomes 50
     32  round off  to tens becomes 30
    65 round off to tens becomes 70
In the next topic we are going to discuss Percentages

Proportions

Here we are going to learn about Proportions (some portion of this topic taken from ICSE class 10 syllabus).
When any two ratios are equal, we say that the two ratios are in proportion. It is the relationship between two ratios whose output is same and constant.
It is represented by a/b  :: c/d
or
a : b :: c : d
Here the ratio of a: b is proportion to c: d. Both symbols :: and = are used to represent the proportionality of two ratios.
If we have a : b : : c : d , it is read as a is to b as  c is to d
In the above statement we have 'a' as First  term, 'b' as second term , 'c' as third term and  'd' as the fourth term.
In this a and d are called extreme terms or extremes and b and c are middle terms and are also called means
If the given four numbers are in proportion, then the product of means is equal to the product of extremes.
In order to check that the two ratios are in proportion, we simply check if the product of extremes and the product of means are equal.
Let us see how to solve proportion with the help of an example:
Example: solve Proportion problem  60 : 105  :: 84 :147 .
Sol: We first take the product of means ie 105 * 84 = 8820
now we take the product of extremes i.e. 60 * 147 = 8820
Here we observe that the product of means = product of extremes. So the two ratios are in proportion.
This can also be checked by converting both ratios in lowest terms, if both the values are same, they are in proportion
Let us try it for the same data:
60 / 105  ,we divide numerator and denominator by 5 and get
= 12 /21 , again dividing by 3 we get
= 4 / 7
Similarly we write 84/147 , dividing & multiplying  by 3 we get
= 28 / 49
Now dividing & multiplying by 7 we get
= 4 /7
So they are in proportion.

In next post we will talk on Estimation of Solutions in Grade VI. For more information on Substitution Method, you can visit our website

Math Blog on Grade VI

Here we are going to discuss one of the most interesting and a bit complex topic of mathematics: operations on fractions, decimals, integers and exponents which are usually studied in grade VI.
Now we will first start with fraction:
Fractions can be defined as the part of whole numbers having both numerator and denominator. ¼, ¾  are some examples of fractions. Generally fractions are of three types i.e. proper fractions, improper fractions and mixed fractions. Let’s see some operations on fractions:

1.    Addition fractions:

While adding two fractions we must take care of that the denominators of the fractions to be added must be same. And if the denominators are not same then find the LCM of the denominators and put each one in its equivalent form. Then simply add the numerators.

                     p/q + r/q = (p + r)/q

2.    Subtraction of fractions:

While subtracting two fractions we must take care of that the denominators of the fractions to be subtracted must be same. And if the denominators are not same then find the LCM of the denominators and put each one in its equivalent form. Then simply subtract the numerators.

                     p/q - r/q = (p - r)/q

3.    Multiplication of fractions:

While multiplying two fractions multiply the numerators together and then multiply the denominator together and simplify them.

                     p/q x r/s = (p x r)/(q x s)

4.    Division of fractions:

While dividing the fractions take reciprocal of the second fraction and multiply both the fractions together.

           p/q ÷ r/s = p/q x s/r = (p x s)/(q x r)

Now move to the next topic i.e. decimals:
Decimal number can be defined as the number which contains decimal point. To understand the decimal numbers we must have knowledge about the place values, which is very important when we write the decimal numbers. (also try fraction to decimal converter)
In the number 234:

• The "4" is at the Units position.
• The "2" is at the Tens position.
• And the "3" is at the Hundreds position.
  

In a decimal number as we move left, each position becomes 10 times bigger and as we move right, each position become 10 times smaller.
Now we move to the next topic i.e. integers:
Integer can be defined as similar to the whole number but integers also contains negative numbers and do not contain fractions.
 Let’s see some examples of integers:

·         Negative Integers =   -1, -2,-3, -4, -5, …  

·         Positive Integers = 1, 2, 3, 4, 5, …  

·         Zero =   0

We can also put the integers like this ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... .
Now we move to the next topic i.e. exponents:
Exponents can be defined as the power or indices of a number. The exponent of a number represents that how many times the number will be use in multiplication. Let’s understand it by an example:
Example: 12² = 12 x 12 = 144
In words: 12² can be called as "12 to the second power", "12 to the power 2" or simply "12 square".

In upcoming posts we will discuss about Proportions. Visit our website for information on syllabus for class 10th ICSE

Percentages in Grade VI

Hello friends, in this math homework help session, I am going to discuss about the percentages, which you study in grade VI. The term percentage comes from Latin word and means for every hundreds. Another definition is percentages is a fraction with any numbers which divide by hundredths. Percentages mean divide by 100 or per hundred. Percentages are described in two ways. The first one is fraction form. Example:-26/100 =26% The second is decimal form. Example:-33=33%.
A model for percents is a square form divided into 100 equal parts.
some examples: 33 of the 100 parts in the square to show 33 hundredths means 33/100 or 0.33 or 33%.
Money is good example for percentages. There are 100 cents in a dollar. Five cents ($.05) are 5 hundredths or (5/100) or 5% of a dollar.
Any type of problems like fractions, decimals deal with the percentages and we can easily solve them as discussed below.
Formula for percentage is used to solve different percentages problems. It is given as:-
is/of=%/100, where formula start from the left side and we can use the cross multiply it means to one multiply numerator of one fraction by denominator of another fraction.
For example:- we can find out the 20 % of 100 is.....
step 1:- is =? of=100 ,%=?
step 2:-is/100 20/100
step 3:-let's assume that a is a single digit.
Step 4:-a/100=20/100
step 5:-now we have to do cross multiplication
a*100 and 100*20
a*100=2000
divide 2000 by 100 to get the a
step 6:-now 2000/100=20 and a=20
step 7:-so,20% of 100 is 20
In this way we can calculate percentages. In the next topic we are going to discuss Proportions.

In next post we will talk on Math Blog on Grade VI. For more information on class 10 ICSE syllabus, you can visit our website

Algebraic Expressions

In this section we will talk about the basic algebra1 like operations related to the algebraic expressions. We will learn here about the process for evaluating algebraic expression in the algebraic mathematics. All the algebra in this article will be around the level of grade VI of CBSE math Syllabus.

In the mathematics, pupils learn about some standard fractions, decimals, exponential, and some other algebraic math. We form the algebraic expressions with the help of these algebraic components. In algebra we generally learn about the linear equations, linear function, graphs, positive linear function, factors, greatest common factor (GCF) etc.  In the field of evaluation of algebraic expressions and equations we learn about balancing equation and also about the solution of them.For practicing you can refer number sets algebra 1 worksheets,
In algebra, if we add some thing with some other quantity then they also form an expression. Any of the expressions is the first step to solve an algebraic expression. The algebraic expressions are the way to show any problem in the word form which contains some variables in it and tells the story of the problem in the expression form. It’s not always necessary that an algebraic expression contains an operation in it. An algebraic expression may contains some of the variables, some operations (+, -, x, / ) applied on those variables or some times only variables within it. As per the naming is concerned, the variables are the components of the expressions, the elements whose products form a single term of the algebraic expression, are called as factors and the numerical factors present with the variables are called as the coefficient of the expressions.
A simple example: the algebraic expression 5x + 6y – 8z = xyz. While evaluating algebraic expression, we have four different terms in the expression. 5 is coefficient of the x and in the R.H.S. term x, y, z individually are the factors of that term.If you want more information on algebra, click here.
The elements whose product forms a term of an algebraic expression are called the factors of that term.  The numerical factor of a term containing a variable is called the coefficient of the term.
The evaluation of algebraic expressions includes the formation of simple algebraic expressions with the help of some data. For example, while evaluating algebraic expressions we write an algebraic equation. Let n be any number then we can write expressions of several type according to the condition given. For example:
1.      Number “n” is increased by 45 can be written as n + 45;
2.      Number is decreased by 36 as n - 36;
3.      Number is get multiplied by 6 as 6 * n;
In some other form If n = 4 then (- 4) n + n² = ?
If x = 3, y = 4 and z = 2 then expression 2 x + 3 y + z² =?.
Similarly we can make a number of algebraic expressions. The expressions are categorized according to the number of terms contained in them into monomials, binomials, and trinomials.

In above content we have discussed about algebraic expressions and if anyone want to know about Proportions then they can refer to Internet and text books for understanding it more precisely. You can also refer Grade 7 blog for further reading on inequalities

Evaluating formulas

Hello children, in this section we are going to learn some of the concepts of algebra 2 help. We will discuss how to evaluate formulas for algebra. Chiefly variables are used for evaluating formulas.
The term variable means something that can vary or change. In other words, we can say that value of variable is not fix i.e. it keeps changing. Now we discuss the steps to evaluate formulas for algebra:
Step 1: For evaluating formulas first of all we measure what kind of variables are used in that geometric figure like rectangular is combination of length and width, then we assume variable for length and width like l = length and w = width.
Step 2: After measurement of variables, we evaluate what kind of operation is to be performed between variables like in area of rectangle, multiplication operation perform between length and width. So, area of rectangle = l * w .
Now we take some example to understand process of evaluating formulas –
Perimeter of square:
Step 1: We all know that all side of square are equal than there is only one variable exist length and we assume that l = length.
Step 2 : Now, we all know that perimeter of polygon is equal to sum of all side’s length . So perimeter of square is = l + l + l + l = 4 l.
So, formula to evaluate perimeter of square is 4l.
Area of triangle:
Step 1: We all know that there are two variable lengths and height exist in triangle. So, we assume that length = l and height = h.
Step 2 : Now we all know that area of triangle is half of multiplication of length and height . So, area of triangle = 1 * l * h.
                                                                                                                                                                                   2
Perimeter of rectangle:
Step 1: We all know that there are two variable lengths and width exist in rectangle. So, we assume that l = length and w =width.
Step 2: Now, we all know that perimeter of polygon is equal to sum of all side’s length. So perimeter of rectangle is = l + b + l + b = 2 l + 2b
So, formula to evaluate perimeter of rectangle is 2l + 2b.
Area of circle :
Step 1: We all know that circle has only one variable which is radius. So we assume R = radius .
Step 2: Now we all know that area of circle is equal to mulplication of pi and square of r. So, area of circle = Π * r².
Similarly we can evaluate formula for each and every figure of geometry like perimeter of hexagon = 6 * l, area of cube = 6 * l * l etc.

In upcoming posts we will discuss about Algebraic Expressions. Visit our website for information on ICSE class 12 syllabus

  

math blog on grade VI

Mainly when we study about relationship of operators on numbers then this study is called the algebra. In grade V of icse board we learn how to use operations like addition, subtraction, multiplication, associative property of addition and multiplication worksheets and division on numbers, but in grade VI, we have to learn some new topics from algebra 2, but here we discuss topics about Equations which shows linear relationships.
So, first of all we discuss what is linear? Linear means one side is equal to other side like: x = 9
This means this equation is represent value of x is equal to 9.
Now we discuss what is relationship? Basically when one quantity is related to other quantity or we can say that one quantity is depend on other quantity then this dependency is known as a relation.(want to Learn more about algebra, click here),
Linear relationship is a way to represent relationship between two or more quantities in linear form. As we know velocity depends on acceleration and time and these three quantity are related with each other, then we define relationship in mathematical equation way –
v = at + c
And this equation as a representation between these linear relative quantities is called an Equations for linear relationships. We can call Equations for linear relationships as linear equations in short. So, we further proceed to understand linear equation:
 1)  Linear equation with two variables and one constant:
We take an example to understand about linear equation of two variables and one constant like we have two variables x, y, constant c and m is slope then linear equation between x and y is -
 y = mx + c
This is standard linear equation and this equation is also known as a line equation. So , when we draw graph between two variables x and y, then it  shows straight line and each point of straight line shows result of linear equation like we take an example -

Here slope m = 2 and constant c = 1
2) linear equation with three constants and two variables : We take an example to understand linear equation of three constants  and two variables like we have A , B and C three constants and two variables x and y , then linear equation is -
                  Ax + By = C  
Now, we will discuss about graphical representation of these type of linear equation : it shows straight line graph and each point of  straight line shows result of linear equation . So, graphical representation of above equation Ax + By = C is -

To determine the y intercept of the line,
Put x= 0, solve for y = C/B =b;
To determine the x intercept of the line,
Put y = 0, solve for x = C/A = a;
This graph shows us how we get x intercept and y intercept from linear equations like Ax + By = C.
This is all about the algebra and if anyone want to know about Algebraic Expressions then they can refer to Internet and text books for understanding it more precisely. Read more maths topics of different grades such as Arithmetic sequences in the upcoming blogs.

Math Blog on Identity and inverse properties

In Grade VI, while studying about the whole numbers and properties of whole numbers worksheets, we are going to learn identity and inverse properties of the whole numbers which falls under maharashtra board.
Children while discussing the properties of whole numbers, Additive and multiplicative inverse and identities cannot be overseen and for this we can take Math help online.
Identity of addition or identity of multiplication is any number which when added or multiplied to the given number does not change the value of the number, i.e. the number remains same.
We take the case of additive identity property math we say that 0 (zero) is the additive identity, i.e. if we add 0 to any whole number, the number remains same.
3 + 0 = 0
9 + 0 = 0
Here we observe that when 0 is added to 3 and 9, in both the cases the result is the original number, so 0 is the additive identity.
Similarly in case of multiplicative identity property math we have 1 as a multiplicative identity, which means that if we multiply any number by 1 the number remains same.
For e.g.: 4 * 1 = 4
 And 8 * 1 = 8
Here we observe that when both numbers 4 and 8 are multiplied by 1, the number remains unchanged. So 1 is the multiplicative Identity.
Inverse property of addition says that the number and its inverse added gives the additive identity as the result i.e. 0
 So additive inverse of 4 is -4,
So 4 + (-4) = 0 as 4 is added to -4 it results to zero.
 Similarly multiplicative inverse identity means what number to be multiplied to the given number to get multiplicative identity as the result.
  eg multiplicative inverse of 4 is 1 / 4. So 4 * (1/4) is 1.

In the next blog we are going to discuss Estimation of Solutions in Grade VI and if anyone want to know about Formulas for measurement then they can refer to Internet and text books for understanding it more precisely.

Ratios and rates in Grade VI

Previously we have discussed about is a repeating decimal a rational number  and now we will discuss about the fundamentals of Ratios and Rates which comes under karnataka education board and merely describes that how a number is related to another number we will also understand how to do math questions related to it..

A simple representation of the ratio is given below

  A:B   or    A/B .

The ratio can be understood by the following example:

If we write 1:10 in a number system then we can say that the second number is 10 times greater than the first number.(Know more about Ratios in broad manner here,)

Example:

Determine the value of x if y=3 and their ratio is x:y=1:2

Then the solution is given below:

Step 1: divide the value of y  by their corresponding ratios = 3/2=1.5

Step 2: now multiply result by another number ratio value = 1.5*1=1.5

Step 3: then the final answer we obtain i.e. x=1.5.

Now we will discuss about the Rates.

Rates are a process of money calculation.

Rate is determined to weigh one quantity of different unit against another.

Suppose first person gives some of his money to second person for a particular fix time period. Then extra money is paid by the second person to the first person after that time period.

This extra money is known as Interest money and this can be calculated with the help of pre agreement percentages which are known as Rates.

This rate s concept can easily be understood by the following example:

Example:

Suppose Jerry gave 1000$ to Tom at the 10% rate for one year .Then what amount tom have to

pay to jerry?

Then the solution is 1000*10/100=100$

Thus at the end of the year, Jerry will have to pay total amount of $1100 to Tom.

This is the sufficient information to understand the concept of Ratio and Rates and if anyone want to know about Proportions then they can refer to Internet and text books for understanding it more precisely. You can also refer 7th Grade blog for further reading on Properties of lines.