BUSINESSmaths MMS

Sinimole K.R.

BUSINESS MATHEMATICS

““Statistics are the numbers of the Statistics are the numbers of the state: state: The numbers or symbols that formally The numbers or symbols that formally summarize our observations of the summarize our observations of the world”world”

DefinitionDefinition

Business statistics Business Statistics is a science assisting

you to make business decisions under uncertainties based on some numerical and measurable scales.

Decision making processes must be based on data, neither on personal opinion nor on belief.

Statistics in business

The need for statistical information in smooth functioning of an undertaking increases along with its size. No business large or small, public or private, can flourish in these days of large-scale production and cut –throat competition without the help of statistics.

Cont..Today's good decisions are driven by data. In

all aspects of our lives, and importantly in the business context, an amazing diversity of data is available for inspection and analytical insight.

Business managers and professionals are increasingly required to justify decisions on the basis of data. They need statistical model-based decision support systems. Statistical skills enable them to intelligently collect, analyze and interpret data relevant to their decision-making.

Statistical concepts and statistical thinking enable them to:

Solve problems in a diversity of contexts. Add substance to decisions. Reduce guesswork.

In shortIn your future life and career, you will need

to be able to make good use of such information to make sound decisions.

StatisticsArt and Science of Collecting and Understanding DATA:

DATA = Recorded Information e.g., Sales, Productivity, Quality, Costs, Return, …

Why? Because you want:Best use of imperfect information:

e.g., 50,000 customers, 1,600 workers, 386,000 transactions,…

Good decisions in uncertain conditions: e.g., new product launch: Fail? OK? Make you rich?

Competitive Edge e.g., for you in the job market!

Activities of Statistics1. Designing the study:

First stepPlan for data-gatheringSampling

2. Exploring the data:First step (once you have data)Look at, describe, summarize the dataAre you on the right track?

Activities of Statistics (continued)

3. Modeling the dataA framework of assumptions and equationsParameters represent important aspects of the

dataHelps with estimation and hypothesis testing

4. Estimating an unknown:Best “guess” based on dataWrong - by how much?Confidence interval - “we’re 95% sure that

the unknown is between …”

Activities of Statistics (continued)

5. Hypothesis testing:Data decide between two possibilitiesDoes “it” really work? [or is “it” just randomly

better?]Is financial statement correct? [or is error

material?]

Data MiningSearch for patterns in large data sets

Businesses data: marketing, finance, production .. Collected for some purpose, often useful for others From government or private companies

Makes use of Statistics – all the basic activities, and

Prediction, classification, clustering Computer science – efficient algorithms

(instructions) for Collecting, maintaining, organizing, analyzing data

Optimization – calculations to achieve a goal Maximize or minimize (e.g. sales or costs)

Descriptive Statistics

&

Inferential Statistics

Inferential help us to

Formulate & test hypotheses

Descriptive helps us to

Observe more

Acutely

SingleSingle NumericalNumerical

Summaries of a batch

Summaries of a batch

Descriptive Statistics

•It references to a given batch of data rather than to a more general population or class. Such as an average.•Leads to formulate laws or test hypotheses

used to describe

the basic features

of the data

in a study

Characterize your data in simple but informative ways

Descriptive Statistics

Consider the following statements 1. A sample of ten students selected (one by one ) from a class of hundred students includes boys and girls in the order: GBGGBGBGGB.2. Six girls and four boys were included in a sample of ten students from a class of hundred students.

The first of the above statements provides information (data) regarding the sex of ten students chosen in a sample in its raw form.

After making some computations, this information is then put in a comprehensive manner in the second statement.

The data in its original form as in (1) are called statistics.

The data in its secondary form as in (2) are also called statistics.

We can thus say that we compute statistics (data as in(2)) from statistics (data in original form as in (1)) with the help of statistics.

Or we can say thatThe science of statistics deals with the collection

of data called statistics, their classification, presentation of data with the help of tables, graphs, diagrams, the analysis involving computations of data from the classified data and the interpretation of the computed data called statistics which leads one to the conclusions and recommendations that can be implemented in practice.

Importance of relevant information.

It is very important for a statistician to ensure that any inference that he draws or any recommendations that he makes should be practical.

Suppose for instance you are furnished with the data about the minimum temperature recorded in Mumbai on 1st , 2nd , 3rd and 4th day of January last year as 200c , 190c, 180c, 170c respectively.

Example

You are asked to forecast the minimum temperature that may be recorded on 1st of May of the same year.

If you carefully study the data, you find that there is a decreasing trend, the decrease being 10c per day.

Hence one may expect that after 120 days from 1st January, i.e. on first May, the temperature would be 200c – 120* 10c= -1000c.

Example

But it is practically impossible. Your estimate should be at least practically feasible.

For obtaining such an estimate, one requires the relevant information.

For instance, the above case the information supplied to you is irrelevant. What you need is the information about the temperature recorded on 1st May of last few years.

Example

Uses of statisticsStatistical methods can be fruitfully applied to any

problem of decision making where numerical data are available or can be made available.

Therefore, in business, industry and economics; the statistical techniques are applicable to problems like

maintenance of trends of population, production of agricultural and non-agricultural

industries,prices, internal and external trades, gross national product, taxation laws and rates; preparation of budgets, computation of consumer price indices from time to

time to revise the wage structures,preparation of price policies of new products,scheduling of the projects, quality control etc.

Functions of statisticsThe science of statistics is used to perform

the following functions:Statistics help in developing sound methods

of collecting data so that the data collected can be used to draw the valid inference regarding the desired objectives.

It presents the information in numerical form.

It helps in simplifying the complex data by way of classification / tabulation/ graphical representation.

It can be used to study the relationship between two or more factors.

The data regarding a characteristic for a series of past periods can be used to forecast its value for a future period.

Limitations of statistics.It is very powerful science to study

quantitative data. Qualitative data cannot be studied with the help of statistics.

It is used to draw conclusions regarding a group of units rather than single unit.

SOURCES OF DATA.The sources of data can be classified into

two broad categories viz. primary sources and secondary sources.

A source of data is primary for one who creates such a source by collecting the data through direct investigation and the same source of data becomes secondary to anyone else who makes use of it.

The primary data are to be collected through personal investigation by asking questions where as the secondary data are the published reports in journals, news papers, periodicals etc.

SOURCES OF DATA.

you collect the data yourself using methods such as interviews and questionnaires. The key point here is that the data you collect is unique to you and your research and, until you publish, no one else has access to it.

Primary Data

questionnairesinterviewsfocus group interviewsobservationcase-studiesdiariescritical incidentsportfolios.

There are many methods of collecting primary data and the main methods include:

What is 'interpreting data'?Data means information. So interpreting data just

means working out what information is telling you.

Information is sometimes shown in tables, charts and graphs to make the information easier to read. It is important to read all the different parts of the table, chart or graph.

PRESENTATION OF DATA

Classification of Data

Geographical area-wise, e.g. cities, districts, etc.

Chronological on the basis of time. Qualitative

according to some attributes.Quantitative

in terms of Magnitudes

classificationClassification is the process of arranging

things in groups or classes according to their resemblance and affinities.

It serves the following purposes1. eliminating unnecessary details2. brings out clearly the point of similarity 3. allows the comparisons & drawing of inferences

Cont..The characteristic of a population may be

broadly divided into two categories variables (quantitative) & attributes (qualitative)

Cont..Attributes – not capable of being described

numerically eg. Sex, nationality, color of eye, etc,

When classified on the basis of attribute, groups are differentiated either by the absence or presence of the attribute or by its differing qualiites.

Cont..Variables- numerically describedEg. Height, weight, turnover, age, export

etc..

Quantitative data The data which assumes numerical values are called

quantitative data.

Qualitative data The data representing class to which a unit

belongs with respect to a qualitative character (reli뉧

For example,Consider the following for a sample of ten

students chosen from a class of hundred students.1.Their ages in completed years are

17,19,18,19,21,20,18,18,19,20.2. the sample included boys and girls in the following

order: B G B B B G B G B.The age values in (1) are quantitative in nature.

Whereas sex codes B,G in case (2) represent qualitative data.

Variables May be Continuous or Discrete

Discrete and Continuous Data.

Discrete dataIf the data values are integer in nature,

such data values are called discrete. Examples

1,2,3,4,5,6. Nominal &Ordinal

Continuous Data.If the data values are real in nature (any

numerical with in a certain range), such data values are called continuous. Examples

12.5, 13.6, 14.7 ,16.8, 17.1. Variables like age, height, weight, rainfall records

always lead to continuous data Interval or ratio

Data types

Nominal

Qualitative / categorical

Ratio

Quantitative/ Numerical

Discrete/ continuousdiscrete

Ordinal Interval

Nominal &OrdinalCan be classified into categories or groups and

labels to describe them.No order implied in the groupsEg. Yes or No, Pass or Fail, metropolitan, urban,

semi-urban etc….Ordinal is essentially same as nominal except there

is now an order within the groups into which data is classified.

Eg. Ratings of hotels, restaurants, and movies etc..We can say 5 star is better than 4 star but not by

howmuch

Interval &RatioCan be measured on a numerical scale.

However zero pt does not mean absence of something.

Eg. Temperature, time, longitude, latitude etc…

Ratio-can be measured on numerical scale, but here zero pt does mean the absence of what is being measured.

Height , weight, life, price, sales, revenue etc….

Organizing the quantitative dataFirst stepArranging the collected data to prepare an

array.Array – arranging the values by ascending or

descending order

Arranging DataSecond stepOrganize data into similar categories or

classes and then count the number of observations that falls into each category . This method produces a frequency distribution.

The frequency distributionA frequency distribution is a table that

organizes data into classes, that is, into groups of values describing one characteristic of the data.

A frequency distribution shows the number of observations from the data set that fall into each of the classes.

Interpreting data

Tally marks and frequency tables

Tallying is a way of counting that uses groups of five, like this:

As you can see, using groups of five makes it easier to see the total!

Frequency tables show the totals of the tally marks.

Grouping data

When there are a large number of possible outcomes, we will usually need to group the data. To do this, we first need to decide on our classes.

Cont…Decide on the type and number of classes for

dividing the data. by quantitative or qualitative measureThe range must be divided by equal classesThe width of the interval from the beginning of

one class to the beginning of the next class must be the same for every class.

Width of a class interval

intervals class ofnumber total

datain aluesmallest v -datain luelargest vaafter unit valuenext

intervals class theofwidth

Step 2 &3Sort the data points into classes and count

the number of points in each classIllustrate the data in a chart.

Class intervalsClass limits

The lowest an the highest value that can be included. 20-30

Class intervalsThe difference between upper and lower limit.

Class frequencyThe number of observations corresponding to the

particular class

Class mid pointValue lying the half-way between lower & upper limit.

Exclusive & Inclusive methodUpper limit of one class is the lower limit of

the next class 20-30 30-40 etc.Upper limit of one class is included in that

class itself20-29 30-39 etc.

General rules The number of classes should preferably be

between 5 & 15K= 1 + 3.322 log NWhere k= the approximate number of classesN= total number of observationWhenever possible all classes should be of the

same size.If not possible form open end classes

ExampleImagine that you are carrying out a survey to

determine the number of CDs bought by your classmates in the past year. The possible answers are likely to range from 0 to 50 (about one a week!), so you might draw a tally chart similar to the one below:

Number of CDs Frequency

0- 4  8

5 - 9  5

10 - 14  7

15 - 19  4

20 - 24  1

25 - 29  

30 - 34  1

35 - 39  

40 - 44  

45 - 49  2

more than 49  

TablesA table is a systematic arrangement of

statistical data in columns and rows.Parts of a table

Table number Body of the tableTitle of the table Head noteCaption Foot noteStub

Table……….Caption ….

Column headings. It explains what the column represents

Stub ….Row headings

Head notes ….Brief explanatory statementPlaced below the title enclosed in brackets.

Bar charts

Bar charts are one way of showing the information from a frequency table

Types of bar DiagramsSimple bar diagramsSubdivided bar diagramsMultiple bar diagramsPercentage bar diagramsDeviation barsBroken bars

Simple bar diagramsOnly one variable.

Funds flow in Rs. (crores) of ABC ltd.

0 50 100 150 200 250

1999-00

2000-01

2001-02

2002-03

2003-04

year

s

Funds Flow

Sub divided bar diagrams

distribution of Assistance sanctioned.

0

1000

2000

3000

years

Rs.

(cr

ore

s)Series3Series2Series1

Series3 283.4 473.8 402.8 536.6 686.8

Series2 365.3 484.7 668.6 992.8 1067.4

Series1 233.8 301.8 303.2 365.3 416.4

1999-00 2000-01 2001-02 2002-03 2003-04

Multiple bar diagrams

Corporate secotr profits

1663

13761219

982846

589

10

200

400

600

800

1000

1200

1400

1600

1800

2002-03 2003-04

gross profitsprofit beofre taxprofit after taxretained profits

Percentage bar diagrams

Sales & Net Profits

14% 11% 17%29%

61%

-74%

1 2 3

sales netprofits

Pie chartsPie charts are circles divided into segments,

where each segment represents a fraction of the total amount.

Pie chartsPie charts are circles divided into segments,

where each segment represents a fraction of the total amount.

Pie charts…This pie chart shows the 20 bikes sold at the

bike shop. The segment for Mountain bikes is one half of the chart. This is because 10 Mountain bikes were sold, which is exactly half the number of bikes sold in total (20 bikes).

HistogramsA histogram is a series of rectangles, each

proportional in width to the range of values within a class and proportional in height to the number of items falling in the class.

If the classes are of equal width, then the vertical bars in the histogram are also of equal width.

Histogramsare similar to bar charts apart from the

consideration of areas. In a bar chart, all of the bars are the same width and the only thing that matters is the height of the bar. In a histogram, the area is the important thing.

Example of a Histogram

0

1

2

3

4

5

Histogram

Frequency

Relative Frequency

Percent

0 15 25 35 45 55

Lower Boundary

Bars Touch

Class Freq.15 but < 25 325 but < 35 535 but < 45 2

Count

Example

The next slide contains a real world example of a histogram. It plots the relative frequency of the heights of some students based on the data below.                              Frequency:Height (feet):          (Number of pupils)     :0-2                        0                                2-4                        1                                4-5                        4                                5-6                        8                               6-8                        2                                

Real World Example

This Histogram is courtesy of http://www.gcsemaths.fsnet.co.uk/page5.html

Frequency polygonsTo construct a frequency polygon

Mark the frequencies on the vertical axis and the values of the variable on the horizontal axis.

Plot each frequency by drawing a dot above its midpoint and

Connect the successive dots with straight lines to form a polygon.

Cont.. Add two new classes containing zero

observations.Allow the polygon to reach the horizontal axis

at both end of the distribution.

0

1

2

3

4

5

Polygon

Midpoint

Fictitious Class

0 10 20 30 40 50 60

Class Freq.15 but < 25 325 but < 35 535 but < 45 2

Frequency

Relative Frequency

Percent

Count

Relative frequency polygonA frequency polygon that uses the relative

frequency of data points in each of the classes is called a relative frequency polygon.

Ogives A graph of a cumulative frequency

distribution is called an ogiveA cumulative less-than ogive &A cumulative greater-than ogive.

0%

25%

50%

75%

100%

Cumulative % Polygon (Ogive)

Lower Boundary

Fictitious Class

0 15 25 35 45 55

Class Cum. %15 but < 25 0%25 but < 35 30%35 but < 45 80%45 but < 55 100%

Cumulative %

Errors in Presenting Data

1. Using ‘Chart Junk’

2. No Relative Basis in Comparing Data Batches

3. Compressing the Vertical Axis

4. No Zero Point on the Vertical Axis

‘Chart Junk’

Bad Presentation Good Presentation

1960: $1.00

1970: $1.60

1980: $3.10

1990: $3.80

Minimum Wage Minimum Wage

0

2

4

1960 1970 1980 1990

$

No Relative Basis

Good PresentationA’s by Class A’s by Class

Bad Presentation

0

100

200

300

FR SO JR SR

Freq.

0%

10%

20%

30%

FR SO JR SR

%

Compressing Vertical Axis

Good PresentationQuarterly Sales Quarterly Sales

Bad Presentation

0

25

50

Q1 Q2 Q3 Q4

$

0

100

200

Q1 Q2 Q3 Q4

$

No Zero Point on Vertical Axis

Good PresentationMonthly Sales Monthly Sales

Bad Presentation

0

20

40

60

J M M J S N

$

36

39

42

45

J M M J S N

$

THANK YOUTHANK YOU