Business Statistics for Managers with SPSS[1].pptx

Business Statistics for
Managers
Dr. D. Pradeep Kumar

Statistics for Business and
Economics
Chapter 1
Statistics, Data, &
Statistical Thinking

Contents
1. The Science of Statistics
2. Types of Statistical Applications in Business
3. Fundamental Elements of Statistics
4. Processes
5. Types of Data
6. Collecting Data
7. The Role of Statistics in Managerial Decision
Making

Learning Objectives
1. Introduce the field of statistics
2. Demonstrate how statistics applies to business
3. Establish the link between statistics and data
4. Identify the different types of data and data-
collection methods
5. Differentiate between population and sample data
6. Differentiate between descriptive and inferential
statistics

What Is Statistics?
Why?
1. Collecting Data
e.g., Survey
2. Presenting Data
e.g., Charts & Tables
3. Characterizing Data
e.g., Average
Data
Analysis
Decision-
Making
© 1984-1994 T/Maker Co.
© 1984-1994 T/Maker Co.

What Is Statistics?
Statistics is the science of data. It involves collecting,
classifying, summarizing, organizing, analyzing, and
interpreting numerical information.

1.2
Types of Statistical Applications in
Business

Application Areas
• Economics
• Forecasting
• Demographics
• Sports
• Individual & Team
Performance
• Engineering
• Construction
• Materials
• Business
• Consumer Preferences
• Financial Trends

Statistics: Two Processes
Describing sets of data
and
Drawing conclusions (making estimates, decisions,
predictions, etc. about sets of data based on sampling)

Statistical Methods
Statistical
Methods
Descriptive
Statistics
Inferential
Statistics

Descriptive Statistics
1. Involves
• Collecting Data
• Presenting Data
• Characterizing Data
2. Purpose
• Describe Data
X = 30.5 S2
= 113
0
25
50
Q1 Q2 Q3 Q4
$

1. Involves
• Estimation
• Hypothesis
Testing
2. Purpose
• Make decisions about
population characteristics
Inferential Statistics
Population?

1.3
Fundamental Elements
of Statistics

1. Experimental unit
• Object upon which we collect data
2. Population
• All items of interest
3. Variable
• Characteristic of an individual
experimental unit
4. Sample
• Subset of the units of a population
• P in Population
& Parameter
• S in Sample
& Statistic

1. Statistical Inference
• Estimate or prediction or generalization about a
population based on information contained in a sample
2. Measure of Reliability
• Statement (usually qualified) about the degree of
uncertainty associated with a statistical inference

Four Elements of Descriptive
Statistical Problems
1. The population or sample of interest
2. One or more variables (characteristics of the
population or sample units) that are to be
investigated
3. Tables, graphs, or numerical summary tools
4. Identification of patterns in the data

Five Elements of Inferential
Statistical Problems
1. The population of interest
2. One or more variables (characteristics of the
population units) that are to be investigated
3. The sample of population units
4. The inference about the population based on
information contained in the sample
5. A measure of reliability for the inference

Process
A process is a series of actions or operations that
transforms inputs to outputs. A process produces or
generates output over time.

Process
A process whose operations or actions are unknown or
unspecified is called a black box.
Any set of output (object or numbers) produced by a
process is called a sample.

Types of Data
Quantitative data are measurements that are recorded
on a naturally occurring numerical scale.
Qualitative data are measurements that cannot be
measured on a natural numerical scale; they can only be
classified into one of a group of categories.

Types of Data
Types of
Data
Quantitative
Data
Qualitative
Data

Quantitative Data
Measured on a numeric
scale.
• Number of defective
items in a lot.
• Salaries of CEOs of
oil companies.
• Ages of employees at
a company.
3
52
71
4
8
943
120 12
21

Qualitative Data
Classified into categories.
• College major of each
student in a class.
• Gender of each employee
at a company.
• Method of payment
(cash, check, credit card).
$ Credit

Obtaining Data
1. Data from a published source
2. Data from a designed experiment
3. Data from a survey
4. Data collected observationally

Obtaining Data
Published source:
book, journal, newspaper, Web site
Designed experiment:
researcher exerts strict control over units
Survey:
a group of people are surveyed and their responses are
recorded
Observation study:
units are observed in natural setting and variables of
interest are recorded

Samples
A representative sample exhibits characteristics typical
of those possessed by the population of interest.
A random sample of n experimental units is a sample
selected from the population in such a way that every
different sample of size n has an equal chance of
selection.

Random Sample
Every sample of size n has an equal chance of selection.

1.7
The Role of Statistics in
Managerial Decision Making

Statistical Thinking
Statistical thinking involves applying rational thought
and the science of statistics to critically assess data and
inferences. Fundamental to the thought process is that
variation exists in populations and process data.
A random sample of n experimental units is a sample
selected from the population in such a way that every
different sample of size n has an equal chance of
selection.

Nonrandom Sample Errors
Selection bias results when a subset of the
experimental units in the population is excluded so
that these units have no chance of being selected for
the sample.
Nonresponse bias results when the researchers
conducting a survey or study are unable to obtain data
on all experimental units selected for the sample.
Measurement error refers to inaccuracies in the
values of the data recorded. In surveys, the error may
be due to ambiguous or leading questions and the
interviewer’s effect on the respondent.

Statistical
Computer Packages
1. Typical Software
• SPSS
• MINITAB
• Excel
2. Need Statistical
Understanding
• Assumptions
• Limitations

Key Ideas
Types of Statistical Applications
Descriptive
1. Identify population and sample (collection of
experimental units)
2. Identify variable(s)
3. Collect data
4. Describe data

Key Ideas
Types of Statistical Applications
Inferential
1. Identify population (collection of all
experimental units)
2. Identify variable(s)
3. Collect sample data (subset of population)
4. Inference about population based on sample
5. Measure of reliability for inference

Key Ideas
Types of Data
1. Quantitative (numerical in nature)
2. Qualitative (categorical in nature)

Key Ideas
Data-Collection Methods
1. Observational
2. Published source
3. Survey
4. Designed experiment

Key Ideas
Problems with Nonrandom Samples
1. Selection bias
2. Nonresponse bias
3. Measurement error

The mean, median, and mode are measures of central tendency that are used to identify the core position of a data
set. They are applied in different situations depending on the type of data and the level of measurement:
•Nominal data: The mode is the only appropriate measure of central tendency to use. The mode is the most frequent value in the data set.
•Ordinal data: The median or mode is usually the best choice. The median is the value in the middle of the data set.
•Interval or ratio data: The mean, median, and mode can all be used. The mean is the average value.
•Skewed distribution: The median is often the best measure of central tendency.
•Symmetrical distribution for continuous data: The mean, median, and mode are all equal.
•Data with extreme scores: The median is preferred because a single outlier can have a big effect on the mean.
•Data with missing or undetermined values: The median is preferred.
The mean is the most commonly used measure of central tendency, but the best measure depends on the type of data.

43
Measures of Central
Tendency
Greg C Elvers, Ph.D.

44
Measures of Central Tendency
• A measure of central tendency is a descriptive statistic that describes
the average, or typical value of a set of scores
• There are three common measures of central tendency:
• the mode
• the median
• the mean

45
The Mode
• The mode is the score
that occurs most
frequently in a set of
data
0
1
2
3
4
5
6
75 80 85 90 95
Score on Exam 1
Frequency

46
Bimodal Distributions
• When a distribution
has two “modes,” it is
called bimodal
0
1
2
3
4
5
6
75 80 85 90 95
Score on Exam 1
Frequency

47
Multimodal Distributions
• If a distribution has
more than 2 “modes,”
it is called multimodal
0
1
2
3
4
5
6
75 80 85 90 95
Score on Exam 1
Frequency

48
When To Use the Mode
• The mode is not a very useful measure of central
tendency
• It is insensitive to large changes in the data set
• That is, two data sets that are very different from each other
can have the same mode
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
0
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100

49
When To Use the Mode
• The mode is primarily used with nominally scaled
data
• It is the only measure of central tendency that is
appropriate for nominally scaled data

50
The Median
• The median is simply another name for the 50th
percentile
• It is the score in the middle; half of the scores are larger than the median and
half of the scores are smaller than the median

51
How To Calculate the Median
• Conceptually, it is easy to calculate the median
• There are many minor problems that can occur; it is best to let a computer do
it
• Sort the data from highest to lowest
• Find the score in the middle
• middle = (N + 1) / 2
• If N, the number of scores, is even the median is the average of the middle
two scores

52
Median Example
• What is the median of the following scores:
10 8 14 15 7 3 3 8 12 10 9
• Sort the scores:
15 14 12 10 10 9 8 8 7 3 3
• Determine the middle score:
middle = (N + 1) / 2 = (11 + 1) / 2 = 6
• Middle score = median = 9

53
Median Example
• What is the median of the following scores:
24 18 19 42 16 12
• Sort the scores:
42 24 19 18 16 12
• Determine the middle score:
middle = (N + 1) / 2 = (6 + 1) / 2 = 3.5
• Median = average of 3rd
and 4th
scores:
(19 + 18) / 2 = 18.5

Median Example for
Discrete frequency
x: 1 2 3 4 5 6 7 8 9
F: 8 10 11 16 20 25 15 9 6
x f CF
1 8 8
2 10 8+10=18
3 11 18+11=29
4 16 45
5 20 65
6 25 90
7 15 105
8 9 114
9 6 120
The median class is 65
N=Σ fi =120
N/2= 120/2=60
The CF just greater than (N/2=60)
is 65
Median=5
Median for continuous frequency distribution
Wages : 2000-3000 3000-4000 4000-5000 5000-6000 6000-7000
No.of workers : 3 5 20 10 5
wages
no.of
workers cf
2000-3000 3 3
3000-4000 5 8
4000-5000 20 28
5000-6000 10 38
6000-7000 5 43
N=Σ fi =43
N/2= 43/2=21.5
The CF just greater than (N/2=21.5) is 28
The corresponding interval is 4000-5000
Median= L+h/2(N/2-c.f)
L = limit of the median class
f = frequency of Median class
h =Magnitude of Median class
CF = The cf of the class preceeding the median class
Median= 4000+(1000/2)(21.5-8)
4000+500(13.5)
4675

55
When To Use the Median
• The median is often used when the distribution of scores is either
positively or negatively skewed
• The few really large scores (positively skewed) or really small scores
(negatively skewed) will not overly influence the median

56
The Mean
• The mean is:
• the arithmetic average of all the scores
(X)/N
• the number, m, that makes (X - m) equal to 0
• the number, m, that makes (X - m)2
a minimum
• The mean of a population is represented by the Greek letter ; the
mean of a sample is represented by X

57
Calculating the Mean
• Calculate the mean of the following data:
1 5 4 3 2
• Sum the scores (X):
1 + 5 + 4 + 3 + 2 = 15
• Divide the sum (X = 15) by the number of scores (N = 5):
15 / 5 = 3
• Mean = X = 3

Calculating the Mean for discrete data
x= 1 2 3 4 5 6 7
Fi= 5 9 12 17 14 10 6
Mean= Xifi/ fi
=299/73
=4.06
xi fi fi*xi
1 5 5
2 9 18
3 12 36
4 17 68
5 14 70
6 10 60
7 6 42

59
When To Use the Mean
• You should use the mean when
• the data are interval or ratio scaled
• Many people will use the mean with ordinally scaled data too
• and the data are not skewed
• The mean is preferred because it is sensitive to every score
• If you change one score in the data set, the mean will change

60
Relations Between the Measures of Central
Tendency
• In symmetrical
distributions, the median
and mean are equal
• For normal distributions,
mean = median = mode
• In positively skewed
distributions, the mean is
greater than the median
In negatively skewed
distributions, the mean is
smaller than the median

61
Measures of Dispersion
Greg C Elvers, Ph.D.

62
Definition
• Measures of dispersion are descriptive statistics that describe how
similar a set of scores are to each other
• The more similar the scores are to each other, the lower the measure of
dispersion will be
• The less similar the scores are to each other, the higher the measure of
dispersion will be
• In general, the more spread out a distribution is, the larger the measure of
dispersion will be

63
• Which of the
distributions of scores
has the larger
dispersion?
0
25
50
75
100
125
1 2 3 4 5 6 7 8 9 10
0
25
50
75
100
125
1 2 3 4 5 6 7 8 9 10
The upper distribution
has more dispersion
because the scores are
more spread out
That is, they are less
similar to each other

64
• There are three main measures of dispersion:
• The range
• The semi-interquartile range (SIR)
• Variance / standard deviation

65
The Range
• The range is defined as the difference between the largest score in
the set of data and the smallest score in the set of data, XL - XS
• What is the range of the following data:
4 8 1 6 6 2 9 3 6 9
• The largest score (XL) is 9; the smallest score (XS) is 1; the range is XL -
XS = 9 - 1 = 8

66
When To Use the Range
• The range is used when
• you have ordinal data or
• you are presenting your results to people with little or no knowledge of
statistics
• The range is rarely used in scientific work as it is fairly insensitive
• It depends on only two scores in the set of data, XL and XS
• Two very different sets of data can have the same range:
1 1 1 1 9 vs 1 3 5 7 9

67
The Semi-Interquartile Range
• The semi-interquartile range (or SIR) is defined as the difference of
the first and third quartiles divided by two
• The first quartile is the 25th
percentile
• The third quartile is the 75th
percentile
• SIR = (Q3 - Q1) / 2

68
SIR Example
• What is the SIR for the
data to the right?
• 25 % of the scores are
below 5
• 5 is the first quartile
• 25 % of the scores are
above 25
• 25 is the third quartile
• SIR = (Q3 - Q1) / 2 = (25 -
5) / 2 = 10
2
4
6
 5 = 25th
%tile
8
10
12
14
20
30
 25 = 75th
%tile
60

69
When To Use the SIR
• The SIR is often used with skewed data as it is insensitive to the
extreme scores

70
Variance
• Variance is defined as the average of the square
deviations:
 
N
X
2
2  




71
What Does the Variance Formula Mean?
• First, it says to subtract the mean from each of the scores
• This difference is called a deviate or a deviation score
• The deviate tells us how far a given score is from the typical, or average, score
• Thus, the deviate is a measure of dispersion for a given score

72
• Why can’t we simply take the average of the
deviates? That is, why isn’t variance defined as:
 
N
X
2  



This is not the formula
for variance!

73
• One of the definitions of the mean was that it always made the sum
of the scores minus the mean equal to 0
• Thus, the average of the deviates must be 0 since the sum of the
deviates must equal 0
• To avoid this problem, statisticians square the deviate score prior to
averaging them
• Squaring the deviate score makes all the squared scores positive

74
• Variance is the mean of the squared deviation scores
• The larger the variance is, the more the scores deviate, on average,
away from the mean
• The smaller the variance is, the less the scores deviate, on average,
from the mean

75
Standard Deviation
• When the deviate scores are squared in variance, their
unit of measure is squared as well
• E.g. If people’s weights are measured in pounds, then the
variance of the weights would be expressed in pounds2
(or
squared pounds)
• Since squared units of measure are often awkward to
deal with, the square root of variance is often used
instead
• The standard deviation is the square root of variance

76
Standard Deviation
• Standard deviation = variance
• Variance = standard deviation2

77
Computational Formula
• When calculating variance, it is often easier to use
a computational formula which is algebraically
equivalent to the definitional formula:
 
 
N
N
N X
X
X  







2
2
2
2
2
is the population variance, X is a score,  is the
population mean, and N is the number of scores

78
Computational Formula Example
X X2
X- (X-)2
9 81 2 4
8 64 1 1
6 36 -1 1
5 25 -2 4
8 64 1 1
6 36 -1 1
 = 42  = 306  = 0  = 12

79
Computational Formula Example
 
2
6
12
6
294
306
6
6
306 42
2
2
2
2










N
N
X
X


80
Variance of a Sample
• Because the sample mean is not a perfect estimate
of the population mean, the formula for the
variance of a sample is slightly different from the
formula for the variance of a population:
 
1
N
X
X
s
2
2


 
s2
is the sample variance, X is a score, X is the
sample mean, and N is the number of scores

81
Measure of Skew
• Skew is a measure of symmetry in the distribution
of scores
Positive Skew Negative Skew
Normal (skew = 0)

82
Measure of Skew
• The following formula can be used to determine
skew:
 
 
N
N
X
X
X
X
s 2
3
3
 
 


83
Measure of Skew
• If s3
< 0, then the distribution has a negative skew
• If s3
> 0 then the distribution has a positive skew
• If s3
= 0 then the distribution is symmetrical
• The more different s3
is from 0, the greater the skew in the
distribution

84
Kurtosis
(Not Related to Halitosis)
• Kurtosis measures whether the scores are spread
out more or less than they would be in a normal
(Gaussian) distribution
Mesokurtic (s4
= 3)
Leptokurtic (s4
> 3) Platykurtic (s4
< 3)

85
Kurtosis
• When the distribution is normally distributed, its kurtosis equals 3
and it is said to be mesokurtic
• When the distribution is less spread out than normal, its kurtosis is
greater than 3 and it is said to be leptokurtic
• When the distribution is more spread out than normal, its kurtosis is
less than 3 and it is said to be platykurtic

86
Measure of Kurtosis
• The measure of kurtosis is given by:
 
N
N
X
X
X
X
s
4
2
4

 

















87
s2
, s3
, & s4
• Collectively, the variance (s2
), skew (s3
), and kurtosis (s4
) describe the
shape of the distribution

Karl Pearson’s coefficient of skewness Bowley’s coefficient of skewness
It is based on mean, mode and standard deviation. It is based on quartiles.
It is the usual method of finding coefficient of skewness. It is usually used when difference between
quartiles are given.
Skewness = mean – mode Skewness = Q3 + Q1 – 2Median
Coefficient of Skewness by Karl Pearson’s method = mean-
mode / standard deviation
Coefficient of Skewness by Bowley’s method =
Q3 + Q1 – 2Median / Q3 - Q1
Tip
Coefficient of Skewness by Karl Pearson’s method = mean- mode / standard deviation coefficient of
Skewness by Bowley’s method = Q3 + Q1 – 2Median / Q3 - Q1
Explanation
Final Answer
Karl Pearson’s method: It is the usual method of finding the coefficient of skewness. It is based on mean,
mode and standard deviation.
Coefficient of Skewness by Karl Pearson’s method = mean- mode / standard deviation.Bowley’s method: It
is usually used
when the difference between quartiles are given. It is based on quartiles. Coefficient of Skewness by
Bowley’s method = Q3 + Q1 – 2Median / Q3 - Q1

Caluculate karlpearsons co-efficient for following data
X: 20 30 40 50 60 70
f: 8 12 20 10 6 4
Skp=M-M0/
Mean=  Fixi/  Fi
=2460/60=41
Mode=40
Standard deviation
=13.7
skp=41-40/13.7=0.07
X Fi XiFi X2
X2
F
20 8 160 400 3200
30 12 360 900 10800
40 20 800 1600 32000
50 10 500 2500 25000
60 6 360 3600 21600
70 4 280 4900 19600
  



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