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Applied Business Forecasting and
Regression Analysis
Review lectures
Examining Distributions
Introduction

Descriptive Statistics

Methods that organize and summarize data aid in
effective presentation and increased understanding.


Bar charts, tabular displays, various plots of economic data,
averages and percentages.
Often the individuals or objects studied by an
investigator come from a much larger collection, and
the researcher’s interest goes beyond just data
summarization.
Introduction

Population


The entire collection of individuals or objects
about which information is desired.
Sample

A subset of the population selected in some
prescribed manner for study.
Introduction

Inferential Statistics



Involves generalizing from a sample to the population
from which it was selected.
This type of generalization involves some risk, since a
conclusion about the population will be reached based
on the basis of available, but incomplete, information.
An important aspect in the development of inference
techniques involves quantifying the associated risks.
Individuals and variables

Individuals



are the objects described by a set of data.
They may be people, but they may also be
business firms, common stocks, or other
objects.
A Variable


is any characteristic of an individual.
A variable can take different values for
different individuals.
Categorical & Quantitative Variables



A Categorical Variable places an individual
into one of several groups or categories.
A Quantitative Variable takes numerical
values for which arithmetic operations such
as adding and averaging make sense.
The distribution of a variable tell us what
values it takes and how often it takes these
values.
Example
Example
Discrete and Continuous Variable

With numerical data (quantitative
variables), it is useful to make a further
distinction.


Numerical data is discrete if the possible values
are isolated points on the number line.
Numerical data is continuous if the set of
possible values form an entire interval on the
number line.
Stem plot
To make a stem plot:

1.
2.
3.
Separate each observation into a stem consisting of all
but the final (rightmost) digit and a leaf, the final
digit. Stems may have as many digits as needed, but
each leaf contains only a single digit.
Write the stems in a vertical column with the smallest
at the top, and draw a vertical line at the right of this
column.
Write each leaf in the row to the right of its stem, in
increasing order out from the stem.
Stem plot
Frequency Distribution


A frequency distribution for categorical data
is a table that displays the categories,
frequencies, and relative frequencies.
Example

The increasing emphasis on exercise has
resulted in an increase of sport related injuries.
A listing of the 82 sample observations would
look something like this:
F, Sp, Sp, Co, F, L, F, Ch, De, L, Sp, Di, St, Cn,…
Frequency Distribution

The following coding is used:

Sp = Sprain, St = Strain, Di = dislocation,
Co = Contusion, L = laceration,
Cn = Concussion, F = fracture,
Ch = chronic, De = dental
Frequency Distribution
Categories
Sprain
Contusion
Fracture
Strain
Laceration
Chronic
Dislication
Concussion
Dental
Total
Frequency
22
18
17
9
6
4
3
2
1
Relative Frequency
0.268
0.22
0.207
0.11
0.073
0.049
0.037
0.024
0.012
82
1
Bar Graph
Frequency Distribution for Type of Injury
25
20
Count
15
10
5
0
Sprain
Contusion
Fracture
Strain
Laceration
Chronic
Dislication
Concussion
Dental
Pie Chart
Frequency Distribution for type of Injury
4%
2% 1%
5%
27%
7%
Sprain
Contusion
Fracture
Strain
Laceration
11%
Chronic
Dislication
Concussion
Dental
22%
21%
Frequency Distribution for Discrete
Numerical Data



Discrete numerical data almost always
results from counting.
In such cases, each observation is a whole
number.
For example, if the possible values are 0, 1,
2, 3, …, then these are listed in column, and
a running tally is kept as a single pass is
made through the data
Frequency Distribution for Discrete
Numerical Data

Example

A sample of 708 bus drivers employed by
public corporations was selected, and the
number of traffic accidents in which each was
involved during a 4-year period was
determined. A listing of the 708 sample
observations would look something like this:
3, 0, 6, 0, 0, 2, 1, 4, 1, …
Frequency Distribution for Discrete
Numerical Data
Number of Accidents
0
1
2
3
4
5
6
7
8
9
10
11
Frequency
117
157
158
115
78
44
21
7
6
1
3
1
Relative Frequency
0.165
0.22
0.223
0.162
0.11
0.062
0.03
0.01
0.008
0.001
0.004
0.001
Total
708
0.998
Bar Graph
Frequency Distribution for Number of Accidents by Bus Drivers
180
160
140
Count
120
100
80
60
40
20
0
1
2
3
4
5
6
7
Number of Accidents
8
9
10
11
12
Frequency Distributions for
Continuous Data


The difficulty with continuous data, such as
observations on the unemployment rate by
state, is that there is no natural categories.
Therefore we define our own categories. by
marking off some intervals on horizontal
unemployment rate axis as picture below.
1.00
9.00
Frequency Distributions for
Continuous Data


If the smallest rate were 1.5%, and the
largest was 8.9%, we might use the intervals
of width 1% with the first one starting at 1
and the last one ending at 9.
Each data value should fall in exactly one of
these intervals.
Frequency Distributions for
Continuous Data
Frequency Distributions for
Continuous Data
Unemployment rate Intervals
[1, 2)
[2, 3)
[3, 4)
[4, 5)
[5, 6)
[6, 7)
[7, 8)
[8, 9)
Total
Frequency
2
13
21
10
3
1
0
1
Relative Frequency
0.039
0.255
0.412
0.196
0.059
0.020
0.000
0.020
51
1.000
Histograms




Mark the boundaries of the class intervals
on a horizontal axis.
Draw a vertical scale marked with either
relative frequencies or frequencies.
The rectangle corresponding to a particular
interval is drawn directly above the interval.
The height of each rectangle is then the
class frequency or relative frequency.
Histograms
Histograms
Examining a Distribution



In any graph of data, look for overall pattern
and for striking deviation from that pattern.
You can describe the overall pattern of a
histogram by its shape, center, and spread.
An important kind of deviation is an outlier,
an individual value that falls outside the
overall pattern.
Symmetric & Skewed Distribution



A distribution is symmetric if the right and left
sides of the histogram are approximately mirror
images of each other.
A distribution is skewed to the right if the right
side of the histogram ( containing the half of the
observations with larger values) extends much
farther out than the left side.
It is skewed to the left if the left side of the
histogram extends much farther out than the right
side.
Symmetric Distribution
Skewed to the Right
Symmetric Distribution
Numerical Summary Measures

Describing the center of a data set.



Mean
Median
Describing the variability in a data set.


Variance, standard deviation
Quartiles
The Mean X

To find the mean of a set of observations, add
their values and divide by the number of
observations. If the n observations are
x1 , x2 ,, xn , their mean is
X

x1  x2   xn
n
In a more compact notation,
x

X
i
n
The Median
The Median M is the midpoint of a distribution,
the number such that half of the observations are
smaller and the other half are larger. To find the
median of a distribution:

1.
2.
3.
Arrange all observations in order of size, from
smallest to largest.
If the number of observations n is odd, the median M
is the center observation in the ordered list.
If the number of observations n is even, the median
M is the mean of the two center observations in the
ordered list.
The Quartiles Q1 and Q3
To calculate the quartiles:

1.
2.
3.
Arrange the observations in increasing order and
locate the median M in the ordered list of
observations.
The first quartile Q1 is the median of the observations
whose position in the ordered list is to the left of the
location of the overall median.
The third quartile Q3 is the median of the
observations whose position in the ordered list is to
the right of the location of the overall median.
The Five Number Summary and
Box-Plot

The five number summary of a distribution
consists of the smallest observation, the first
quartile, the median, the third quartile, and
the largest observation, written in order
from smallest to largest. In symbols, the
five number summary is
Minimum
Q1
M
Q3
Maximum
The Five Number Summary and
Box-Plot

A box-plot is a graph of the five number
Summary.




A central box spans the quartiles.
A line in the box marks the median.
Lines extend from the box out to the smallest
and largest observations.
Box-plots are most useful for side-by-side
comparison of several distributions.
Example
The Standard Deviation s

The Variance s2 of a set of observations is the
average of the squares of the deviations of the
observations from their mean. In symbols, the
variance of n observations x , x ,, x is
1
2
( x1  x ) 2  ( x2  x ) 2  ( xn  x ) 2
s 
n 1
2
or, more compactly,
2
(
x
)

i
2
2
x

i
( xi  x )

2
n
s 

n 1
n 1
n
The Standard Deviation s

The standard deviation s is the square root
of the variance s2:
( x )
x 

(
x

x
)

n
s

2
2
i
n 1
i
2
i
n 1
Choosing a Summary

The five number summary is usually better
than the mean and standard deviation for
describing a skewed distribution or a
distribution with extreme outliers. Use x ,
and s only for reasonably symmetric
distributions that are free of outliers.
Strategies for Exploring Data

Plot the data


Make a graph, usually a histogram or a stemplot.
Look at the distribution of the variable for:




overall pattern (shape, center, spread).
striking deviations such as outliers.
Calculate a numerical summary to briefly
describe center and spread.
Describe the overall pattern with a smooth
curve.
Density Curves


Sometimes the overall pattern (the
distribution of the variable) of a large
number of observations is so regular that we
can describe it by a smooth curve, called
Density curve.
The curve is a mathematical model for the
distribution.
Density Curve


Histogram of the city
gas mileage (miles per
gallon) of 856, 2001
model year motor
vehicle.
The smooth curve,
density curve, shows
the overall shape of
the distribution.
Density Curve

The proportion of cars
with gas mileage less
than 20 from the
histogram is
384
 .449  44.9%
856
Density Curve


The proportion of cars
with gas mileage less
than 20 from the
density curve is .410
The area under the
density curve gives a
good approximation of
areas given by
histogram.
Density Curve

A density curve is a curve that




Is always on or above the horizontal axis.
Has area exactly 1 underneath it.
A density curve describes the overall
pattern of a distribution.
The area under the curve and above any
range of values is the proportion of all
observations that fall in that range.
Median and mean of a Density
Curve

The median of a
density curve is the
point that divides the
area under the curve in
Half.
Median and Mean of a Density
Curve

The mean of a density
curve is the balance
point, at which the
curve would balance if
made of solid material.
Median and Mean of a Density
Curve


The median and mean
are the same for a
symmetric density
curve.
They both are at the
center of the curve.
Median and Mean of a Density
Curve

The mean of a skewed
curve is pulled away
from the median in the
direction of the long
tail.
Normal Density Curve

These density curves,
called normal curves,
are




Symmetric
Single peaked
Bell shaped
Normal curves
describe normal
distributions.
Normal Density Curve



The exact density curve for a particular
normal distribution is described by giving
its mean  and its standard deviation .
The mean is located at the center of the
symmetric curve and it is the same as the
median.
The standard deviation  controls the spread
of a normal curve.
Normal Density Curve
The 68-95-99.7 Rule


Although there are many normal curve, They all
have common properties. In particular, all Normal
distributions obey the following rule.
In a normal distribution with mean  and standard
deviation :



68% of the observations fall within  of the mean .
95% of the observations fall within 2 of .
99.7% of the observations fall within 3 of .
The 68-95-99.7 Rule
The 68-95-99.7 Rule
Standard Normal Distribution

The standard Normal
distribution is the
Normal distribution
N(0, 1) with mean
 = 0 and standard
deviation  =1.
The standard Normal Table


What is the area under
the standard normal
curve between z = 0
and z = 2.3?
Compact notation:
p(0  z  2.3)

P = .9893 - .5 =.4893
Finding the area under a normal curve
1.
2.
3.
4.
State the problem in terms of the observed
variable x.
Standardize x to restate the problem in terms of
a standard normal variable z
Draw a picture to show the area under the
standard Normal curve.
Find the required area under the standard
Normal curve Using table A and the fact that the
total area under the curve is 1.
Example
The annual rate of return on stock indexes (which
combine many individual stocks) is approximately
Normal. Since 1954, the Standard & Poor’s 500
stock index has had a mean yearly return of about
12%, with standard deviation of 16.5%. Take this
Normal distribution to be the distribution of yearly
returns over a long period. The market is down for
the year if the return on the index is less than zero.
In what proportion of years is the market down?
Example

State the problem


Call the annual rate of return for Standard & Poor’s
500-stocks Index x. The variable x has the N(12, 16.5)
distribution. We want the proportion of years with
X < 0.
Standardize

Subtract the mean, then divide by the standard
deviation, to turn x into a standard Normal z:
x0
x  12 0  12

16.5
16.5
z  .73
Example


Draw a picture to show
the standard normal
curve with the area of
interest shaded.
Use the table


The proportion of
observations less than
- 0.73 is .2327.
The market is down on an
annual basis about
23.27% of the time.
Example

What percent of years have annual return
between 12% and 50%?

State the problem
12  x  50

Standardize
12  12 x  12 50  12


16.5
16.5
16.5
0  z  2.30
Example


Draw a picture.
Use table.

The area between 0
and 2.30 is the area
below 2.30 minus the
area below 0.
0.9893- .50 = .4893
Finding a Value when Given a
Proportion


Sometimes we may want to find the
observed value with a given proportion of
observations above or below it.
To do this, use table A backward. Find the
given proportion in the body of the table,
read the corresponding z from the left
column and top row, then unstandardize to
get the observed value.
Example

Miles per gallon ratings of compact cars
(2001 model year) follow approximately the
N(25.7, 5.88) distribution. How many miles
per gallon must a vehicle get to place in the
top 10% of all 2001 model year compact
cars?
Example

We want to find the miles
per gallon rating x with
area 0.1 to its right under
the Normal Curve with
mean 25.7 and standard
deviation 5.88. That is the
same as finding the miles
per gallon rating x with
area 0.9 to its left.
Example

Look in the body of
Table A for the entry
closest to 0.9. It is
0.8997. This is the
entry corresponding to
z = 1.28.
Example

Unstandardize to transform the solution
from the z back to the original x scale.
x

z
x  25.7
 1.28
5.88
x  25.5  (1.28)( 5.88)  33.2
Standard Normal Distribution

If a variable x has any normal distribution N(, )
with mean  and standard deviation , then the
standardized variable
z

x

has the standard Normal distribution.
This standardized value is often called z-score.
The standard Normal Table


Table A is a table of area
under the standard Normal
curve. The table entry for
each value z is the area
under the curve to the left
of z.
Or you can use the applet
at the following site.
http:/www.stat.sc.edu~west/applet
s/normaldemo.html
The standard Normal Table


What is the area under
the standard normal
curve to the right of
z = - 2.15?
Compact notation:
p ( z  2.15)

P = 1 - .0158 =.9842