Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY : The CBSE class XI syllabus for the academic year 2016-17 has been issued by Central Board of Secondary Education, New Delhi. Keeping in mind the need for detailed and well-defined sections in the subject, the syllabus of CBSE Class 11 is comprehensive, yet concise.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY – syllabus
UNIT VI : STATISTICS AND PROBABILITY
1. Statistics
- Measure of dispersion, range, mean deviation, variance and standard deviation of ungrouped/grouped data.
- Analysis of frequency distributions with equal means but different variances.
2. Probability
- Random experiments: outcomes, sample spaces (set representation).
- Events: Occurrence of events, ‘not’, ‘and’ & ‘or’ events, exhaustive events, mutually exclusive events.
- Axiomatic (set theoretic) probability, connections with the theories of earlier classes.
- Probability of an event, probability of ‘not’, ‘and’, & ‘or’ events.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Introduction:
In the modern world of computers and information technology, the importance of statistics is very well recogonised by all the disciplines. Statistics has originated as a science of statehood and found applications slowly and steadily in Agriculture, Economics, Commerce, Biology, Medicine, Industry, planning, education and so on. As on date there is no other human walk of life, where statistics cannot be applied.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Origin and Growth of Statistics:
The word ‘ Statistics’ and ‘ Statistical’ are all derived from the Latin word Status, means a political state. The theory of statistics as a distinct branch of scientific method is of comparatively recent growth. Research particularly into the mathematical theory of statistics is rapidly proceeding and fresh discoveries are being made all over the world.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Meaning of Statistics:
Statistics is concerned with scientific methods for collecting, organising, summarising, presenting and analysing data as well as deriving valid conclusions and making reasonable decisions on the basis of this analysis. Statistics is concerned with the systematic collection of numerical data and its interpretation. The word ‘ statistic’ is used to refer to 1. Numerical facts, such as the number of people living in particular area. 2. The study of ways of collecting, analysing and interpreting the facts.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Definitions:
Statistics is defined differently by different authors over a period of time. In the olden days statistics was confined to only state affairs but in modern days it embraces almost every sphere of human activity. Therefore a number of old definitions, which was confined to narrow field of enquiry were replaced by more definitions, which are much more comprehensive and exhaustive. Secondly, statistics has been defined in two different ways – Statistical data and statistical methods. The following are some of the definitions of statistics as numerical data. 1. Statistics are the classified facts representing the conditions of people in a state. In particular they are the facts, which can be stated in numbers or in tables of numbers or in any tabular or classified arrangement. 2. Statistics are measurements, enumerations or estimates of natural phenomenon usually systematically arranged, analysed and presented as to exhibit important interrelationships among them.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Definitions by A.L. Bowley:
Statistics are numerical statement of facts in any department of enquiry placed in relation to each other. – A.L. Bowley Statistics may be called the science of counting in one of the departments due to Bowley, obviously this is an incomplete definition as it takes into account only the aspect of collection and ignores other aspects such as analysis, presentation and interpretation. Bowley gives another definition for statistics, which states ‘statistics may be rightly called the scheme of averages’ . This definition is also incomplete, as averages play an important role in understanding and comparing data and statistics provide more measures.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Definition by Croxton and Cowden:
Statistics may be defined as the science of collection, presentation analysis and interpretation of numerical data from the logical analysis. It is clear that the definition of statistics by Croxton and Cowden is the most scientific and realistic one.
According to this definition there are four stages:
1. Collection of Data: It is the first step and this is the foundation upon which the entire data set. Careful planning is essential before collecting the data. There are different methods of collection of data such as census, sampling, primary, secondary, etc., and the investigator should make use of correct method.
2. Presentation of data: The mass data collected should be presented in a suitable, concise form for further analysis. The collected data may be presented in the form of tabular or diagrammatic or graphic form.
3. Analysis of data: The data presented should be carefully analysed for making inference from the presented data such as measures of central tendencies, dispersion, correlation, regression etc.,
4. Interpretation of data: The final step is drawing conclusion from the data collected. A valid conclusion must be drawn on the basis of analysis. A high degree of skill and experience is necessary for the interpretation.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Functions of Statistics:
There are many functions of statistics. Let us consider the following five important functions.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Condensation:
Generally speaking by the word ‘ to condense’ , we mean to reduce or to lessen. Condensation is mainly applied at embracing the understanding of a huge mass of data by providing only few observations. If in a particular class in Chennai School, only marks in an examination are given, no purpose will be served. Instead if we are given the average mark in that particular examination, definitely it serves the better purpose. Similarly the range of marks is also another measure of the data. Thus, Statistical measures help to reduce the complexity of the data and consequently to understand any huge mass of data.
Comparison:
Classification and tabulation are the two methods that are used to condense the data. They help us to compare data collected from different sources. Grand totals, measures of central tendency measures of dispersion, graphs and diagrams, coefficient of correlation etc provide ample scope for comparison. If we have one group of data, we can compare within itself. If the rice production (in Tonnes) in Tanjore district is known, then we can compare one region with another region within the district. Or if the rice production (in Tonnes) of two different districts within Tamilnadu is known, then also a comparative study can be made. As statistics is an aggregate of facts and figures, comparison is always possible and in fact comparison helps us to understand the data in a better way.
Forecasting:
By the word forecasting, we mean to predict or to estimate before hand. Given the data of the last ten years connected to rainfall of a particular district in Tamilnadu, it is possible to predict or forecast the rainfall for the near future. In business also forecasting plays a dominant role in connection with production, sales, profits etc. The analysis of time series and regression analysis plays an important role in forecasting.
Estimation:
One of the main objectives of statistics is drawn inference about a population from the analysis for the sample drawn from that population. The four major branches of statistical inference are 1. Estimation theory 2. Tests of Hypothesis 3. Non Parametric tests 4. Sequential analysis In estimation theory, we estimate the unknown value of the population parameter based on the sample observations. Suppose we are given a sample of heights of hundred students in a school, based upon the heights of these 100 students, it is possible to estimate the average height of all students in that school.
Tests of Hypothesis:
A statistical hypothesis is some statement about the probability distribution, characterising a population on the basis of the information available from the sample observations. In the formulation and testing of hypothesis, statistical methods are extremely useful. Whether crop yield has increased because of the use of new fertilizer or whether the new medicine is effective in eliminating a particular disease are some examples of statements of hypothesis and these are tested by proper statistical tools.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
PROBABILITY
In everyday life, we come across statements such as
(1) It will probably rain today.
(2) I doubt that he will pass the test.
(3) Most probably, Kavita will stand first in the annual examination.
(4) Chances are high that the prices of diesel will go up.
(5) There is a 50-50 chance of India winning a toss in today’s match. The words ‘probably’, ‘doubt’, ‘most probably’, ‘chances’, etc., used in the statements above involve an element of uncertainty.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
For example, in (1), ‘probably rain’ will mean it may rain or may not rain today. We are predicting rain today based on our past experience when it rained under similar conditions. Similar predictions are also made in other cases listed in (2) to (5). The uncertainty of ‘probably’ etc can be measured numerically by means of ‘probability’ in many cases. Though probability started with gambling, it has been used extensively in the fields of Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather Forecasting, etc.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
“Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.” Most people have some vague ideas about what probability of an event means. The interpretation of the word probability involves synonyms such as chance, odds, uncertainty, prevalence, risk, expectancy etc.
There are many distinct interpretations of the word probability. A complete discussion of these interpretations will take us to areas such as philosophy, theory of algorithm and randomness, religion, etc. Thus, we will only focus on two extreme interpretations. One interpretation is due to the so-called objective school and the other is due to the subjective school. The subjective school defines probabilities as subjective assignments based on rational thought with available information. Some subjective probabilists interpret probabilities as the degree of belief. Thus, it is dicult to interpret the probability of an event. The objective school defines probabilities to be “long run” relative frequencies. This means that one should compute a probability by taking the number of favorable outcomes of an experiment and dividing it by total numbers of the possible outcomes of the experiment, and then taking the limit as the number of trials becomes large.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Counting Techniques
There are three basic counting techniques. They are multiplication rule, permutation and combination.
Karnataka Class 11 Commerce Maths UNIT VI STATISTICS AND PROBABILITY
Multiplication Rule.
If E1 is an experiment with n1 outcomes and E2 is an experiment with n2 possible outcomes, then the experiment which consists of performing E1 first and then E2 consists of n1n2 possible outcomes.
Example . Find the possible number of outcomes in a sequence of two tosses of a fair coin. Answer: The number of possible outcomes is 2 · 2 = 4.
Permutation
Consider a set of 4 objects. Suppose we want to fill 3 positions with objects selected from the above 4. Then the number of possible ordered arrangements is 24.
Combination
In permutation, order is important. But in many problems the order of selection is not important and interest centers only on the set of r objects. Let c denote the number of subsets of size r that can be selected from n different objects. The r objects in each set can be ordered in ^{n}P_{r} ways.
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