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Introduction and
Descriptive Statistics
WHAT IS STATISTICS?
STATISTICS The science of collecting,
organizing, presenting, analyzing, and interpreting
data to assist in making more effective decisions.



Statistics is a science that helps us make better decisions in
business, economics and finance as well as in other fields.
Statistics teaches us how to summarize, analyze, and draw
meaningful inferences from data that then lead to improve
decisions.
These decisions that we make help us improve the running,
for example, a department, a company, the entire economy,
etc.
Using Statistics (Two Categories)

Descriptive Statistics





Collect
Organize
Summarize
Display
Analyze

Inferential Statistics
 Predict and forecast
value of population
parameters
 Test hypothesis about
value of population
parameter based on
sample statistic
 Make decisions
Samples and Populations

A population consists of the set of all
measurements for which the investigator is
interested.

A sample is a subset of the measurements selected
from the population.

A census is a complete enumeration of every item
in a population.
Why Sample?
Census of a population may be:
 Impossible
 Impractical
 Too costly
Parameter Versus Statistic
PARAMETER A measurable characteristic of a
population.
STATISTIC A measurable characteristic of a
sample.
Types of Data - Two Types

Qualitative Categorical or
Nominal:
Examples are-

Quantitative Measurable or
Countable:
Examples are-
 Color
 Temperatures
 Gender
 Salaries
 Nationality
 Number
of points
scored on a 100
point exam
Scales of Measurement
•
Nominal Scale - groups or classes
 Gender
•
Ordinal Scale - order matters
 Ranks
•
(top ten videos)
Interval Scale - difference or distance matters –
has arbitrary zero value.
 Temperatures (0F, 0C),
•
Likert Scale
Ratio Scale - Ratio matters – has a natural zero
value.
 Salaries
Population Mean
For ungrouped data, the population mean is the sum of all the population values divided by the
total number of population values. The sample mean is the sum of all the sample values divided
by the total number of sample values.
EXAMPLE:
The Median
MEDIAN The midpoint of the values after they have been
ordered from the smallest to the largest, or the largest to the
smallest.
1.
2.
3.
4.
PROPERTIES OF THE MEDIAN
There is a unique median for each data set.
It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when
such values occur.
It can be computed for ratio-level, interval-level, and ordinal-level data.
It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class.
EXAMPLES:
The ages for a sample of five college students are:
21, 25, 19, 20, 22
The heights of four basketball players, in inches, are:
76, 73, 80, 75
Arranging the data in ascending order gives:
Arranging the data in ascending order gives:
73, 75, 76, 80.
19, 20, 21, 22, 25.
Thus the median is 21.
Thus the median is 75.5
The Mode
MODE The value of the observation that appears
most frequently.
Measures of Dispersion



A measure of location, such as the mean or the median, only describes the center of the data. It is valuable
from that standpoint, but it does not tell us anything about the spread of the data.
For example, if your nature guide told you that the river ahead averaged 3 feet in depth, would you want to
wade across on foot without additional information? Probably not. You would want to know something about
the variation in the depth.
A second reason for studying the dispersion in a set of data is to compare the spread in two or more
distributions.

RANGE

MEAN DEVIATION

VARIANCE AND
STANDARD DEVIATION
EXAMPLE – Mean Deviation
EXAMPLE:
The number of cappuccinos sold at the Starbucks location in the Orange Country Airport between 4 and 7
p.m. for a sample of 5 days last year were 20, 40, 50, 60, and 80. Determine the mean deviation for the
number of cappuccinos sold.
Step 1: Compute the mean
x
x
n

20  40  50  60  80
 50
5
Step 2: Subtract the mean (50) from each of the observations, convert to positive if difference is negative
Step 3: Sum the absolute differences found in step 2 then divide by the number of observations
Variance and Standard Deviation
VARIANCE The arithmetic mean of the squared deviations from the mean.
STANDARD DEVIATION The square root of the variance.




The variance and standard deviations are nonnegative and are zero only if all
observations are the same.
For populations whose values are near the mean, the variance and standard
deviation will be small.
For populations whose values are dispersed from the mean, the population
variance and standard deviation will be large.
The variance overcomes the weakness of the range by using all the values in the
population
EXAMPLE – Population Variance and Population
Standard Deviation
The number of traffic citations issued during the last twelve months in Beaufort County, South Carolina, is reported
below:
What is the population variance?
Step 1: Find the mean.
 
x
N

19  17  ...  34  10
348

 29
12
12
Step 2: Find the difference between each observation and the mean, and square that difference.
Step 3: Sum all the squared differences found in step 3
Step 4: Divide the sum of the squared differences by the number of items in the population.
2 
 ( X  )
N
2

1,488
 124
12
Sample Variance and
Standard Deviation
Where :
s 2 is the sample variance
X is the value of each observatio n in the sample
X is the mean of the sample
n is the number of observatio ns in the sample
EXAMPLE
The hourly wages for a sample of
part-time employees at Home
Depot are: $12, $20, $16, $18,
and $19.
What is the sample variance?
The Arithmetic Mean and Standard
Deviation of Grouped Data
EXAMPLE:
Determine the arithmetic mean vehicle selling
price given in the frequency table below.
EXAMPLE
Compute the standard deviation of the vehicle selling prices in the frequency table
below.
Group Data and the Histogram

Dividing data into groups or classes or intervals

Groups should be:

Mutually exclusive
 Not overlapping - every observation is assigned to only one
group

Exhaustive
 Every observation is assigned to a group

Equal-width (if possible)
 First or last group may be open-ended
Frequency Distribution

Table with two columns listing:


Each and every group or class or interval of values
Associated frequency of each group
 Number of observations assigned to each group
 Sum of frequencies is number of observations




N for population
n for sample
Class midpoint is the middle value of a group or class or
interval
Relative frequency is the percentage of total observations
in each class

Sum of relative frequencies = 1
Example : Frequency Distribution
x
Spending Class ($)
0 to less than 100
100 to less than 200
200 to less than 300
300 to less than 400
400 to less than 500
500 to less than 600
f(x)
Frequency (number of customers)
f(x)/n
Relative Frequency
30
38
50
31
22
13
0.163
0.207
0.272
0.168
0.120
0.070
184
1.000
• Example of relative frequency: 30/184 = 0.163
• Sum of relative frequencies = 1
Cumulative Frequency Distribution
x
Spending Class ($)
0 to less than 100
100 to less than 200
200 to less than 300
300 to less than 400
400 to less than 500
500 to less than 600
F(x)
Cumulative Frequency
30
68
118
149
171
184
F(x)/n
Cumulative Relative Frequency
0.163
0.370
0.641
0.810
0.929
1.000
The cumulative frequency of each group is the sum of the
frequencies of that and all preceding groups.
Histogram

A histogram is a chart made of bars of different heights.


Widths and locations of bars correspond to widths and locations of data
groupings
Heights of bars correspond to frequencies or relative frequencies of data
groupings
Histogram Example
Frequency Histogram
Relative Frequency Histogram
Chebyshev’s Theorem and Empirical Rule
Quartiles, Deciles and Percentiles

The standard deviation is the most widely used measure of dispersion.

Alternative ways of describing spread of data include determining the location of values
that divide a set of observations into equal parts.

These measures include quartiles, deciles, and percentiles.

To formalize the computational procedure, let Lp refer to the location of a desired percentile.
So if we wanted to find the 33rd percentile we would use L33 and if we wanted the median,
the 50th percentile, then L50.

The number of observations is n, so if we want to locate the median, its position is at (n +
1)/2, or we could write this as (n + 1)(P/100), where P is the desired percentile
Percentiles - Example
EXAMPLE
Listed below are the commissions earned last month by a sample of 15 brokers at Salomon Smith Barney’s Oakland,
California, office.
$2,038
$1,758
$1,940
$1,721
$2,311
$1,637
$2,054
$2,097
$2,406
$2,047
$1,471
$2,205
$1,460
$1,787 $2,287
Locate the median, the first quartile, and the third quartile for the commissions earned.
Step 1: Organize the data from lowest to largest value
$1,460
$2,038
$2,406
$1,471
$2,047
$1,637
$2,054
$1,721
$2,097
$1,758
$2,205
$1,787
$2,287
Step 2: Compute the first and third quartiles. Locate L25 and L75 using:
25
75
4
L75  (15  1)
 12
100
100
Therefore, the first and third quartiles are located at the 4th and 12th
L25  (15  1)
positions, respective ly
L25  $1,721
L75  $2,205
$1,940
$2,311
Measures of Variability or Dispersion

Range
 Difference

Interquartile Range
 Difference

between maximum and minimum values
between third and first quartile (Q3 - Q1)
Variance
 Average*of

Standard Deviation
 Square

the squared deviations from the mean
root of the variance
Definitions of population variance and sample variance differ slightly
.
Skewness



Another characteristic of a set of data is the shape.
There are four shapes commonly observed: symmetric, positively skewed, negatively skewed, bimodal.
The coefficient of skewness can range from -3 up to 3.
 A value near -3, indicates considerable negative skewness.
 A value such as 1.63 indicates moderate positive skewness.
 A value of 0, which will occur when the mean and median are equal, indicates the distribution is symmetrical and that
there is no skewness present.
The Relative Positions of the Mean,
Median and the Mode
Methods of Displaying Data
Pie Charts
 Categories represented as percentages of total
 Bar Graphs
 Heights of rectangles represent group frequencies
 Frequency Polygons
 Height of line represents frequency
 Ogives
 Height of line represents cumulative frequency
 Time Series Plots
 Represents values over time
• Stem-and-Leaf Displays
 Quick listing of all observations
 Conveys some of the same information as a histogram
• Box Plots
 Median
 Lower and upper quartiles
 Maximum and minimum

Pie Chart
Figure 1-10: Twentysomethings split on job satisfication
Category
Don't like my job but it is on my career path
Job is OK, but it is not on my career path
Enjoy job, but it is not on my career path
My job just pays the bills
Happy with career
6.0%
Do not like my job, but it is on my career path
Happy with career
19.0%
33.0%
Job OK, but it is not on my career path
19.0%
Enjoy job, but it is not on my career path
23.0%
My job just pays the bills
Bar Chart
Figure 1-11: SHIFTING GEARS
Quartely net income for General Motors (in billions)
1.5
1.2
0.9
0.6
0.3
0.0
1Q
2003
2Q
3Q
C4
4Q
1Q
2004
Frequency Polygon and Ogive
Relative Frequency Polygon
0.3
Ogive
1.0
0.2
0.5
0.1
0.0
0.0
0
10
20
Sales
30
40
50
0
10
20
30
40
50
Sales
(Cumulative frequency or
relative frequency graph)
Time Series Plot
M o n th ly S t e e l P r o d u c t io n
Millio n s o f To n s
8 .5
7 .5
6 .5
5 .5
M o n th
J FMAM J J A S OND J FMAM J J A SOND J F MAM J J AS O
Stem-and-Leaf Display


Stem-and-leaf display is a statistical technique to present a set of data. Each
numerical value is divided into two parts. The leading digit(s) becomes the stem and
the trailing digit the leaf. The stems are located along the vertical axis, and the leaf
values are stacked against each other along the horizontal axis.
Two disadvantages to organizing the data into a frequency distribution:
(1) The exact identity of each value is lost
(2)
Difficult to tell how the values within each class are distributed.
EXAMPLE
Listed in Table 4–1 is the number of 30-second radio advertising spots purchased by each of the 45 members of
the Greater Buffalo Automobile Dealers Association last year. Organize the data into a stem-and-leaf
display. Around what values do the number of advertising spots tend to cluster? What is the fewest
number of spots purchased by a dealer? The largest number purchased?
Boxplot - Example
Step1: Create an appropriate scale along the horizontal axis.
Step 2: Draw a box that starts at Q1 (15 minutes) and ends at Q3 (22
minutes). Inside the box we place a vertical line to represent the median (18 minutes).
Step 3: Extend horizontal lines from the box out to the minimum value (13
minutes) and the maximum value (30 minutes).
Box Plot – Buffalo Automobile Example –
SPSS output
180
160
140
120
100
80
N=
45
NUMBERS
Scatter Plots
• Scatter Plots are used to identify and report
any underlying relationships among pairs of
data sets.
• The plot consists of a scatter of points, each
point representing an observation.
Describing Relationship between Two
Variables – Scatter Diagram Examples
Describing Relationship between Two Variables –
Scatter Diagram Example
In the data from AutoUSA presented in
the file whitner.sav, the information
concerned the prices of 80 vehicles
sold last month at the Whitner
Autoplex lot in Raytown, Missouri.
The data shown include the selling
price of the vehicle as well as the age
of the purchaser.
Is there a relationship between the selling
price of a vehicle and the age of the
purchaser?
Would it be reasonable to conclude that
the more expensive vehicles are
purchased by older buyers?
Describing Relationship between Two Variables – Scatter Diagram
SPSS Example
Vehicle Age Vs. Selling Price
40
30
20
10
20
AGE
30
40
50
60