Statistics For Management

1. Introduction

2. Statistical Survey

3. Classification, Tabulation & Presentation of data

4. Measures used to summarise data

5. Probabilities

6. Theoretical Distributions

7. Sampling & Sampling Distributions

8. Estimation

9. Testing of Hypothesis in case of large & small samples

10. Chi-Square

11. F-Distribution and Analysis of variance (ANOVA)

12. Simple correlation and Regression

13. Business Forecasting

14. Time Series Analysis

15 . Index Numbers

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Q: What’s the definition of Statistics ?

A : Statistics are usually defined as:

1. A collection of numerical data that measure something.

2. The science of recording, organising, analysing and

reporting quantitative information.

Q: What are the different components of statistics ?

A: There are four components as per Croxton & Cowden

1. Collection of Data.

2. Presentation of Data

3. Analysis of Data

4. Interpretation of Data

Q: What’s the use of Correlation & Regression ?

A: Correlation & Regression is a statistical tools, are used to

measure strength of relationships between two variables.

Q. What is the need for Statistics?

Statistics gives us a technique to obtain, condense, analyze

and relate numerical data. Statistical methods are of a

supreme value in education and psychology.

Q. How is statistics used in everyday life?

Statistics are everywhere, election predictions are statistics,

anything food product that says they x% more or less of a

certain ingredient is a statistic. Life expectancy is a statistic.

If you play card games card counting is using statistics.

There are tons of statistics everywhere you look.

Statistical Survey

1. What is statistical survey?

Statistical surveys are used to collect quantitative

information about items in a population. A survey may focus

on opinions or factual information depending on its purpose,

and many surveys involve administering questions to

individuals. When the questions are administered by

a researcher, the survey is called a structured interview or

a researcher-administered survey. When the questions are

administered by the respondent, the survey is referred to as

a questionnaire or a self-administered survey.

2. What are the advantages of survey?

Efficient way of collecting information

Wide range of information can be collected

Easy to administer

Cheaper to run

3. What are the disadvantages of survey?

Responses may be subjective

Motivation may be low to answer

Errors due to sampling

If the question is not specific, it may lead to vague data.

4. What are the various modes of data collection?

Telephone

Mail

Online surveys

Personal survey

Mall intercept survey

5. What is sampling?

“Sampling” basically means selecting people/objects from a

“population” in order to test the population for something.

For example, we might want to find out how people are

going to vote at the next election. Obviously we can’t ask

everyone in the country, so we ask a sample.

Classification, Tabulation & Presentation of data

1. What are the types of data collection?

Qualitative Data

Nominal, Attributable or Categorical data

Ordinal or Ranked data

Quantitative or Interval data

Discrete data

Continuous measurements

2. What is tabulation of data?

Tabulation refers to the systematic arrangement of the

information in rows and columns. Rows are the horizontal

arrangement. In simple words, tabulation is a layout of

figures in rectangular form with appropriate headings to

explain different rows and columns. The main purpose of the

table is to simplify the presentation and to facilitate

comparisons.

3. What is presentation of data?

Descriptive statistics can be illustrated in an understandable

fashion by presenting them graphically using statistical and

data presentation tools.

4. What are the different elements of tabulation?

Tabulation:

Table Number

Title

Captions and Stubs

Headnotes

Body

Source

5. What are the forms of presentation of the data?

Grouped and ungrouped data may be presented as :

Pie Charts

Frequency Histograms

Frequency Polygons

Ogives

Boxplots

Measures used to summarise data

1. What are the measures of summarizing data?

Measures of Central tendency: Mean, median, mode

Measures of Dispersion: Range, Variance, Standard

Deviation

2. Define mean, median, and mode?

Mean: The mean value is what we typically call the

“average.” You calculate the mean by adding up all of the

measurements in a group and then dividing by the number

of measurements.

Median: Median is the middle most value in a series when

arranged in ascending or descending order

Mode: The most repeated value in a series.

3. Which measure of central tendency is to be used?

The measure to be used differs in different contexts. If your

results involve categories instead of continuous numbers,

then the best measure of central tendency will probably be

the most frequent outcome (the mode). On the other hand,

sometimes it is an advantage to have a measure of central

tendency that is less sensitive to changes in the extremes of

the data.

4. Define range, variance and standard deviation?

The range is defined by the smallest and largest data values

in the set.

Variance: The variance (σ2) is a measure of how far each

value in the data set is from the mean.

Standard Deviation: it is the square root of the variance.

5. How can standard deviation be used?

The standard deviation has proven to be an extremely useful

measure of spread in part because it is mathematically

tractable.

Probablity

1. What is Probability?

Probability is a way of expressing knowledge or belief that

an event will occur or has occurred.

2. What is a random experiment?

An experiment is said to be a random experiment, if it’s out-

come can’t be predicted with certainty.

3. What is a sample space?

The set of all possible out-comes of an experiment is called

the sample space. It is denoted by ‘S’ and its number of

elements are n(s).

Example; In throwing a dice, the number that appears at top

is any one of 1,2,3,4,5,6. So here:

S ={1,2,3,4,5,6} and n(s) = 6

Similarly in the case of a coin, S={Head,Tail} or {H,T} and

n(s)=2.

4. What is an event? What are the different kinds of

event?

Event: Every subset of a sample space is an event. It is

denoted by ‘E’.

Example: In throwing a dice S={1,2,3,4,5,6}, the

appearance of an event number will be the event E={2,4,6}.

Clearly E is a sub set of S.

Simple event: An event, consisting of a single sample point

is called a simple event.

Example: In throwing a dice, S={1,2,3,4,5,6}, so each of

{1},{2},{3},{4},{5} and {6} are simple events.

Compound event: A subset of the sample space, which has

more than on element is called a mixed event.

Example: In throwing a dice, the event of appearing of odd

numbers is a compound event, because E={1,3,5} which has

’3′ elements.

5. What is the definition of probability?

If ‘S’ be the sample space, then the probability of occurrence

of an event ‘E’ is defined as:

P(E) = n(E)/N(S) =

number of elements in ‘E’

number of elements in sample space ‘S’

Theoretical Distributions

1. What are theoretical distributions?

Theoretical distributions are based on mathematical

formulae and logic. It is used in statistics to define statistics.

When empirical and theoretical distributions correspond,

you can use the theoretical one to determine probabilities of

an outcome, which will lead to inferential statistics.

2. What are the various types of theoretical

distributions?

Rectangular distribution (or Uniform Distribution)

Binomial distribution

Normal distribution

3. Define rectangular distribution and binomial

distribution?

Rectangular distribution: Distribution in which all possible

scores have the same probability of occurrence.

Binomial distribution: Distribution of the frequency of events

that can have only two possible outcomes.

4. What is normal distribution?

The normal distribution is a bell-shaped theoretical

distribution that predicts the frequency of occurrence of

chance events. The probability of an event or a group of

events corresponds to the area of the theoretical distribution

associated with the event or group of event. The distribution

is asymptotic: its line continually approaches but never

reaches a specified limit. The curve is symmetrical: half of

the total area is to the left and the other half to the right.

5. What is the central limit theorem?

This theorem states that when an infinite number of

successive random samples are taken from a population, the

sampling distribution of the means of those samples will

become approximately normally distributed with mean μ and

standard deviation σ/√ N as the same size (N) becomes

larger, irrespective of the shape of the population

distribution.

Sampling & Sampling Distributions

1. What is sampling distribution?

Suppose that we draw all possible samples of size n from a

given population. Suppose further that we compute a

statistic (mean, proportion, standard deviation) for each

sample. The probability distribution of this statistic is called

Sampling Distribution.

2. What is variability of a sampling distribution?

The variability of sampling distribution is measured by its

variance or its standard deviation. The variability of a

sampling distribution depends on three factors:

N: the no. of observations in the population.

n: the no. of observations in the sample

The way that the random sample is chosen.

3. How to create the sampling distribution of the

mean?

Suppose that we draw all possible samples of size n from a

population of size N. Suppose further that we compute a

mean score for each sample. In this way we create the

sampling distribution of the mean.

We know the following. The mean of the population (μ) is

equal to the mean of the sampling distribution (μx). And the

standard error of the sampling distribution (σx) is determined

by the standard deviation of the population (σ), the

population size, and the sample size. These relationships are

shown in the equations below:

μx = μ      and      σx = σ * sqrt( 1/n – 1/N )

4. What is the sampling distribution of the population?

In a population of size N, suppose that the probability of the

occurence of an event (dubbed a “success”) is P; and the

probability of the event’s non-occurence (dubbed a “failure”)

is Q. From this population, suppose that we draw all possible

samples of size n. And finally, within each sample, suppose

that we determine the proportion of successes p and

failures q. In this way, we create a sampling distribution of

the proportion.

5. Show the mathematical expression of the sampling

distribution of the population.

We find that the mean of the sampling distribution of the

proportion (μp) is equal to the probability of success in the

population (P). And the standard error of the sampling

distribution (σp) is determined by the standard deviation of

the population (σ), the population size, and the sample size.

These relationships are shown in the equations below:

μp = P      and      σp = σ * sqrt( 1/n – 1/N ) = sqrt[ PQ/n -

PQ/N ]

where σ = sqrt[ PQ ].

Estimation

1. When will the sampling distribution be normally

distributed?

Generally, the sampling distribution will be approximately

normally distributed if any of the following conditions apply.

The population distribution is normal.

The sampling distribution is symmetric, unimodal,

without outliers, and the sample size is 15 or less.

The sampling distribution is moderately skewed,

unimodal, without outliers, and the sample size is between

16 and 40.

The sample size is greater than 40, without outliers.

2. Get the variability of the sample mean.

Suppose k possible samples of size n can be selected from a

population of size N. The standard deviation of the sampling

distribution is the “average” deviation between the ksample

means and the true population mean, μ. The standard

deviation of the sample mean σx is:

σx = σ * sqrt{ ( 1/n ) * ( 1 – n/N ) * [ N / ( N - 1 ) ] }

where σ is the standard deviation of the population, N is the

population size, and n is the sample size. When the

population size is much larger (at least 10 times larger) than

the sample size, the standard deviation can be approximated

by:

σx = σ / sqrt( n )

3. How can standard error of the population

calculated?

When the standard deviation of the population σ is unknown,

the standard deviation of the sampling distribution cannot

be calculated. Under these circumstances, use the standard

error. The standard error (SE) provides an unbiased

estimate of the standard deviation. It can be calculated from

the equation below.

SEx = s * sqrt{ ( 1/n ) * ( 1 – n/N ) * [ N / ( N - 1 ) ] }

where s is the standard deviation of the sample, N is the

population size, and n is the sample size. When the

population size is much larger (at least 10 times larger) than

the sample size, the standard error can be approximated by:

SEx = s / sqrt( n )

4. How to find the confidence interval of the mean?

Identify a sample statistic. Use the sample mean to

estimate the population mean.

Select a confidence level. The confidence level describes

the uncertainty of a sampling method. Often, researchers

choose 90%, 95%, or 99% confidence levels; but any

percentage can be used.

Specify the confidence interval. The range of the

confidence interval is defined by the sample

statistic + margin of error. And the uncertainty is denoted

by the confidence level.

Testing of Hypothesis in case of large & small samples

1. What is a statistical hypothesis?

A statistical hypothesis is an assumption about a

population parameter. This assumption may or may not be

true.

2. What are the types of statistical hypothesis?

There are two types of statistical hypotheses.

Null hypothesis. The null hypothesis, denoted by H0, is

usually the hypothesis that sample observations result

purely from chance.

Alternative hypothesis. The alternative hypothesis,

denoted by H1 or Ha, is the hypothesis that sample

observations are influenced by some non-random cause.

3. What is hypothesis testing?

Statisticians follow a formal process to determine whether to

reject a null hypothesis, based on sample data. This process

is called hypothesis testing.

4. Define the steps of hypothesis testing?

Hypothesis testing consists of four steps.

State the hypotheses. This involves stating the null and

alternative hypotheses. The hypotheses are stated in such

a way that they are mutually exclusive. That is, if one is

true, the other must be false.

Formulate an analysis plan. The analysis plan describes

how to use sample data to evaluate the null hypothesis.

The evaluation often focuses around a single test statistic.

Analyze sample data. Find the value of the test statistic

(mean score, proportion, t-score, z-score, etc.) described

in the analysis plan.

Interpret results. Apply the decision rule described in the

analysis plan. If the value of the test statistic is unlikely,

based on the null hypothesis, reject the null hypothesis.

5. What are decision errors?

Two types of errors can result from a hypothesis test.

Type I error. A Type I error occurs when the researcher

rejects a null hypothesis when it is true. The probability of

committing a Type I error is called the significance level.

This probability is also called alpha, and is often denoted

by α.

Type II error. A Type II error occurs when the researcher

fails to reject a null hypothesis that is false. The

probability of committing a Type II error is called Beta,

and is often denoted by β. The probability

of not committing a Type II error is called the Power of

the test.

6. How to arrive at a decision on hypothesis?

The decision rules can be taken in two ways – with reference

to a P-value or with reference to a region of acceptance.

P-value. The strength of evidence in support of a null

hypothesis is measured by the P-value. Suppose the test

statistic is equal to S. The P-value is the probability of

observing a test statistic as extreme as S, assuming the

null hypotheis is true. If the P-value is less than the

significance level, we reject the null hypothesis.

Region of acceptance. The region of acceptance is a

range of values. If the test statistic falls within the region

of acceptance, the null hypothesis is not rejected. The

region of acceptance is defined so that the chance of

making a Type I error is equal to the significance level.The

set of values outside the region of acceptance is called

the region of rejection. If the test statistic falls within

the region of rejection, the null hypothesis is rejected. In

such cases, we say that the hypothesis has been rejected

at the α level of significance.

7. Explain one-tailed and two-tailed tests?

A test of a statistical hypothesis, where the region of

rejection is on only one side of the sampling distribution, is

called a one-tailed test. For example, suppose the null

hypothesis states that the mean is less than or equal to 10.

The alternative hypothesis would be that the mean is greater

than 10. The region of rejection would consist of a range of

numbers located located on the right side of sampling

distribution; that is, a set of numbers greater than 10.

A test of a statistical hypothesis, where the region of

rejection is on both sides of the sampling distribution, is

called a two-tailed test. For example, suppose the null

hypothesis states that the mean is equal to 10. The

alternative hypothesis would be that the mean is less than

10 or greater than 10. The region of rejection would consist

of a range of numbers located located on both sides of

sampling distribution; that is, the region of rejection would

consist partly of numbers that were less than 10 and partly

of numbers that were greater than 10.

What is Chi Sqare in Statistics?

Suppose Sachin plays 100 tests, and 20 times he made 50. Is

he a good player ?

In statistics, the chi-square test calculates how well a series

of numbers fits a distribution. In this module, we only test

for whether results fit an even distribution. It doesn’t simply

say “yes” or “no”. Instead, it gives you a confidence interval,

which sets upper and lower bounds on the likelihood that the

variation in your data is due to chance.

There are basically two types of random variables and they

yield two types of data:numerical and categorical.

A chi square (X2) statistic is used to investigate whether

distributions of categorical variables differ from one another.

Basically categorical variable yield data in the categories

and numerical variables yield data in numerical form.

Responses to such questions as “What is your major?” or Do

you own a car?” are categorical because they yield data such

as “biology” or “no.” In contrast, responses to such

questions as “How tall are you?” or “What is your G.P.A.?”

are numerical. Numerical data can be either discrete or

continuous.

Datatype Questiontype Possible answer

Categorical Where are you from ? India / USA / UK / Any country

Numerical How tall are you ? 70 inches

F-Distribution and Analysis of variance (ANOVA)

1. What is ANOVA?

Analysis of variance (ANOVA) is a collection of statistical

models and their associated procedures in which the

observed variance is partitioned into components due to

different sources of variation. ANOVA provides a statistical

test of whether or not the means of several groups are all

equal.

2. What are the assumption in ANOVA?

The following assumptions are made to perform ANOVA:

Independence of cases – this is an assumption of the

model that simplifies the statistical analysis.

Normality – the distributions of the residuals are normal.

Equality (or “homogeneity”) of variances,

called homoscedasticity — the variance of data in groups

should be the same. Model-based approaches usually

assume that the variance is constant. The constant-

variance property also appears in the randomization

(design-based) analysis of randomized experiments, where

it is a necessary consequence of the randomized design

and the assumption of unit treatment

additivity (Hinkelmann and Kempthorne): If the responses

of a randomized balanced experiment fail to have constant

variance, then the assumption of unit treatment

additivity is necessarily violated. It has been shown,

however, that the F-test is robust to violations of this

assumption.

3. What is the logic of ANOVA?Partitioning of the sum of squares

The fundamental technique is a partitioning of the total sum

of squares (abbreviated SS) into components related to the

effects used in the model. For example, we show the model

for a simplified ANOVA with one type of treatment at

different levels.

So, the number of degrees of freedom (abbreviated df) can

be partitioned in a similar way and specifies the chi-square

distribution which describes the associated sums of squares.

4. What is the F-test?

The F-test is used for comparisons of the components of the

total deviation. For example, in one-way, or single-factor

ANOVA, statistical significance is tested for by comparing

the F test statistic

where

 I = number of treatments

and

 nT = total number of cases

to the F-distribution with I − 1,nT − I degrees of freedom.

Using the F-distribution is a natural candidate because the

test statistic is the quotient of two mean sums of squares

which have a chi-square distribution.

5. Why is ANOVA helpful?

ANOVAs are helpful because they possess a certain

advantage over a two-sample t-test. Doing multiple two-

sample t-tests would result in a largely increased chance of

committing a type I error. For this reason, ANOVAs are

useful in comparing three or more means.

Simple correlation and Regression

1. What is correlation?

Correlation is a measure of association between two

variables. The variables are not designated as dependent or

independent.

2. What can be the values for correlation coefficient?

The value of a correlation coefficient can vary from -1 to +1.

A -1 indicates a perfect negative correlation and a +1

indicated a perfect positive correlation. A correlation

coefficient of zero means there is no relationship between

the two variables.

3. What is the interpretation of the correlation

coefficient values?

When there is a negative correlation between two variables,

as the value of one variable increases, the value of the other

variable decreases, and vise versa. In other words, for a

negative correlation, the variables work opposite each other.

When there is a positive correlation between two variables,

as the value of one variable increases, the value of the other

variable also increases. The variables move together.

4. What is simple regression?

Simple regression is used to examine the relationship

between one dependent and one independent variable. After

performing an analysis, the regression statistics can be used

to predict the dependent variable when the independent

variable is known. Regression goes beyond correlation by

adding prediction capabilities.

5. Explain the mathematical analysis of regression?

In the regression equation, y is always the dependent

variable and x is always the independent variable. Here are

three equivalent ways to mathematically describe a linear

regression model.

y = intercept + (slope  x) + error

y = constant + (coefficient x) + error

y = a + bx + e

The significance of the slope of the regression line is

determined from the t-statistic. It is the probability that the

observed correlation coefficient occurred by chance if the

true correlation is zero. Some researchers prefer to report

the F-ratio instead of the t-statistic. The F-ratio is equal to

the t-statistic squared.

Business Forecasting

1. What is forecasting?

Forecasting is a prediction of what will occur in the future,

and it is an uncertain process. Because of the uncertainty,

the accuracy of a forecast is as important as the outcome

predicted by the forecast.

2. What are the various business forecasting

techniques?

3. How to model the Causal time series?

With multiple regressions, we can use more than one

predictor. It is always best, however, to be parsimonious,

that is to use as few variables as predictors as necessary to

get a reasonably accurate forecast. Multiple regressions are

best modeled with commercial package such as SAS or

SPSS. The forecast takes the form:

Y = 0 + 1X1 + 2X2 + . . .+ nXn,

where 0 is the intercept, 1, 2, . . . n are coefficients

representing the contribution of the independent variables

X1, X2,…, Xn.

4. What are the various smoothing techniques?

Simple Moving average: The best-known forecasting

methods is the moving averages or simply takes a certain

number of past periods and add them together; then divide

by the number of periods. Simple Moving Averages (MA) is

effective and efficient approach provided the time series is

stationary in both mean and variance. The following formula

is used in finding the moving average of order n, MA(n) for a

period t+1,

MAt+1 = [Dt + Dt-1 + ... +Dt-n+1] / n

where n is the number of observations used in the

calculation.

Weighted Moving Average: Very powerful and economical.

They are widely used where repeated forecasts required-

uses methods like sum-of-the-digits and trend adjustment

methods. As an example, a Weighted Moving Averages is:

Weighted MA(3) = w1.Dt + w2.Dt-1 + w3.Dt-2

where the weights are any positive numbers such that: w1 +

w2 + w3 = 1.

5. Explain exponential smoothing techniques?

Single Exponential Smoothing: It calculates the smoothed

series as a damping coefficient times the actual series plus 1

minus the damping coefficient times the lagged value of the

smoothed series. The extrapolated smoothed series is a

constant, equal to the last value of the smoothed series

during the period when actual data on the underlying series

are available.

Ft+1 =  Dt + (1 - ) Ft

where:

Dt is the actual value

Ft is the forecasted value

 is the weighting factor, which ranges from 0 to 1

t is the current time period.

Double Exponential Smoothing: It applies the process

described above three to account for linear trend. The

extrapolated series has a constant growth rate, equal to the

growth of the smoothed series at the end of the data period.

6. What are time series models?

A time series is a set of numbers that measures the status of

some activity over time. It is the historical record of some

activity, with measurements taken at equally spaced

intervals (exception: monthly) with a consistency in the

activity and the method of measurement.

Time Series Analysis

1. What is time series forecasting?

The time-series can be represented as a curve that evolve

over time. Forecasting the time-series mean that we extend

the historical values into the future where the measurements

are not available yet.

2. What are the different models in time series

forecasting?

Simple moving average

Weighted moving average

Simple exponential smoothing

Holt’s double Exponential smoothing

Winter’s triple exponential smoothing

Forecast by linear regression

3. Explain simple moving average and weighted moving

average models?

Simple Moving average: The best-known forecasting

methods is the moving averages or simply takes a certain

number of past periods and add them together; then divide

by the number of periods. Simple Moving Averages (MA) is

effective and efficient approach provided the time series is

stationary in both mean and variance. The following formula

is used in finding the moving average of order n, MA(n) for a

period t+1,

MAt+1 = [Dt + Dt-1 + ... +Dt-n+1] / n

where n is the number of observations used in the

calculation.

Weighted Moving Average: Very powerful and economical.

They are widely used where repeated forecasts required-

uses methods like sum-of-the-digits and trend adjustment

methods. As an example, a Weighted Moving Averages is:

Weighted MA(3) = w1.Dt + w2.Dt-1 + w3.Dt-2

where the weights are any positive numbers such that: w1 +

w2 + w3 = 1.

4. Explain the exponential smoothing techniques?

Single Exponential Smoothing: It calculates the smoothed

series as a damping coefficient times the actual series plus 1

minus the damping coefficient times the lagged value of the

smoothed series. The extrapolated smoothed series is a

constant, equal to the last value of the smoothed series

during the period when actual data on the underlying series

are available.

Ft+1 = a Dt + (1 - a) Ft

where:

Dt is the actual value

Ft is the forecasted value

a is the weighting factor, which ranges from 0 to 1

t is the current time period.

Double Exponential Smoothing: It applies the process

described above three to account for linear trend. The

extrapolated series has a constant growth rate, equal to the

growth of the smoothed series at the end of the data period.

Triple exponential Smoothing: It applies the process

described above three to account for nonlinear trend.

5. How should one forecast by linear regression?

Regression is the study of relationships among variables, a

principal purpose of which is to predict, or estimate the

value of one variable from known or assumed values of other

variables related to it.

Types of Analysis

Simple Linear Regression: A regression using only one

predictor is called a simple regression.

Multiple Regression: Where there are two or more

predictors, multiple regression analysis is employed.

Index Numbers

1. What are index numbers?

Index numbers are used to measure changes in some

quantity which we cannot observe directly. E.g changes in

business activity.

2. Describe the classification of index numbers?

Index numbers are classified in terms of the variables that

are intended to measure. In business, different groups of

variables in the measurement of which index number

techniques are commonly used are i) price ii) quantity iii)

value iv) Business activity

3. What are simple and composite index numbers?

Simple index numbers: A simple index number is a

number that measures a relative change in a single variable

with respect to a base.

Composite index numbers: A composite index number is a

number that measures an average relative change in a group

of relative variables with respect to a base.

4. What are price index numbers?

Price index numbers measure the relative changes in the

prices of commodities between two periods. Prices can be

retail or wholesale.

5. What are quantity index numbers?

These index numbers are considered to measure changes in

the physical quantity of goods produced, consumed, or sold

of an item or a group of items.

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