Business Statistics Paper I Notes.
Welcome to our comprehensive collection of notes for the Business Statistics! my aim is to provided you with the knowledge you need as you begin your journey to comprehend the essential ideas of this subject.
Statistics is a science of collecting, Presenting, analyzing, interpreting data to make informed business decisions. It forms the backbone of modern-day business practices, guiding organizations in optimizing processes, identifying trends, and predicting outcomes.
I will explore several important topics through these notes, such as:
1. Introduction to Statistics. : meaning definition and scope of Statistics.
2. Data collection methods.
3. Sampling techniques.
4. Measures of central tendency : Mean, Median, Mode.
5. Measures of Dispersion : Relative and Absolute Measures of dispersion, Range, Q.D., Standard deviation, Variance. coefficient of variation.
6.Analysis of bivariate data: Correlation, Regression.
These notes will serve as you guiding light throughout your study journey, helping to you grasp the principles of business statistics, and apply them to real-world scenarios, and develop a strong foundation for data-driven decision-making. Remember, the knowledge gained here will not only enhance your academic performance but also pave the way for a successful in career or in the dynamic world of business. Happy learning!
Introducing to Statistics.
Statistics in sense, is old as the human society itself. the word "Statistics" seems to have been derived from the latin word 'status' or the Italian word 'statista' or the german word statistik each of which means a political state. in India, an efficient system of collecting an official and administrative statistics existed more than 2000 year's ago. in particular during the reign of chandra gupta maurya.
Sir Ronald A. Fisher (1890 - 1962) known as 'Father of Statistics' they applying Statistics in various field such as, genetics, biometry, education, agriculture etc.
Definition: Sir R. A. Fisher Defined Statistics as "The science of statistics is a essentially a branch of applied mathematics and may be regarded as mathematics applied to observational data."
Or "Statistics May be defined as the collection, presentation, analysis and interpretation of numerical data ."
we see this terms as one by one
i. Collection of data: Collection of data is a first step in statistical method, statistics based on the collected data, for which maximum care should be taken at this stage. in this step for set the objective and collecting data related to objective to draw inference about that objective. or collect information regarding the objective of study.
ii. Presentation of data : this is an second stage in a statistical investigation is to classify this data and put in form of tables, the data collected are in the form of figures and this figures are arranged in properly. to get better understanding of data.
iii. Analysis of data: this is a important step in statistical analysis to analysis this collected data to derived the results. this calculated as measures of central tendency, measure of variation, correlation are calculated these single figures tell us much about the problem.
iv. Interpretation of data : this is the last stage in the process and therefore it is the most difficult part, requiring high degree of skill and experience. this result are in the form of numbers and need to be transforms into statements.
Importance of Scope of Statistics :
In the modern time, statistics as more importance in many fields.
1. Statistics and Planning: Statistics is important in planning. in the modern age which is termed as the age of planning almost all over the world, governments, particularly for planning for the economic development. Planning often involves predicting future trends and outcomes. Statistics provides various forecasting methods, such as time series analysis and predictive modeling, which help estimate future values based on historical data.
2. Economics: Statistics is widely used in economics to analyze economic data, and calculate economic indicators, and make predictions about economic trends. It helps economists summarize and interpret data through descriptive statistics, such as mean and standard deviation. Regression analysis allows them to study relationships between economic variables, Time series analysis aids in understanding trends and cyclical movements in economic data. Additionally, statistics is crucial in economic forecasting, financial analysis, and conducting cost-benefit analysis for policy decisions. Overall, statistics plays a vital role in guiding economic research and policy-making processes.
3. Business: In business, statistics plays a critical role in decision-making and improving performance. It is used for market research, sales analysis, financial evaluation, and quality control. Statistics helps businesses understand consumer preferences, optimize inventory levels, set prices, and personalize marketing efforts. It aids in risk management, forecasting, and strategic planning.
Limitations of statistics
1.It is not useful for individual cases.
2. it cannot used for qualitative data.
3. the results are true on average only.
4. statistical results may be biased.
5. Many statistical methods rely on specific assumptions about the data.
Collection of Data :
First we see the meaning of data , here data means the information collected for statistical study is called data. in may be in form of numbers, words etc. and person who collect data for investigation the problem is called as investigator.
there are two type / methods to collect data
1. Primary data 2. Secondary Data.
1. Primary Data : the data collected by investigator himself is called as Primary Data. or the data collected directly from individual is known as Primary data. it is original data hence it is more reliable than any other data. but in some cases we it is not possible to collect original data, hence go to secondary data. this primary data is called as raw data. e.g. data collected through personal interview, or survey.
2. Secondary Data: The data has been collected by someone for their purpose we use this data for our study is known as secondary data. or the data collected by other and this data we used foe study in called secondary data. e.g. the data published by government, data published in news papers.
For collecting any type of data primary of secondary there are some methods are used.
Method of Collection of Primary Data :
There are various methods to collect primary data.
i. Direct Personal interview: in this method the investigator collect the necessary information form individual or persons. e.g. we want to collect the marks of student for that we meet each students and collect the marks form the students.
ii. Indirect Investigation: Indirect investigation, also known as indirect measurement or indirect observation, refers to a research or data collection approach where the information of interest is not directly measured or observed. Instead, researchers use other observable variables to infer estimate the value of the target variable.
iii. Questionnaire Method: in this method we collect the required information using the question - answer form. we create the list of questions to collect the information is known as questionnaire.
we collecting data to sending the trained persons with questionnaire. this persons known as enumerators. or we sending the questionnaire by post.
a) By post: we sending the prepared questionnaire to persons and requesting to fill or provide the necessary information in the questionnaire. e.g. if we want to collect the information related to their name we sending the questionnaire by post to all students to collect information.
b) Through Enumerators : this is another method to collecting information using questionnaire, in this method we send enumerators with questionnaire to collect the information in particular area or selected area for study.
c) Local Correspondents: Local correspondents are individuals or representatives who are hired or appointed by organizations, businesses, media outlets, or research institutions to gather data and report on specific events or information in a particular geographical area or region. These local correspondents play a crucial role in collecting primary data from their respective locations.
Method of collecting Secondary Data:
The secondary data can be collected from the following sources:
i. Government Publications: The government bodies collect the data on various problems and publish then, such publications are used as source of secondary data.
ii. government bodies: the some government bodies publish the data , such data is the secondary data, like W.H.O. publish data related to health. N.S.O.
iii. Publications of autonomous institutes: Various research institute publish there research articles with data, such institution data is source od secondary data.
iv. Journals and Magazines, news papers: various journals, Magazines publish data related to social and economic problems, is a source of data.
above all the sources collet information it can be used a secondary data.
Qualitative and Quantitative Data :
i. Qualitative Data : a characteristic which can not be measured or expressed in the form of well define quality is called quality. e.g. Gender, Beauty etc. these are qualities. a data collected or arranged according to well defined quality is called Qualitative data. it is also known as categorical data. e,g, data collected as male and female.
ii. Quantitative Data : a characteristic which can be measured or expressed in the form of numbers is called quality. e.g. Income, Height etc. a data collected according to any quantity is called as quantitative data. e.g. Number of students.
Data are classified n to two type as i. Qualitative data and ii. Quantitative data. and again the quantitative data are classified in to two type as Discrete and Continuous data.
Discrete variable: A variable which assume only some specific value or only integer values is called discrete variable e.g. no of pens, number of children's. the data arranging according to discrete variable is called Discrete Data.
Continuous Data: A variable which can assume or take any value in interval is called continuous variable. e.g. Height, Etc. and the data arranged according to continuous variable is called Continuous Data.
Sampling Methods:
Census and Sampling Method:
There are two methods of data collection:
i. Census Method ii. Sampling Method
but we first define the some terms as
Population: In Statistics, the group of individuals under study is called 'population'. The number of individuals in belonging to a population is known as Population Size and it is denoted as "N".
e.g. if we are interested to study the income of male in Sangli district then the population will be the all males in Sangli. therefore the population may be group of individual or object like animate or inanimate or peoples or cars. the population is may be finite or infinite.
now we see the Census Method:
Definition of Census Method: The process of collecting data or information from every member of the population. i.e. we collect data from entire population is called the Census Method. or 100% inspection or complete enumeration. the Census Method is suitable when the population is limited. or when the greater accuracy is expected that case we use Census Method to collect Data.
There are some limitations of Census Method:
i. Census Method provide Reliable result.: In Census method we study each and every individual in Population. the data collected from every individual it is expensive and time consuming and it required large amount of time and manpower.
ii. There is in some situation where Census Method is possible but impracticable. e.g. blood test.
iii. If the population is infinite then census method can't used.
these are the limitations of Census Method that can be over come using Sampling Method
Sampling Method:
First we see the sample means :
Sample : A finite Sub-Group of population is called a sample, and the number of individuals in Sample is called the sample size and it is denoted as "n".
Sampling : Sampling is a Process of collecting data from selected sample. it is also called sampling method. A part of population is studied is called sampling. from this sample we draw a inference about the entire population. for that the selected sample is unbiased and sufficiently large.
Advantages of Sampling Method :
Followings are the advantages of sampling method to overcome the drawback of Census method.
i. Less Time: If the population is large and the study of population required a lot of time. not only for collection but also for analysing the data. as compared to sample. therefore the sample is required the less time as compared to the population.
ii. Less Cost : The cost of collection of data on each unit in case of population is likely to be more as compared to sampling method.
iii. Reliability: The collection of data in sample survey is more reliable than that of complete enumeration.
iv. Detailed Information: The sample contain the small size of members therefore we studied it carefully and detailed information can be collected.
v. Necessity: Some situation where the sampling is necessity, when we study the destructive sampling where the quality of an object can be determined only by destroying the object in the process of testing. testing the explosive.
vi. if the population is very large or spread over the large geographical areas. when we use only the sampling methods.
Remark: there are some limitations of sampling
i. proper care should be taken in planning of the sample survey, other wise the result may be might bee inaccurate.
ii. if the time and money is not important factor then Census method is better than sampling method.
Following are the Methods of Sampling.
i. Simple Random Sampling
ii. Stratified Random Sampling
iii. Systematic Sampling
iv. Multistage Sampling
v. Cluster Sampling
the Selection of sampling method dependence on the available information about the Population and Nature of data.
Now we Discuss only two methods. i. Simple Random Sampling and ii. Stratified Random Sampling.
i. Simple Random Sampling (SRS):
Simple Random Sampling it is denoted as SRS. SRS is the easiest and most commonly used method of sampling. in this method each unit of the population has equal chance to select in sample. i.e. 1/N. the simple random sampling method is divided into two types. due to selection procedure of elements in population.
a) Simple Random Sampling with replacement (SRSWR).
b) Simple Random Sampling without replacement (SRSWOR).
a) Simple Random Sampling with replacement (SRSWR): In Simple Random Sampling with replacement (SRSWR). element or unit are selected one by one from the population in such a way that after each drawing the unit is studied completely and then return back to the population before the next unit being drawn. therefore in SRSWR method the population size is remains the same at every draw. this method of sampling called simple random sampling with replacement (SRSWR) this method is used when the population is finite. but in this method the same unit is selected more than once in sample. it is drawback of that sampling method.
b) Simple Random Sampling without replacement (SRSWOR). : It is the another method of sampling in which unit are selected one by one from population without replacement i.e. the unit selected once it not replaced back to the population. this method of selecting the sample is called the Simple Random Sampling without replacement (SRSWOR). in this method the population size is decreases at each draw. this method is used when the population is infinite. and the drawback of SRSWR method is overcome in SRSWOR method.
Let N be the population size and n be the sample size n be following methods are used to drawing sample from population. i. Lottery Method, ii. Random Number.
i. Lottery Method:
Suppose we want to select the sample of size n out of the Population Size N. In this method we write the name or number of all N units on the slip of paper ( or small size of paper having same size, same colour, same shape. and fold it and collect all N chits in box then select the n chits of paper from the box. ( when we select the chits there is no idea to which number in that chits so it is random sample) this method used for prizes of lottery so it is called Lottery Method.
ii. Method Of Random Number:
In this method we give number to each unit in population from 1 to N. ( if the N<99 then we use two digit number as 01, 02, ......,99) then we use random number number book to select random number and the this numbered unit selected form population as sample.
Merit and Demerit of Simple Random Sampling.
Merit:
Sample unit Randomly selected hence each unit has equal chance to select in sample so person bias is removed.
Demerit: Population is large then work for giving number is tedious, and the size of sample in this method is required to be large.
ii. Stratified Random Sampling:
when the is consist of different groups or classes, then simple random sampling does not give proper representation of sample in that case we use Stratified Random Sampling.
Stratification means data divided into classes, e.g age , gender.
In the Stratified Random Sampling items of each group are include into the right proportion, when the total population is known then we divided the population N into K strata's or groups of size N1, N2, N3, ............NK. respectively.
such that ∑Ni = N, we want to take sample of size n units then we select simple random sample without replacement method to select sample of size n1, n2,.... nk. units from the respective group of population. such that ∑ni = n,
here the unit of size ∑ni = n, selecting using stratified random sample, hence the method is known as stratified random sampling.
now the sample selected from each group using following formulae.
there are two formula to select sample i. Proportional Allocation and ii. Optimum Allocation
i. Proportional Allocation :
ni = (n/N) Ni , i = 1, 2, ....k
where ni - i th sample size
N - Population Size
Ni - i th Population Size
ii. Optimum Allocation:
ni = n {(NiSi) / ∑ (NiSi)
where ni - i th sample size
N - Population Size
Ni - i th Population Size
Example: There are 1000 students in college, out of which 500 from commerce, 200 from arts and 300from science. we want to select sample of 100 students.
Solution: Given N= 1000, N1 = 500, N1 =200, N3 =300
n = 100
for finding the sample size we use the Proportional allocation.
n1 = (n/N) N1 = (500/1000) x100
n1 = 50 i.e 50 students from commerce selected from sample.
n2 = (n/N) N2 = (200/1000) x100
n2 = 20 i.e 20 students from commerce selected from sample.
n3 = (n/N) N3 = (300/1000) x100
n3 = 30 i.e 30 students from commerce selected from sample.
n = n1 + n2 +n3 = 50+30+20 = 100 sample size.
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