What is Statistics ?

Statistics is simply the study of numerical data, facts, figures and measurements. Statistics is used to convert raw numerical data into useful information for relevant users.

According to Bowley, “Statistics is a science of Averages”.

He defined statistics as “Numerical statement of facts in any department of enquiry placed in relation to each other”

Most of the information around us is determined with help of statistics –

• Weather Forecasts
• Medical Studies
• Quality Testing
• Stock Markets
• Predicting Emergencies

Business Statistics involves the application of statistical tools in the area of marketing, production, finance, research and development, manpower planning etc. to extract relevant information for the purpose of decision making.

Business managers use statistical tools and techniques to explore almost all areas or business operations of public and private enterprises. On the basis of the statistical technique used, statistics may be broadly divided into two categories:

• Descriptive Statistics – Descriptive statistics makes use of Graphs, tables, charts and other statistical tools to make generalizations or to describe a certain phenomenon.
• Inferential Statistics – All generalization made through descriptive statistics may not necessarily be true and therefore Inferential statistics is used to test the validity of the generalizations made. It involves estimating and validating facts and figures for the purpose of decision making.  

Importance of Statistics 

In Business – It helps to make swift decisions by providing useful information about customer trends and variations, cost customer trends and variations, price customer trends and variations etc.

In Mathematics – It helps in describing measurements and providing accuracy of theories.

In Economics – It helps to find relationship between two variables like demand and supply, cost and revenue, imports and exports and helps to establish relationship between inflation rate, per capita income, income distribution etc.

In Accounts – It helps to discover trends and create projections for next year.

In Physics – It helps to compute distance between objects in space.

Research – It helps in formulating and testing hypothesis.

Government – Government takes help of statistics to make budgets, set minimum wages, estimate cost of living etc.    

Importance of Business Statistics

Business Statistics helps a business to:

• Deal with uncertainties by forecasting seasonal, cyclic and general economic fluctuations 
• Helps in Sound Decision making by providing accurate estimates about costs, demand, prices, sales etc.
• Helps in business planning on the basis of sound predictions and assumptions
• Helps in measuring variations in performance of products, employees, business units etc.
• It allows comparison of two or more products, business units, sales teams etc. 
• Helps in identifying relationship between various variables and their effect on each other like effect of advertisement on sales
• Helps in validating generalizations and theoretical concepts formulated by managers 

Also Check Out: Statistics Handbook

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Table of Contents

Measures of Central Tendency

Measures of Dispersion

Correlation

Regression

Sampling

Test of Hypothesis

Chi-Square Test

Index Numbers

Interpolation

Extrapolation

Measures of Central Tendency – MEAN, MEDIAN, MODE

MEAN – It is the average of a given set of observation.

Ungrouped data

Grouped data

Direct Method:

Shortcut Method:

Combined Mean:

MEDIAN – It is the middle value of an observation

Ungrouped data:

Grouped data:

MODE – It is the value which occur the maximum number of times in a data

Ungrouped data:

Mode is the value which has the highest frequency.

Grouped data:

Relationship between Mean,Median and Mode

Measures of Dispersion

Range: It is the difference between the value of smallest observation and largest observation in a data.

Quartile Deviation:

Average Deviation:

Ungrouped Data:

Grouped Data:

Standard Deviation:

Ungrouped Data:

Assumed Mean Method:

Grouped Data:

S.D. of Natural Numbers:

Variance:

Relationship between Measures of Variation:

Correlation

Karl Pearson`s Co-efficient Method:

In case of Grouped data:

Spearman`s Rank Coefficient:

Regression Analysis

Regression Equation of Y on X:

Regression Equation of X on Y:

If deviations are taken from mean:

If deviations are taken from assumed mean:

Regression Coefficients:

Relation between coefficient of correlation and two regression coefficients:

Index numbers

Sampling

Sample size determination:

Mean:

Proportion:

Test of Hypothesis

Null Hypothesis – H_o

Alternate Hypothesis – H₁

Size of Sample – n

Types of Tests – One tailed, Two Tailed, Right tailed, Left tailed

Process:

• Formulate the hypothesis
• Set the significance level
• Decide the Test Statistic (z,t)
• Find out the critical value
• Make a conclusion

Use Z statistic when sample size is > 30

Use T statistic when sample size is < 30 and/or Standard Deviation is Unknown

Calculation of Z statistic

Mean

Difference of two mean

Counting 

Proportion

Difference of two Proportions

Calculation of T statistic:

Mean:

Difference in two mean

Difference of two means with dependent samples

Chi Square Test

Interpolation

Extrapolation

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Probability Sampling – In this sampling method the probability of each item in the universe to get selected for research is the same. Hence the sample collected through this method is totally random in nature. Therefore it is also known as Random Sampling.

Non-Probability Sampling – In this sampling method the probability of each item in the universe to get selected for research is not the same. Hence the sample collected through method is not random in nature. Therefore it is known as Non-random Sampling.

Sampling Methods/Sampling Techniques

Probability or Random Sampling Methods:

(1) Simple random sampling – This method simply involves the task selecting sampling units randomly out of the sampling frame. A researcher may use the following methods for selecting random samples – Lottery Method, Random Numbers, software etc.

There are two types of random sampling:

• SRSWR – Simple random sampling with replacement
• SRSWOR – Simple random sampling without replacement

(2) Stratified sampling – In this method a heterogeneous population is divided into different small sub-units, which are called stratas. These stratas are homogenous among themselves with respect to a certain factor or characteristic. Items or sampling units are randomly selected from these stratas that together make up the sample.

(3) Systematic sampling – In this type of sampling the first unit is selected randomly and then every K^th item on the source list is selected, which becomes the part of the sample. The value of K is determined by :

K = Total no. of units in population/No. of units in sample

The essence of this method is selection of random items from the source list at a specified interval from the selected unit, hence forming a system for selecting items. The Items may be arranged numerically, alphabetically or in an increasing or decreasing order and then a formula is applied to it.

(4) Cluster sampling – This method is used where the size of population is very large. In this method a homogeneous population is divided into smaller heterogeneous groups and then samples are drawn out at random from these heterogeneous groups. These heterogeneous groups are called clusters. All items belonging to the selected heterogeneous groups become the part of the sample.

(5) Area Sampling – If the clusters are divided on geographical basis, it is termed as area sampling.

(6) Multi-stage sampling – In multistage sampling, sampling is performed at more than 1 step or stage. At first stage units are selected by some random sampling method usually SRSWOR or Systematic sampling and at the second stage again some units are selected out of the previously selected units through some suitable method. It can be understood as an expansion of the cluster sampling method where instead of selecting the entire heterogeneous group, items are drawn randomly from each heterogeneous group to form a sample.

Non-Probability or Non-Random Sampling Methods

(1) Judgement sampling – In this method, the sampling units are chosen by the researcher on the basis of his or her own judgement. The research simply selects the sample which in his opinion will be best for the study.

(2) Quota sampling – In this method of sampling, quotas in form of reservation or percentage are established for different classes of population on the basis of age, gender, nationality etc. A sample is then drawn out on the basis of these quotas.

(3) Panel sampling – In this method regular surveys are taken by a researcher from a panel of experts of a particular domain through questionnaires or schedules. The panelists may or may not know about other during the research process.

(4) Convenience sampling – In convenience sampling, a researcher simply selects the sample and sampling units that are easily available and accessible. No extra efforts are taken by the researcher as he simply chooses the samples on the basis of convenience.

(5) Snowball sampling – This method is used in cases where the population to be studied is rare, therefore it is difficult to find good representative sampling units. In this method the researcher initially selects a sampling unit (a doctor, a musician, a cancer patient depending upon the study) based on his judgement and then starts taking further samples on the basis of directions/advice/referral provided by the first sampling unit.

The researcher starts by interviewing one person or small group of people and then asks them for references. He then collects data from the suggested people and asks them for references and the chain continues until an adequate sample is formed.  

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Population or Universe – It is the subject matter of research study. It refers to the entire group or population of something taken into consideration for the purpose of research. It may be finite or infinite.

Sample – A sample is that portion of the population which is critically analysed during a research study in order to make estimations or draw inferences about the entire population. A sample may be defined as a unit chosen from the entire population which represents all the features or characteristics of the entire population.

Sampling Unit – It refers to one item of a sample. It may be one unit of anything i.e. one consumer, one company, one state, one city etc.

Sampling Frame – The collection of all the items or units of a sample make up the sampling frame. It consists a list of all the items in a universe (only in case of finite universe, where it is possible to list down all items). 

Sampling Design – It is simply a plan for obtaining a sample out of a given population. It lays down a definite plan for obtaining a sample out of the entire universe in terms of sampling objectives, population, sample frame, sample size, sample unit, data collection  etc. It is determined before the step of data collection in order to obtain reliable, relevant and adequate information.

There are two ways in which information can be obtained for sampling:

• Census Survey – When the entire population or universe is taken into consideration for the purpose of research.
• Sample Survey – When only a part of population (sample) is studied.

Sample Size – It is the number of observations that form a sample i.e. the number of items that are selected from the entire population for the purpose of research that form a sample. It is denoted by n. The following points must be kept in mind while selecting a sample size:

• Optimum – It must be optimum in size – Not too large, nor too small.
• Representative – It must represent the entire population.
• Reliable – It must meet the parameters of interest of the research study.

Sampling Errors – It refers to the inaccuracy or errors in the process of collection, analysis and interpretation of sampling data.

Sampling errors arise due to two reasons:

• Systematic or biased or Non-sampling errors – These arise due to use of faulty procedures and techniques in making a sample and lack of experience in research.
• Unsystematic or unbiased or sampling errors – These arise due to the limitations of the sampling process.

Sampling Errors – Sampling errors arise as we study only a small portion of the entire population to draw inferences about the whole population. Hence, there are random variations in the sample values as compared to population values. However if we study the entire population it is believed that errors will be nil. This also means the larger the sample size the smaller the sampling error i.e. sampling error is inversely proportional to the size of sample.

Non-Sampling errors – These errors result due to the following reasons:

• Incorrect sampling frame or source list
• Incorrect data collection techniques
• Bias responses of respondents
• Non-responses and omission errors
• Errors in coding, tabulating, analysing data
• Lack of trained and qualified investigators

Types of Sampling

Probability Sampling – In this type of sampling the probability of each item in the universe to get selected for research is the same. Hence the sample collected through this method is totally random in nature.

Non-Probability Sampling – In this type of sampling the probability of each item in the universe to get selected for research is not the same. Hence the sample collected through method is not random in nature.

Also read: Methods/Techniques of Sampling

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