Theory of Business Statistics

What is Business Statistics

• Business statistics is the science of good decision making in the face of uncertainty and is used in many disciplines such as financial analysis, econometrics, auditing, production and operations including services improvement, and marketing research.

Importance of business statistics

• (i) The planning of operations: This may relate to either special projects or to the recurring activities of a firm over a specified period.

• (ii) The setting up of standards: This may relate to the size of employment, volume of sales, fixation of quality norms for the manufactured product, norms for the daily output, and so forth.

• (iii) The function of control: This involves comparison of actual production achieved against the norm or target set earlier. In case the production has fallen short of the target, it gives remedial measures so that such a deficiency does not occur again.

• Focusing on Big Picture

• Statistical analysis of a representative group of consumers can provide a reasonably accurate, cost-effective snapshot of the market with faster and cheaper statistics than attempting a census of very single customer a company may ever deal with. The statistics can also afford leadership an unbiased outlook of the market, to avoid building strategy on uncorroborated presuppositions.

• Backing Judgments

• Statistics back up assertions. Leaders can find themselves backed into a corner when persuading people to move in a direction or take a risk based on unsubstantiated opinions. Statistics can provide objective goals with stand-alone figures as well as hard evidence to substantiate positions or provide a level of certainty to directions to take the company.

• Making Connections

• Statistics can point out relationships. A careful review of data can reveal links between two variables, such as specific sales offers and changes in revenue or dissatisfied customers and products purchased. Delving into the data further can provide more specific theories about the connections to test, which can lead to more control over customer satisfaction, repeat purchases and subsequent sales volume.

• Ensuring Quality

• Anyone who has looked into continuous improvement or quality assurance programs, such as Six Sigma or Lean Manufacturing, understands the necessity for statistics. Statistics provide the means to measure and control production processes to minimize variations, which lead to error or waste, and ensure consistency throughout the process. This saves money by reducing the materials used to make or remake products, as well as materials lost to overage and scrap, plus the cost of honoring warranties due to shipping defective products.

Limitation of business statistics( material given)• Difficulty of Understanding

• Research has shown that people have a difficult time thinking statically. The idea that a statistic is part of a distribution of possible figures is relatively unintuitive. As such, business owners tend to neglect characteristics such as base rates. Say a company has designed a test to detect fraud that is 99 percent accurate. If the proportion of fraud in the entire population is only 1 in 1,000, the chance that you have detected fraud is much lower. In fact, the probability of fraud existing, given a positive test result, is only 9 percent. Because the base rate of fraud is so low, a positive test result cannot give us much insight into the actual chance that fraud has occurred.

• Frequency

• Statistical tests in business are often conducted from a frequentist approach, which may not be representative of the questions we are asking. In production processes, this often takes the form of a tolerance for error. Say a company produces sheets of metal that are 3 mm thick. The company may say that sheets within the 2.95 mm to 3.05 mm range of thickness are acceptable. If the company is producing 3.02 mm thick sheets, the sheets are acceptable based upon the company's quality standards and, statistically speaking, this may be not be significantly greater than 3 mm. However, overweighting in production could cost the company money.

• Small Sample Sizes• In general, people tend to poorly determine the effect of sample

size when the sample size is small. For example, a foreman may have the choice to complete a small production run of bottles daily or a larger run every other day. The company considers a production run successful when fewer than 1 percent of bottles are defective. All else constant, most think that it is equally likely to exceed the 1 percent threshold using either size of production run. However, in smaller production runs, random fluctuations have a larger effect of the total number of defects. In larger runs, these fluctuations tend to even themselves out.• Outcome Bias• When using statistics as a business diagnostic tool, managers tend

to suffer from outcome bias. For example, managers may use the percentage of defective products to determine if a production process is sound. If many defects are found, managers will usually investigate the process and try to determine the source of the problem. However, it does not make sense to investigate low numbers of defective products. When the defective product count is inconclusive, the manager has to choose whether to investigate. Research has shown that if the manager investigates the defects and finds no systematic problem in production, management will be less satisfied with the manager's performance than if he uncovered a problem. This occurs even though the manager had no notion of the outcome of his investigation before he started it.

What is Classification?

• The process of dividing the data into different groups ( viz. classes) which are homogeneous within but heterogeneous between themselves, is called a classification.• It helps in understanding the salient features of the data and also the

comparison with similar data. For a final analysis it is the best friend of a statistician.

Methods of Classification

• According to attributes or qualities this is divided into two parts :1. Simple classification2. Multiple classification.• According to variable or quantity or classification according to class

intervals. -1. Qualitative Classification :2. Quantitative classification:

• Qualitative Classification : When facts are grouped according to the qualities (attributes) like religion, literacy, business etc., the classification is called as qualitative classification.• Quantitative classification: It is done according to numerical size like

weights in kg or heights in cm. Here we classify the data by assigning arbitrary limits known as class-limits. The quantitative phenomenon under study is called a variable. For example, the population.

Classification of Data

• The process of arranging data into homogenous group or classes according to some common characteristics present in the data is called classification.

• For Example: The process of sorting letters in a post office, the letters are classified according to the cities and further arranged according to streets..

Bases of Classification

There are four important bases of classification:(1) Qualitative Base (2) Quantitative Base (3) Geographical Base (4) Chronological or Temporal Base • (1)Qualitative Base: When the data are classified according to some

quality or attributes such as sex, religion, literacy, intelligence etc…• (2) Quantitative Base:

When the data are classified by quantitative characteristics like heights, weights, ages, income etc…

• (3) Geographical Base: When the data are classified by geographical regions or location, like states, provinces, cities, countries etc… (4) Chronological or Temporal Base: When the data are classified or arranged by their time of occurrence, such as years, months, weeks, days etc… For Example: Time series data

Types of Classification:

• (1) One -way Classification: If we classify observed data keeping in view single characteristic, this type of classification is known as one-way classification.For Example: The population of world may be classified by religion as Muslim, Christians etc…

• (2) Two -way Classification: If we consider two characteristics at a time in order to classify the observed data then we are doing two way classifications.For Example: The population of world may be classified by Religion and Sex.

What is Tabulation?

• It is the process of condensation of the data for convenience, in statistical processing, presentation and interpretation of the information.

Collection of Data

• Data means information. Data collected expressly for a specific purpose are called ‘Primary data’• e.g., data collected by a particular person or organisation from the primary source for his

own use, collection of data about the population by censuses and surveys, etc. • Data collected and published by one organisation and subsequently used by other

organisations are called ‘Secondary data’.• The various sources of collection for secondary data are: newspapers and periodicals;

publications of trade associations; research papers published by university departments, U.G.C. or research bureaus; official publications of central, state and the local and foreign governments, etc

• The collection expenses of primary data are more than secondary data. Secondary data should be used with care. • The various methods of collection of primary data are: (i)

Direct personal investigation (interview/observation); • (ii) Indirect oral investigation; • (iii) Data from local agents• and correspondents; • (iv) Mailed questionnaires; • (v) Questionnaires to be filled in by enumerators;• (vi) Results of experiments, etc. • Data collected in this manner are called ‘raw data’. These

are generally voluminous and have to be arranged properly before use.

What is frequency distribution?

• a frequency distribution is an arrangement of the values that one or more variables take in a sample. Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval, and in this way, the table summarizes the distribution of values in the sample.

Principle of frequency distribution

• decide how many groups you want

• the fewer groups, the less precise your description

• too many groups, and it's too much work, too little summary

• usually 10 to 20 groups is best

• take the range (difference between highest and lowest)

• divide by number of intervals (groups) you want

• the lowest interval (group) should begin with a number that can be divided evenly by the size of the interval width

• make sure that the lowest and highest scores are included in your table

• relationship between these depends on the frequency distribution

What is central tendency?

• the tendency for the values of a random variable to cluster round its mean, mode, or median.• Measures of central tendency are numbers that describe what is

average or typical of the distribution of data.

How to calculate Mean?( with Formula)• Arithmetic the sum divided by the population size, n — used when

the sum is of interest. OR• The arithmetic mean (or simply "mean") of a sample is the sum the

sampled values divided by the number of items in the sample:• Geometric nth root of the product — used when the product is of

interest.• Harmonic n divided by the sum of the reciprocals — used for rates

and ratios

what is measure of variation

• Measure of variation is a measure that describes how spread out or scattered a set of data. It is also known as measures of dispersion or measures of spread.• There are four frequently used measures of variability: the range,

interquartile range, variance, and standard deviation.

What is range?

• In arithmetic, the range of a set of data is the difference between the largest and smallest values• the area of variation between upper and lower limits on a particular

scale.• The Range is the difference between the lowest and highest values.

What is Quartile Deviation?

• In a distribution, partial variance between the upper quartile and lower quartile is known as 'quartile deviation'. Quartile Deviation is often regarded as semi inter quartile range.•

(Upper quartile- lower quartile) / 2

What is standard deviation?

• a quantity expressing by how much the members of a group differ from the mean value for the group.• A measure of the dispersion of a set of data from its mean.

Importance of SD

• The standard deviation is a commonly used statistic, but it doesn’t often get the attention it deserves. • Without standard deviation, you can’t get a handle on whether the

data are close to the average or whether the data are spread out over a wide range • Without the standard deviation, you can’t compare two data sets

effectively.• The standard deviation is also important in finance, where the

standard deviation on the rate of return on an investment is a measure of the volatility of the investment.

What is skewness?

• Skewness is asymmetry in a statistical distribution, in which the curve appears distorted or skewed either to the left or to the right. • Describe asymmetry from the normal distribution in a set of statistical

data. • Skewness can come in the form of "negative skewness" or "positive

skewness", depending on whether data points are skewed to the left (negative skew) or to the right (positive skew) of the data average.

Uses of Skewness

• Skewness is extremely important to finance and investing. • Most sets of data, including stock prices and

asset returns, have either positive or negative skew rather than following the balanced normal distribution (which has a skewness of zero). • By knowing which way data is skewed, one can

better estimate whether a given (or future) data point will be more or less than the mean.

Most advanced economic analysis models study data for skewness and incorporate this into their calculations. • Skewness risk is the risk that a model assumes a

normal distribution of data when in fact data is skewed to the left or right of the mean.

Measures of Skewness(figure )

• A measure of skewness is a single value that indicates the degree and direction of asymmetry.• Interpretation of Measure of Skewness • Direction of Skewness1. Sk = 0: symmetric2. Sk > 0: positively skewed3. Sk < 0: negatively skewed• Degree of Skewness1. The farther |Sk| is from 0, the more skewed the distribution

What is Index?

• An "index", as the term is generally used when referring to statistics, is a series of index numbers expressing a series of numbers as percentages of a single number.• Index numbers are a statistician's way of expressing the difference

between two measurements by designating one number as the "base", giving it the value 100 and then expressing the second number as a percentage of the first.

Limitation of simple index number(in book also)• They are simply rough indications of the relative changes.• The choice of representative commodities may lead to fallacious

conclusions as they are based on samples.• There may be errors in the choice of base periods or weights etc.• Comparisons of changes in variables over long periods are not

reliable.• They may be useful for one purpose but not for other.• They are specialized types of averages and hence aresubject to all

those limitations with which an average suffers from.

Method of constructing price and quantity indices• In book

Problems in constructing price and quantity indices• in the construction of price indices for inflation, the nature of goods

in the economy changes over time as well as their prices• There is no theoretically ideal solution to this problem. In practice for

retail price indices,

Following steps are involved in the construction of a frequency distribution.

• (1) Find the range of the data: The range is the difference between the largest and the smallest values.• (2) Decide the approximate number of classes: Which the data are to

be grouped. There are no hard and first rules for number of classes. Most of the cases we have to classes. H.A. Sturges has given a formula for determining the approximation number of classes.

• Where = Number of Classes Where = Logarithm of the total number of observationsFor Example: If the total number of observations is, the number of classes would be Or classes approximately.• (3) Determine the approximate class interval size: The size of class

interval is obtained by dividing the range of data by number of classes and denoted by class interval size In case of fractional results, the next higher whole number is taken as the size of the class interval.

• (4) Decide the starting point: The lower class limits or class boundary should cover the smallest value in the raw data. It is a multiple of class interval.For Example:,,,, etc… are commonly used.

• (5) Determine the remaining class limits (boundary): When the lowest class boundary of the lowest class has been decided, then by adding the class interval size to the lower class boundary, compute the upper class boundary. The remaining lower and upper class limits may be determined by adding the class interval size repeatedly till the largest value of the data is observed in the class.

• (6) Distribute the data into respective classes: All the observations are marked into respective classes by using Tally Bars (Tally Marks) methods which is suitable for tabulating the observations into respective classes. The number of tally bars is counted to get the frequency against each class. The frequency of all the classes is noted to get grouped data or frequency distribution of the data. The total of the frequency columns must be equal to the number of observations.