Standard distributions

Standard Distributions is defined as a value, which is greater than mean of whole distribution in the given data set and we use following formula for evaluation of Standard Distribution, for more information try this
 Standard Distribution = (X – μ) / σ,
Here ‘X’ refers to score value, which is produce by normal distribution, ‘μ’ is the mean value of the original normal distribution and ‘σ’ is the standard deviation of original normal distribution.
This standard distribution of group value sometimes called as a z score distribution like we have a group of students and they all give a math test. Also you can try Z Score Table Normal Distribution
One person get 70 score in that math test with a mean of 50 and standard deviation of 10, then question arises that what is standard distribution of these group of students. So, for standard distribution or z distribution, we use above formula –
Standard Distribution = (X – μ) / σ
= > S = (70 – 50) / 10,
= > S = 20 / 10,
= > S = 2,
So, standard or z distribution of this group of class is 2 and this standard distribution score suggest that there are 2 standard deviation, whose value are greater than mean value of original distribution. Z distribution or standard distributions are called as a normal distribution, if value of original distribution is normal.
Therefore for calculation of standard distribution, we use following steps–
Step 1: First we calculate original score means value of X, like value of X is equals to 60.
Step 2: After evaluation of original score, now we calculate mean of original distribution mean value of μ.
Step 3: After first 2 steps, now calculate standard deviation of original distribution mean value of σ.
Step 4: after first 3 steps, now we apply following formula–
Standard Distribution = (X – μ) / σ,
This formula produces the value of standard distribution.


In upcoming posts we will discuss about Methods of data representation and Justify the Pythagorean identities. Visit our website for information on secondary education Karnataka

Line of best fit - least squares regression

In this blog we are going to discuss about the topic Line of best fit least squares regression that is defined the line that produces the smallest value of the sum of squares of the residuals. If we define the term residual is explained the vertical distance from a point on a scatter diagram that is to the line of best fit. So it will be least square regression line as the best line of best fit.
Define Regression : Regression produce the least square regression line and line can be expressed as y' = a + b x and ‘a’ and ‘b’ are the constant values and here y' is the predicted value that is based on the value of ‘x’ and ‘y’ that are independent variables.
If we try to find Regression Definition by an example as if the values of constant are a =10 and b = 7 and value of x = 10 then according to the formula the predicted value of y' is calculated 45 that is defined as (10 + 5 * 7). It will be occurring with any two variables ‘x’ and ‘y’. There is one equation produce the “best fit” linking ‘x’ to ‘y’. So according to this there will be one formula that will produce the best and accurate prediction for ‘y’ that is given for ‘x’. So this equation is called as the least square regression equation denotes as the Line of best fit. (visit for more information here)
The equation for the Least square regression of ‘y’ on ‘x’ is defined as:
y = y ' = b (x – x ')
Here ‘b’ is defined as:
b = Px y/P x x = ∑(x – x ') (y – y ') / ∑((x – x ')² = ∑x y – x' y' / ∑(x² / n) – x'² .
This is all about least squares regression.
 In upcoming posts we will discuss about Standard distributions and Fundamental theorem of algebra. Visit our website for information on higher secondary education Karnataka

standard deviation of normally distributed random variable

Standard normal distribution is defined in normal distribution that is the special case of it. This distribution occurs when normal random variable has mean equal to zero and value of standard deviation is equal to one.
In the standard normal distribution the normal random variable is known as the standard score or also called as z – score. There is an expression through which every normal random variable ‘x’ can be transformed in z – score that is defined as follows:
z = (x - μ) / s,
The above formula defines the Standard Deviation Problems of normally distributed random variable.
Where ‘x’ is a variable, ‘μ’ define the mean and ‘s’defines the standard deviation of random variable ‘x’.
According to the above formula random variable is defined by subtracting the mean of the given distribution from the random variable that will be standardize for the distribution and then it will be divided by the standard deviation of the distribution. When we talk about the standardize random variable it has value of mean equal to zero and value of standard deviation equals to one. We can define it through the example if value of random variable equals to 40 means x = 40 and value of mean and standard deviation are 20 and 10 respectively then find the standardize value of x?
Solution: Given x = 40,
Mean (μ) = 20 and,
Standard deviation (s) = 10, then according to the formula
z = (x - μ) / s,
z= (40 –20) / 10,
z = 20 / 10,
z = 2,
So for the given example the standardize value that is also define as z – score of random variable 40 is 2.

In upcoming posts we will discuss about Line of best fit - least squares regression and Trigonometric form of complex numbers. Visit our website for information on Indian Certificate of Secondary Education

Central limit theorem

Central limit theory is defined in the probability it states that in certain conditions many of the distributions get closer to the normal distribution (Click this if you want more information on central limit) . The main feature of it is mean of large number of independent random variables will be normally distributed when each have finite mean and variance.
It also states that Population of mean that is created from the mean of infinite number of random population of size (n) then all of them occurred from given parent population. But in the Central Limit Theorem Examples it is not considered as the parent population and states:
(a) The mean for the population mean is equal to the mean of parent population from which that population mean will drawn.
(b)  According to central limit theorem standard deviation of the population mean is equal to the standard deviation of the parent population that is divided by the square root of size of sample that is equal to n.
(c)  And for Central Limit Theorem, the random distribution of the n size of sample will approximately be normally distributed.
So when there are a₁ , a₂ , a₃...... a_n be the set of ‘N‘ variables that are independent and every x_i have its probability distribution as p (a₁ , a₂ …. a_n) that have the mean ‘µ’ and also the variance that is denoted as s2 then the normal variate form is:
a _norm = (∑ⁿ _i=1 x_i - ∑ⁿ _i=1 µ _i ) / √∑ⁿ _i=1 s² _i .
It has a distribution function that is cumulative that approaches to normal distribution. In general define central limit theorem as the any of set of weal convergence theories.


In upcoming posts we will discuss about standard deviation of normally distributed random variable and Polar/rectangular coordinates. Visit our website for information on ICSE board question papers

variance of discrete random variable

In this session we are going to discuss about the variance of discrete random variable and Discrete Variable .Variance is defined in the statistics that is define the deviation of the data value for the given set of values that is also define as how far value from its mean or average value .
When we talk about the random discrete variable in ICSE board question papers it mean the variable that are randomly chosen and not have the specific order of occurrence .So when we define the variance of discrete random variable it will be define for the variables that not have specific order that is define as follows :
Variance is define the measure of the fairness of the data for the data set. Variance is based on the mean value of the data set that is also called as the average value for the data set.
In a given data set for any discrete random variable v, mean value is define as m and it is denoted as
E | x | = m and where E denotes the average value or mean of random variable v . For more information click here
So the expression for the variance is defined as (s) ² = E (x - m ) ² .
It will be more explained through an examples as :
Example : If there are a given data set S that have the variable as :
variable (set s) : s1 s2 s3 s4 s5 s6
values : 3 4 4 5 6 8
then find the variance of the set values S ?
So , mean m = å ⁶_{i = 1} s_i / n = 30 / 6 = 5
variance as s² = å (x - m) ² / n = 2.67 .
By using formula we calculate the value of variance s² = 2.67 for discrete random variables of the set S .

In upcoming posts we will discuss about Central limit theorem and Trigonometric values of standard angles. Visit our website for information on Probability worksheets

Probability problems with finite sample spaces

Today we will discussing about probability problems of ICSE question papers. Probability is define for calculating the possible outcomes for the sample space of all events .It is used to calculate the all possibilities of occurrence of an event that is based on the occurrence of all other events .When we talk about the Probability problems with finite sample spaces. (You can also see Conditional Probability Problems to get in detail) It will be define as if there is a finite set have the number of possible outcomes as if there are two die that are of different colors as red and green then “ Is the number that is on the red die is greater than the number on the green die ? “ So for give the answer of that particular answer we have to maintain the records for each number on every die that is become more complex, Probability problems of finite sample spaces is defined as Set S then there are total 36 events of ordered pairs , that is defined as (a , b) in which a denotes the number on greed die and y define the number on the red die and a and b are integer number declared in Set . As we know that there are six number on the die that is define in the Set
G = 1 , 2 , 3 , 4 , 5 , 6 and on two die they will be define as S = G * G .
So the sample space S can be written in the Cartesian product as
S = G * G = (a , b) | a Ð„ G and b Ð„ G .
Here G itself is a sample space and for the experiment if one die is rolled then
S = 0 sixes , 2 sixes and S = 0 sixes , (1 , 6) , exactly one six , (6 , 6) . These are example of sets that is not serve the sample space for the two die experiment.

In upcoming posts we will discuss about variance of discrete random variable and Trigonometric equations 0 to 360 degrees. Visit our website for information on Introduction to Probability

Correlation coefficient

Hello students, in this section we are going to discuss the Coefficient Correlation. Correlation coefficient is statistical concept; it is used for setting up the relation between the expected value and real value that came from the statistical experiment. As per ICSE board books when we calculate the correlation coefficient then we find the value and that value describes the correctness between the expected and real value.
The most important point is that the value stays between the -1 to +1. So due to this, two condition arises, one is that, if the correlation coefficient's value is positive then it represents exactly corresponding and duplicate relation between two values. If the correlation coefficient's value is negative then it represents the inequality between the two values.
To define correlation coefficient (visit for more), first you should be able to know the meaning of correlation, it is way in which two or more things are related to each other whereas the coefficient is the multiplicative factor and generally it is a number.
Let’s take a look on the correlation coefficient formula or equation: -
When we have ‘x’ and ‘y’ two variables, then correlation coefficient ‘r’ will be: -
r = n (∑x y) - (∑x ) (∑y ) / √( [ n∑x² ] - (∑x )² ] [ n∑y² - (∑y ) ² ] ),
Where n = number of elements.
∑x = first values list's sum.
∑y = second values list's sum.
∑x y = sum of product of first and second values.
∑x² = sum of square of first values.
∑y² = sum of square of second values.
By this formula we can solve the problems that are related to the correlation coefficients.

In upcoming posts we will discuss about Probability problems with finite sample spaces and blog for math tutor online. Visit our website for information on rational expressions calculator

dispersion

Dispersion, according to Gujarat education board, is the name of the method that is used for finding the difference between the maximum and minimum value that is also defined in terms of range, means if we want to define the range of set of values then it can be denoted as:

range = maximum value - minimum value that is also explained through the example as if there is variable that is denoted as v and it has three different values that are 50 , 51 and 53 then the range for the variable v is defined as the range = maximum value - minimum value
Range = 53 – 51
Range = 2 .
There is one another example in which value of variable v are randomly chosen that are 45 , 76 , 95 then the range defined for the variable v = maximum value - minimum value
Range = 95 – 45
Range = 50
So by using the above two examples we can define dispersion as in the very first example there is little dispersion in the values but in the later example dispersion is bigger in the values.
If we talk about the different types of dispersion than there are basically three types are defined that are IQR (Interquartile range) that defined as IQR = Q3-Q1 that is a process for elimination .IQR is come in existence to overcome the effect of outliers on the range.
Another method of dispersion is Standard Deviation that is more use than any other of dispersion method. Standard deviation is calculated by the variance and it is denoted by sigma (σ²) and the expression is denoted as σ² = ∑ (X – μ )² / N and Standard deviation ( σ ) is σ = ∑ [(X - μ)² / N ]^1/2

In upcoming posts we will discuss about Correlation coefficient and area of an ellipse calculator. Visit our website for information on Binomial Probability Formula

quartiles

Hello students, in this session we are going to discuss the quartiles from Gujarat secondary education board. Quartiles are the partition values. They are the same like the mean and median, mean, median divides the whole series into some equal parts and partition values also divides the series into certain equal parts. Besides quartiles, there are also two methods, such as deciles and percentiles. But here we have to discuss the quartiles (for more information visit here).
The quartile definition: - quartiles separate the whole series into 4 equal parts. And the four equal parts are denoted by the 3 quartiles. Four equal parts means 25%, 25% 25%, 25% and the quartiles can be represented by Q,
Q1 = first quartile called the lower quartile and have first 25% of data.
Q2 = second quartile called the median and have 50% of data.
Q3 = first quartile called the upper quartile and have last 25% of data.
Note: - To define the quartiles we have to first arrange the data into ascending order
Let’s take an example.
We have data set 8, 7, 87, 34, 78, 5, 65, 67, 23, 9, 11
Solution: - First of all we have to sort this data 5, 7, 8, 9, 11, 23, 34, 65, 67, 78, 87
Q1 = 8
Q2 = 23
Q3 = 67
We can also follow the formulas to obtain Q1, Q2 and Q3,
Quartiles in individual series
Q1 = (n+1^th )/4 item
Q2 = (n+1^th) /2 item
Q3 =3 (n+1^th) /4 item
Quartiles in Discrete series
Q1 = size of (n+1^th )/4 item
Q2 = size of (n+1^th) /2 item
Q3 =size of 3 (n+1^th) /4 item
We can also calculate the Q1, Q2 and Q3 for the continuous series by applying their formulas

In upcoming posts we will discuss about dispersion and Learn factoring of Two Degree Polynomials. Visit our website for information on Binomial Probability Distribution

Applications of Probability and Statistics

In this blog we are going to discuss about the Applications of Probability and Statistics that is very much used in day today application as budget estimation or planning or future forecasting etc. It is also useful to read about combined events probability
Possibilities of occurrence of an event that is base on occurrence of other events or other independent events which are not based on any other events defined through the portability and it will be define in form of expression if event e is occurring then
Probability p (e) = number of favorable events / total number of events.
This phenomena is describe the probability of occurrence of an event when there are number of possible events in the sample space .Probability is an important concept in quantitative analysis .It is also used in calculation of complex results or unpredictable result .
When we talk about the application of statistics (taken from Gujarat board textbooks) first we know about the meaning of statistics that is define as the collection , analysis and data interpretation .When there is a data that may use in several statistical theory and various methods of discipline .there are several applications of statistics that are as follows :
(1) : It will give the simple and instant information on the topic it is centers .
(2) : Tools of the statistics are very useful in research area and also in studies in different fields that are as economics ,social science , medicine , business and many other fields .
(3) : It will use in different types of representation of collected and organized data that are based on the different charts and diagrams or graphs .
(4) : it will also used in the analysis of the data that will helps in making decision on that data or results .

In upcoming posts we will discuss about quartiles and Radical Equations in Grade XII. Visit our website for information on Correlation Coefficient Formula

Mean

In this blog we are going to discuss about the Mean that is a part of statistics .Sometimes in Statistics Mean is explained by doing the average of the numbers that are given in the data set. If we denote the average as an expression is define as the sum of all the numbers that are divided by the total number of data in data set. If there is number in data set are a₁ , a₂ , a₃ , a₄ then
The average of these number = (a₁ + a₂ + a₃ + a₄) / 4 .
According to CBSE board syllabus basically there are three types of mean that are defined as follows:
(I) arithmetic mean
(II) Geometric mean and
(II) Harmonic mean
Arithmetic mean is one of the type of mean that is normally is called as the average.
We can define it by an example as if there is a range of data in the data set as 40, 20, 35, 27, 13 then the arithmetic mean or average of the data of the data set is define as
Average = (40 + 20 + 35 + 27 +13) / 5 = 155 / 5 = 31.
In other type of mean , geometric mean is define the average of numbers not just by adding that data but also multiplying these numbers .These types of mean (for more on mean visit here) are define in calculation of growth rates , birth rates etc .
It is define through the expression as geometric mean g = (∏ⁿ _i=0 g _i) ^{1 / n}.
We can define it by an example as if number in the data set then g = (2 .3.4 .5) ^{1 / 4} = (120)^{1/ 4.}
We use the harmonic mean when we want to define some relation in the units of a set of numbers.


In upcoming posts we will discuss about Applications of Probability and Statistics and Geometry and Measurement. Visit our website for information on How to Make a Bar Graph

Mean

Statistics is the very important branch of mathematics. In statistics we deal with some kind of measures of central tendency such as mean, median and mode. In this blog we are going to discuss the mean of CBSE 10th syllabus. The mean in statistics is the very important and most commonly used central tendency measure. We can calculate the mean for both types of data that are ungrouped and grouped data. Mean of Ungrouped data,
Mean = sum of the observations / number of observations,
-Find the mean of (I) First 10 natural numbers (ii) first 10 whole numbers?
Solution : - (I) first 10 natural number are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
x' = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) / 10
x' = 55 / 10 = 5.5.
(ii) First 10 whole numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
x' = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 / 10,
x' = 45 / 10 = 4.5.
-Mean of grouped data (without class interval)
Mean for direct method = sum of the values of all observations / number of observations, (visit for more information on mean)
or Ƹ f _i x _I / Ƹ f _I,
Mean for step deviation = x' = a + Æ¸ f _i (x_{1 –} a) / Ƹ f _i = a + Ƹ f _i d _I / Ƹ f _i.
-Mean of grouped data (with class interval),
Mean for direct method = upper class limit + lower class limit / 2,
Mean for step deviation = x' = a + Æ¸ f _i (x_{1 –} a / h) * h / Ƹ f _i = a + h (Ƹ f _i u _I / Ƹ f _I)
By applying the above formula we can also calculate the mean.


In upcoming posts we will continue our discussion about Mean and Euclidean/non-Euclidean geometries in Grade XII. Visit our website for information on How to Graph a Circle

Conditional probability

We can easily solve the problem related to Conditional probability of CBSE syllabus for class 10, but for that firstly we need to have a very good knowledge of events. We can define event by a very simple example if we have are choosing any random card from a deck of card then it is an event. For conditional probability we need to have two events. Let’s suppose if we are having two events ‘a’ and ‘b’, then there are two conditions, first is that, the events are related to each other and second event are not related to each other. The conditional probability definition is the probability of a when b is occurred. So whenever we need to calculate the conditional probability our first task is to check that the event is occurred or not.  You can also play conditional probability worksheets to improve you skills. With the help of formula given below we can calculate the conditional probability of two events as,
If p(a)=0, then p(b/a)=p(b ∩ a)/p(a)   p(a/b)  =p(a ∩ b)/p(b).
With the help of this formula we can calculate the conditional probability of the event. If we are having two event which are having no relation with each other then this type of event are called as mutually exclusive event, then p(b ∩ a)=0, so for this type of case probability will be zero. If we are drawing a card from a deck of 52 card and the card we drawn is a red card then we need to calculate the probability that the card drawn is an ace? We can solve this problem with conditional probability, because we are having two events, first event will be x and p(x) will be 26. Now we need to calculate p (x∩ y) that will be two as number of aces are 2 in black card, so required probability will be 2/26 or we can write it as 1/13.


In upcoming posts we will discuss about Mean and Formal/informal proofs. Visit our website for information on How to Make a Histogram

Measures of central tendency

Hello students, in this blog we are going to discuss the measures of central tendency that are the very important concept in the statistics. The most important central tendency measures are mean, median and mode. We calculate the mean, median and mode for the ungrouped data and we can calculate the mean, median and mode also for grouped data. Let’s discuss the all measure one by one.
Mean: - It means average of the whole numbers, like we have 9 numbers then we will add the 9 numbers together and divide by the 9, 9 is because total number are 9. For example: -
8, 7, 6, 8, 7, 5, 7, 6, 5, 6
Then add the all number and divide the total sum by 10, because 10 numbers are there.
8 + 7 + 6 + 8 + 7 + 5 + 7 + 6 + 5 + 6 / 10,
6.5 is mean,
This example was for the ungrouped data. We can calculate the mean when we have grouped data, to do this we have three methods they are: -
-Direct method.
-Shortcut method.
-Step-deviation method.
The all three methods have their respected formulas.
Medan (from CBSE Books): - Median is the middle value in the number distribution. Like we have 7 numbers then the median will be 4^th number and if we have 6 number then add the 3^rd and 4^th number and divide it by the 2, then he result will be median. For example 6, 7, 5, 4, 2 then the median is 5.
Mode: - The mode or mode value of a distribution is that value of the variable for which the frequency is maximum. For example 2, 6, 7, 2, 5, in this 2 is mode, because it is occurring at 2 times.


In upcoming posts we will discuss about Conditional probability and Conditional statements. Visit our website for information on Confidence Interval Formula

Permutations and combinations

In every area we use Permutations and combinations which are the part of Algebra 2, but generally the permutations and combinations are use in mathematics filed. They are the fundamental principles of counting. The main difference between the permutations and combinations according to board of secondary education Andhra Pradesh is, when the orders means a lot then it is said to be permutations and if the order means nothing then it is said to be combinations. Just take examples for both: -
A permutation is also an ordered combination. Permutations are of two types: -
-Repetition is allowed.
-No repetition.
Example of Permutations: - If the 893 is the password combination of any id then it cannot be 398. Because to apply the 398 we cannot open the id, to open the id we have to insert the 8-9-3 in exact order. In this order matters alot.
Combination is also two types: -
-Repetition is allowed.
-No repetition
Example of Combination:- If I am saying that my vegetable salad is a combination of onions, tomatoes and chilies, then the order means nothing to us, we do not take care the order, either they could be in order like onions, tomatoes and chilies or chilies, tomatoes and onions. It will be same vegetarian salad. No matters what is the order.
Permutations and combinations formula,
Permutations formula for repetition: -n ^r.
Permutations formula without repetition: - n ! / (n –r) !,
Combination formula for repetition : - (n + r – 1) / r = (n + r – 1) ! / r ! (n – 1) !,
Permutations formula without repetition: - n ! / r ! (n – r) ! = (n / r),
By applying the above formulas we can solve the any type of permutations and combination problems.


In upcoming posts we will discuss about Measures of central tendency and Geometric proofs in Grade XII. Visit our website for information on Ogive