Permutations and combinations

Hello students, in this blog we are going to discuss the topic of permutations and combinations. These two topics are extensively used in mathematics.
Permutations: - The different arrangement of a given number of things by taking some or all at a time, are called the permutations. Let’s take a close look on the permutations.
All permutations made with the letter ‘a’, ‘b’, ‘c’ by taking two at a time are (ab, ba, ac, ca, bc, cb).
 All permutations made with the letter ‘a’, ‘b’, ‘c’ by taking all at a time are (abc, acb, bac, bca, cab, cba).
Combinations: As name suggest of the combination that it is concept in which every number of separate groups are made by the when we include few or all number of an object, then it is known as the combination.
Suppose we want to select two out of three boys A, B, C. Then possible selections are AB, BC and CA.
Note that AB and BA represent the same selections.
Formula for permutation and combination are:
For permutation: “P r = n (n - 1) (n - 2)……(n –r + 1) = n! / (n - r) !,
Example for this: 1. 6 p2 =(6 * 5) = 30.                  2. 7 P 3 = (7 * 6 * 5) = 210.
 For combination: “C r = n ! / (r !) (n - r) ! = (n - 1) (n - 2)……to r factors / r !,
Example. ¹¹ C ₄ = (11 * 10 * 9 * 8) / (4 * 3 * 2 * 1) = 330, (know more about icse board papers, here)
So applying both the formulas we can solve any permutation and combination problems.
In the next session we will discuss about Types of events. 

conditional probability

In mathematics, probability can be considered as a branch of mathematics. The concept of probability deals with the calculation of the likelihood occurrence of any given event. The term probability describes their value in the range between 0 and 1. In the concept of probability the value one refers to the certainty of the occurrence of any particular event. In the same aspect the value zero refers to the impossibility of occurrence of any event. The basic behind using the concept of probability in mathematics is that when the lots of situations happens and we are not able to predict the outcome of the particular event then the concept of probability is used in this case to analyze the different possibilities and how likely each of them is. (know more about cbse board papers, here)
   There are various concepts are defined that helps in understanding the probability more clearly. Form that concepts conditional probability is one of them. In the probability theory, the conditional probability is a concept which defines the conditional relationship between two occurring events. It means to say that the conditional probability of ‘X’ given ‘Y’ is the probability of X if Y is known to occur. In the mathematical notation we can represent the conditional probability as P (X / Y).  It means to say that the conditional probability of occurrence an event Y is only when the occurrences of the event ‘X’ is already done.  In probability theory the definition of conditional probability can be represented by using formula for conditional probability:
       P (X / Y) = P (X and Y) / P (Y),
In the above definition of probability we need to remember one thing that the value of the P (Y) must be greater than zero. In most of the cases the P (X / Y) can be visualized as probability of event ‘X’ when the sample space of the event ‘Y’ is restricted. In the next session we will discuss about Permutations and combinations. 

Data representation methods

In statistics we have several methods of data representation that are used for represent the data in a specified manner that means once data is collected by the person or group of persons that is related with the specific experiment is then it will be presented in the way that user of data can understand it very easily and it is not used only for present scenario but also it will be used in future for making some decisions or for comparing the results of different years , so there are some well defined methods of data representation in statistics that helps the user of data to make efficient and effective decision very efficiently in very short span of time .These Data representation methods are as follows :
Data will be represented in the tabular form that will support to understand the data with its entire attributes.
For representing the data there one more technique that is known as graphs or pie charts that will help in understanding the way how data is changed in the particular environment or for specific experiment .
We can also draw some charts for representation of data.
The graphs are also categorized in some types that are line graph or histogram or bar graph or pyramids etc.
So for presenting the data there should some step that must be follow by the user for making the graphs or charts as: (know more about cbse sample papers for class 10, here)
First of all a suitable scale will be selected by the user.
After selection of scale follow all the design as index, directions and titles etc.
After it user select the appropriate way for representing the data that data will represent in two dimensional ways or in three dimensional ways.
In the next session we will discuss about conditional probability. 

Tools of data analysis

Hello students, in this blog we are going to discuss about the tools of data analysis but before discussing the tools of data analysis statistics. We should be familiar with the data analysis term. Data analysis is the process of summarizing the information by using the following steps,
-Inspecting the data.
-Cleaning the data.
-Transforming the data.
-And last modeling the data.
There are various tools that analyze the data more clearly and more precisely. Some of them are: -
1. Histogram: - This is the most common technique for data analyzing. This is the graphic technique that shows the frequency distribution.
2. Survey: - Usually the surveys are organized by the various NGOs to collect the data. This is done publicly so that we can know the behavior, opinions and knowledge of the public.
3. Scatter diagram: - In this type of data analysis tool we create the diagram for the numerical data.
4. Series of experiments: - In this type data analysis tool, we predefine the some series of experiments on the particular data or information. Then data passes through the various series of experiments and meaningful data is drawn.
5. Box and whisker Plot: - Various types of data sets are plotted on the single graph.
6. Check sheet: - It is like ms excel sheet that stores the data in tabular format, so that data can easily be understand by everyone. It is a prepared form for data collecting and data analyzing.
7. Control chart: - It is used to control the data process. It compares the data that change from one state to another. (know more about cbse board papers, here)
But mostly used data analysis tool is histogram.
This bog will be helpful for Grade IX students to understand the Tools of data analysis.
In the next session we will discuss about Data representation methods. 

Mean

In statistical mathematics, the concept arithmetic mean which is simply known as mean is a special kind of average. Before studying the concept of mean we need to clearly understand the concept of average. An average is the most basic concept of mathematics from which we all are familiar with them from a very early time. Average is the mathematical concept in which we add the multiple but similar kinds of objects after that divide them by the total number of objects.
In mathematical definition we can define the mean as mean is a mathematical concept which has the central tendency of a collection of numbers taken as the sum of the numbers which is divided by the actual size of the collection. The concept of arithmetic mean has the wide area for use like economics, sociology and history. In probability theory, we perform the concept of mean on the weighted value of the variables. In probability these concept are as mean in probability. They are most widely used in the academic field to some extent. The arithmetic mean can be consider as a set of similar kind of data is found by taking the sum of the data and then dividing the sum by the total numbers of values in the set. According to the above definition we can say that a mean can be consider as an average. In the mathematical notation we can define the arithmetic mean as given below: (know more about cbse books, here)
                  A = 1 / r ∑_j=1^r  aj
In the above notation ‘A’ refers as a arithmetic mean. ‘aj’ refers to the sum of all the similar objects. Now we show you how to perform the arithmetic by the example:
Example: suppose in the class there are nine students which obtain the marks in the unit test. The numbers are: 18, 14, 16, 14, 13, 21, 13, 13, 13.  Now we need to calculate the value of the mean?
Solution: In the above marks are given of nine students:
                   18, 14, 16, 14, 13, 21, 13, 13, 13
Now arithmetic mean will be
              18+14+16+14+13+21+13+13+13 / 9 = 15
Form the above calculation we can say that the mean value of nine students marks is 15 and In the next session we will discuss about Tools of data analysis. 

Probability and Statistics

In the mathematical field, there are lots of concepts and lots of theories are defined to solve the problem, which are related to math’s subject. In the real, there are lots of problem related to performing calculation, these calculations are based on several aspect of the mathematics. During the study of mathematics, we can divide the mathematics into several categories that are: arithmetic, geometry, discrete mathematics and statistics. All have the different work area to apply the concept of this mathematics. Here we give you a brief introduction to probability and statistics. Probability and statistics are two different academic topics to study. Statistics is the way to perform the process of analyzing the fact and figures and generate the output. Statistical analysis includes the topic of probability distribution. In the probability and statistic we generally include the experimental part of the statistic subject. This subject asks to perform the repeated calculation of same thing again. The concept of probability and statistics helps the students of Grade IX to understand the basic of this topic. So to remove the problem of repeated calculation statistic subject provide lots of formulas to perform the calculation in the simpler manner and this repeated type of calculation generally known as probability.
In below we show you the perfect example that helps in understanding the concept of probability and statistics. Suppose we toss a coin, now the possibility of getting head is 50% and the possibility of getting tail is also 50%. The formula for performing the probability is: (know more about cbse sample papers 2013, here)
In the mathematics, the numbers of ways to get ‘H’ heads or ‘T’ tails in ‘n’ no of toss is spoken as number of combination of n things taken ‘H’ or ‘T’ at a time. This can be written as:
                                                   (H or T / n)
So, to calculate the probability ‘P’ of given values of ‘H’ and ‘T’ can be performed by:
           Probability =(‘H’ or ‘T’ / n) = (no of ways for an event/ total number of possible outcome)
The probabilities can be described in between the value of 0 to 1. In the next session we will discuss about Mean. 

How to calculate Median

Median is a special kind of average which is used to calculate the average value of the given objects value. In statistics, median can be described as the numerical value which separates the any given sample values into two half. In the concept of statistical average, we generally studied about the three most popular concept that are mean, median and mode. All these are similar kind of average. Here we are going to discussing about the median. Median can be specifies as the middle value in the list of values. In the concept of median, when we perform the total of the list is odd, and then we select the middle entry in the list after sorting the list into the increasing order. Sometime the concept of mean is not performing very well when the sum of the given list is odd. If in case the total of the list are even, the median is equal to the sum of the two middle numbers divided by two. But the operation of median can only be performed by sorting the list into increasing order. There are some of the step that needs to be following while performing the median on the list values. (know more about cbse board papers, here)
These steps are considered as a formula for calculating the median.
(a)     First of all arrange the values in the increasing order.
(b)     After that select the middle value from the list. These values consider as a median of the list.
(c)     If in case the middle value contains two values then divides the sum of these two middle values by two.
Formula of median = L + H / 2
In above L refers to lowest middle value and H refers to larger middle value.
Now we show you the concept of median by performing the example in the below:
Example: Suppose there are five children who have the different amount in there pocket. These are 9, 3, 44, 17 and 15. Now we need to calculate the median of these values?
Solution: In above question given that amount of five children are 9, 3, 44, 17 and 15.
So, first we need to arrange them in increasing order that is:
               3, 9, 15, 17, 44
By following the above sequence of number the median is: 15
In the next session we will discuss about Probability and Statistics.