Sampling of statistics and its techniques

Statistics for Business and
Economics
Chapter 2
Methods for Describing
Sets of Data

Contents
1. Describing Qualitative Data
2. Graphical Methods for Describing
Quantitative Data
3. Summation Notation
4. Numerical Measures of Central Tendency
5. Numerical Measures of Variability
6. Interpreting the Standard Deviation

Contents
7. Numerical Measures of Relative Standing
8. Methods for Detecting Outliers: Box Plots
and z-scores
9. Graphing Bivariate Relationships
10. The Time Series Plot
11. Distorting the Truth with Descriptive
Techniques

Learning Objectives
1. Describe data using graphs
2. Describe data using numerical measures

2.1
Describing Qualitative Data

Key Terms
A class is one of the categories into which
qualitative data can be classified.
The class frequency is the number of
observations in the data set falling into a
particular class.
The class relative frequency is the class
frequency divided by the total numbers of
observations in the data set.
The class percentage is the class relative
frequency multiplied by 100.

Data Presentation
Data
Presentation
Qualitative
Data
Quantitative
Data
Summary
Table
Stem-&-Leaf
Display
Frequency
Distribution
Histogram
Bar
Graph
Pie
Chart
Pareto
Diagram
Dot
Plot

Summary Table
1. Lists categories & number of elements in category
2. Obtained by tallying responses in category
3. May show frequencies (counts), % or both
Row Is
Category
Tally:
|||| ||||
|||| ||||
Major Count
Accounting 130
Economics 20
Management 50
Total 200

0
50
100
150
Acct. Econ. Mgmt.
Major
Bar Graph
Vertical Bars
for Qualitative
Variables
Bar Height
Shows
Frequency or %
Zero Point
Percent
Used
Also
Equal Bar
Widths
Frequency

Econ.
10%
Mgmt.
25%
Acct.
65%
Pie Chart
1. Shows breakdown of
total quantity into
categories
2. Useful for showing
relative differences
3. Angle size
• (360°)(percent)
Majors
(360°) (10%) = 36°
36°

Pareto Diagram
Like a bar graph, but with the categories arranged by
height in descending order from left to right.
0
50
100
150
Acct. Mgmt. Econ.
Major
Vertical Bars
for Qualitative
Variables
Bar Height
Shows
Frequency or %
Zero Point
Percent
Used
Also
Equal Bar
Widths
Frequency

Summary
Bar graph: The categories (classes) of the qualitative
variable are represented by bars, where the height of
each bar is either the class frequency, class relative
frequency, or class percentage.
Pie chart: The categories (classes) of the qualitative
variable are represented by slices of a pie (circle). The
size of each slice is proportional to the class relative
frequency.
Pareto diagram: A bar graph with the categories
(classes) of the qualitative variable (i.e., the bars)
arranged by height in descending order from left to
right.

Thinking Challenge
You’re an analyst for IRI. You want to show the
market shares held by Web browsers in 2006.
Construct a bar graph, pie chart, & Pareto diagram
to describe the data.
Browser Mkt. Share (%)
Firefox 14
Internet Explorer 81
Safari 4
Others 1

0%
20%
40%
60%
80%
100%
Firefox Internet
Explorer
Safari Others
Bar Graph Solution*
Market
Share
(%)
Browser

Pie Chart Solution*
Market Share
Safari, 4%
Firefox,
14%
Internet
Explorer,
81%
Others,
1%

Pareto Diagram Solution*
0%
20%
40%
60%
80%
100%
Internet
Explorer
Firefox Safari Others
Market
Share
(%)
Browser

2.2
Graphical Methods for Describing
Quantitative Data

Dot Plot
1. Horizontal axis is a scale for the quantitative variable,
e.g., percent.
2. The numerical value of each measurement is located
on the horizontal scale by a dot.

Stem-and-Leaf Display
1. Divide each observation
into stem value and leaf
value
• Stems are listed in
order in a column
• Leaf value is placed in
corresponding stem
row to right of bar
2. Data: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41
26
2 144677
3 028
4 1

Frequency Distribution
Table Steps
1. Determine range
2. Select number of classes
• Usually between 5 & 15 inclusive
3. Compute class intervals (width)
4. Determine class boundaries (limits)
5. Compute class midpoints
6. Count observations & assign to classes

Frequency Distribution Table
Example
Raw Data: 24, 26, 24, 21, 27 27 30, 41, 32, 38
Boundaries
(Lower + Upper Boundaries) / 2
Width
Class Midpoint Frequency
15.5 – 25.5 20.5 3
25.5 – 35.5 30.5 5
35.5 – 45.5 40.5 2

Relative Frequency &
% Distribution Tables
Percentage
Distribution
Relative Frequency
Distribution
Class Prop.
15.5 – 25.5 .3
25.5 – 35.5 .5
35.5 – 45.5 .2
Class %
15.5 – 25.5 30.0
25.5 – 35.5 50.0
35.5 – 45.5 20.0

0
1
2
3
4
5
Histogram
Frequency
Relative
Frequency
Percent
0 15.5 25.5 35.5 45.5 55.5
Lower Boundary
Bars
Touch
Class Freq.
15.5 – 25.5 3
25.5 – 35.5 5
35.5 – 45.5 2
Count

2.3
Summation Notation

Summation Notation
Most formulas we use require a summation of numbers.
Sum the measurements on the variable that appears to the
right of the summation symbol, beginning with the 1st
measurement and ending with the nth measurement.
xi
i=1
n
∑

Summation Notation
For the data
xi
2
= x1
2
i=1
5
∑ + x2
2
+ x3
2
+ x4
2
+ x5
2
= 52
+ 32
+ 82
+ 52
+ 42
= 25 + 9 + 64 + 25 +16 = 139
x1 = 5, x2 = 3, x3 = 8, x4 = 5, x5 = 4

2.4
Numerical Measures
of Central Tendency

Thinking Challenge
... employees cite low pay --
most workers earn only
$20,000.
... President claims average
pay is $70,000!
$400,000
$400,000
$70,000
$70,000
$50,000
$50,000
$30,000
$30,000
$20,000
$20,000

Two Characteristics
The central tendency of the set of
measurements–that is, the tendency of the data to
cluster, or center, about certain numerical values.
Central Tendency
(Location)

Two Characteristics
The variability of the set of measurements–that
is, the spread of the data.
Variation
(Dispersion)

Standard Notation
Measure Sample Population
Mean X 
Size n N

Mean
1. Most common measure of central tendency
2. Acts as ‘balance point’
3. Affected by extreme values (‘outliers’)
4. Denoted where
x
x
n
x x x
n
i
i
n
n
 
  


1 1 2 …
x

Mean Example
Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
x
x
n
x x x x x x
i
i
n
 
    

    



1 1 2 3 4 5 6
6
10 3 4 9 8 9 117 6 3 7 7
6
8 30
. . . . . .
.

Median
1. Measure of central tendency
2. Middle value in ordered sequence
• If n is odd, middle value of sequence
• If n is even, average of 2 middle values
3. Position of median in sequence
4. Not affected by extreme values
Positioning Point 


n 1
2

Median Example
Odd-Sized Sample
• Raw Data: 24.1 22.6 21.5 23.7 22.6
• Ordered: 21.5 22.6 22.6 23.7 24.1
• Position: 1 2 3 4 5
Positioning Point
Median






n 1
2
5 1
2
3 0
22 6
.
.

Median Example
Even-Sized Sample
• Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
• Ordered: 4.9 6.3 7.7 8.9 10.3 11.7
• Position: 1 2 3 4 5 6
Positioning Point
Median








n 1
2
6 1
2
3 5
7 7 8 9
2
8 30
.
. .
.

Mode
1. Measure of central tendency
2. Value that occurs most often
4. May be no mode or several modes
5. May be used for quantitative or qualitative
data

Mode Example
• No Mode
Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
• One Mode
Raw Data: 6.3 4.9 8.9 6.3 4.9 4.9
• More Than 1 Mode
Raw Data: 21 28 28 41 43 43

Thinking Challenge
You’re a financial analyst
for Prudential-Bache
Securities. You have
collected the following
closing stock prices of new
stock issues: 17, 16, 21, 18,
13, 16, 12, 11.
Describe the stock prices
in terms of central
tendency.

Central Tendency Solution*
Mean
x
x
n
x x x
i
i
n
 
  

      



1 1 2 8
8
17 16 21 18 13 16 12 11
8
15 5
…
.

Median
• Raw Data: 17 16 21 18 13 16 12 11
• Ordered: 11 12 13 16 16 17 18 21
• Position: 1 2 3 4 5 6 7 8
Positioning Point
Median








n 1
2
8 1
2
4 5
16 16
2
2
16
.

Mode
Raw Data: 17 16 21 18 13 16 12
11
Mode = 16

Summary of
Central Tendency Measures
Measure Formula Description
Mean x i / n Balance Point
Median (n+1)
Position
2
Middle Value
When Ordered
Mode none Most Frequent

Shape
1. Describes how data are distributed
2. Measures of Shape
• Skew = Symmetry
Right-Skewed
Left-Skewed Symmetric
Mean
Mean =
= Median
Median
Mean
Mean Median
Median Median
Median Mean
Mean

2.5
Numerical Measures
of Variability

Range
1. Measure of dispersion
2. Difference between largest & smallest
observations
Range = xlargest – xsmallest
3. Ignores how data are distributed
7
7 8
8 9
9 10
10 7
7 8
8 9
9 10
10
Range = 10 – 7 = 3 Range = 10 – 7 = 3

Variance &
Standard Deviation
1. Measures of dispersion
2. Most common measures
3. Consider how data are distributed
4 6 10 12
x = 8.3
4. Show variation about mean (x or μ)
8

Standard Notation
Measure Sample Population
Mean x 
Standard
Deviation s 
Variance s
2
2
Size n N

Sample Variance Formula
n – 1 in denominator!
s2
=
xi −x
( )
2
i=1
n
∑
n −1
=
x1 −x
( )
2
+ x2 −x
( )
2
+L + xn −x
( )
2
n −1

Sample Standard Deviation
Formula
s = s2
=
xi −x
( )
2
i=1
n
∑
n −1
=
x1 −x
( )
2
+ x2 −x
( )
2
+L + xn −x
( )
2
n −1

Variance Example
Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
s
x x
n
x
x
n
s
i
i
n
i
i
n
2
2
1 1
2
2 2 2
1
8 3
10 3 8 3 4 9 8 3 7 7 8 3
6 1
6 368



 

     


 
 
( )
( ) ( ) ( )
where .
. . . . . .
.
…

Thinking Challenge
• You’re a financial analyst
for Prudential-Bache
Securities. You have
collected the following
closing stock prices of
new stock issues: 17, 16,
21, 18, 13, 16, 12, 11.
• What are the variance
and standard deviation
of the stock prices?

Variation Solution*
Sample Variance
Raw Data: 17 16 21 18 13 16 12
11
s
x x
n
x
x
n
s
i
i
n
i
i
n
2
2
1 1
2
2 2 2
1
15 5
17 15 5 16 15 5 11 15 5
8 1
1114



 

     


 
 
( )
( ) ( ) ( )
where .
. . .
.
…

Variation Solution*
Sample Standard Deviation
s = s2
=
xi −x
( )
2
i=1
n
∑
n −1
= 11.14 ≈3.34

Summary of
Variation Measures
Measure Formula Description
Range Xlargest – Xsmallest Total Spread
Standard Deviation
(Sample)
Dispersion about
Sample Mean
Standard Deviation
(Population)
Dispersion about
Population Mean
Variance
(Sample)
Squared Dispersion
about Sample Mean
xi −x
( )
2
i=1
n
∑
n −1
xi −µx
( )
2
i=1
n
∑
N
xi −x
( )
2
i=1
n
∑
n −1

2.6
Interpreting the
Standard Deviation

Interpreting Standard Deviation:
Chebyshev’s Theorem
• Applies to any shape data set
• No useful information about the fraction of data in the
interval x – s to x + s
• At least 3/4 of the data lies in the interval
x – 2s to x + 2s
• At least 8/9 of the data lies in the interval
x – 3s to x + 3s
• In general, for k > 1, at least 1 – 1/k2
of the data lies in
the interval x – ks to x + ks

Chebyshev’s Theorem
s
x 3
− s
x 3
+
s
x 2
− s
x 2
+
s
x +
x
s
x −
No useful information
At least 3/4 of the data
At least 8/9 of the data

Chebyshev’s Theorem Example
• Previously we found the mean
closing stock price of new stock
issues is 15.5 and the standard
deviation is 3.34.
• Use this information to form an
interval that will contain at least
75% of the closing stock prices of
new stock issues.

Chebyshev’s Theorem Example
At least 75% of the closing stock prices of new stock issues will lie within 2 standard deviations of
the mean.
x = 15.5 s = 3.34
(x – 2s, x + 2s) = (15.5 – 2∙3.34, 15.5 + 2∙3.34)
= (8.82, 22.18)

Empirical Rule
• Applies to data sets that are mound shaped and
symmetric
• Approximately 68% of the measurements lie
in the interval
• Approximately 95% of the measurements lie
in the interval
• Approximately 99.7% of the measurements lie
in the interval
x −sto x + s
x −2sto x + 2s
x −3sto x + 3s

Empirical Rule
x – 3s x – 2s x – s x x + s x +2s x + 3s
Approximately 68% of the measurements
Approximately 95% of the measurements
Approximately 99.7% of the measurements

Empirical Rule Example
Previously we found the mean
closing stock price of new
stock issues is 15.5 and the
standard deviation is 3.34. If
we can assume the data is
symmetric and mound shaped,
calculate the percentage of the
data that lie within the intervals
x + s, x + 2s, x + 3s.

Empirical Rule Example
• Approximately 95% of the data will lie in the interval
(x – 2s, x + 2s),
(15.5 – 2∙3.34, 15.5 + 2∙3.34) = (8.82, 22.18)
• Approximately 99.7% of the data will lie in the interval
(x – 3s, x + 3s),
(15.5 – 3∙3.34, 15.5 + 3∙3.34) = (5.48, 25.52)
• According to the Empirical Rule, approximately 68%
of the data will lie in the interval (x – s, x + s),
(15.5 – 3.34, 15.5 + 3.34) = (12.16, 18.84)

2.7
Numerical Measures
of Relative Standing

Numerical Measures of
Relative Standing: Percentiles
• Describes the relative location of a
measurement compared to the rest of the data
• The pth
percentile is a number such that p% of
the data falls below it and (100 – p)% falls
above it
• Median = 50th
percentile

Percentile Example
• You scored 560 on the GMAT exam. This
score puts you in the 58th
percentile.
• What percentage of test takers scored lower
than you did?
• What percentage of test takers scored higher
than you did?

Percentile Example
• What percentage of test takers scored lower
than you did?
58% of test takers scored lower than 560.
• What percentage of test takers scored higher
than you did?
(100 – 58)% = 42% of test takers scored
higher than 560.

Numerical Measures of
Relative Standing: z–Scores
• Describes the relative location of a
measurement compared to the rest of the data
• Measures the number of standard deviations
away from the mean a data value is located
• Sample z–score Population z–score
z =
x −x
s
z =
x −µ
σ

Z–Score Example
• The mean time to assemble a
product is 22.5 minutes with a
standard deviation of 2.5 minutes.
• Find the z–score for an item that
took 20 minutes to assemble.
• Find the z–score for an item that
took 27.5 minutes to assemble.

Z–Score Example
x = 20, μ = 22.5 σ = 2.5
x – μ 20 – 22.5
σ
z = =
2.5
= –1.0
x = 27.5, μ = 22.5 σ = 2.5
x – μ 27.5 – 22.5
σ
z = =
2.5
= 2.0

Interpretation of z–Scores for
Mound-Shaped Distributions
of Data
1. Approximately 68% of the measurements
will have a z-score between –1 and 1.
2. Approximately 95% of the measurements
3. Approximately 99.7% of the measurements
(see the figure on the next slide)

Interpretation of z–Scores

2.8
Methods for Detecting Outliers:
Box Plots and z-Scores

Outlier
An observation (or measurement) that is unusually large
or small relative to the other values in a data set is called
an outlier. Outliers typically are attributable to one of
the following causes:
1. The measurement is observed, recorded, or entered
into the computer incorrectly.
2. The measurement comes from a different
population.
3. The measurement is correct but represents a rare
(chance) event.

Quartiles
Measure of noncentral tendency
25%
25% 25%
25% 25%
25% 25%
25%
Q
Q1
1 Q
Q2
2 Q
Q3
3
Split ordered data into 4 quarters
Lower quartile QL is 25th
percentile.
Middle quartile m is the median.
Upper quartile QU is 75th
percentile.
Interquartile range: IQR = QU – QL

Quartile (Q2) Example
• Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
• Ordered: 4.9 6.3 7.7 8.9 10.3 11.7
• Position: 1 2 3 4 5 6
Q2 is the median, the average of the two middle
scores (7.7 + 8.9)/2 = 8.8

• Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
• Ordered: 4.9 6.3 7.7 8.9 10.3 11.7
• Position: 1 2 3 4 5 6
QL is median of bottom half = 6.3

• Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7
• Ordered: 4.9 6.3 7.7 8.9 10.3 11.7
• Position: 1 2 3 4 5 6
QU is median of bottom half = 10.3

Interquartile Range
1. Measure of dispersion
2. Also called midspread
3. Difference between third & first quartiles
• Interquartile Range = Q3 – Q1
4. Spread in middle 50%

Thinking Challenge
• You’re a financial analyst for
Prudential-Bache Securities.
You have collected the
following closing stock prices
of new stock issues: 17, 16,
21, 18, 13, 16, 12, 11.
• What are the quartiles, Q1
and Q3, and the interquartile
range?

Q1
Raw Data: 17 16 21 18 13 16 12
11
Ordered: 11 12 13 16 16 17 18
21
Position: 1 2 3 4 5 6 7 8
QL is the median of the bottom half, the average
of the two middle scores (12 + 13)/2 = 12.5
Quartile Solution*

Quartile Solution*
Q3
Raw Data: 17 16 21 18 13 16 12
11
Ordered: 11 12 13 16 16 17 18
21
Position: 1 2 3 4 5 6 7 8
QU is the median of the bottom half, the average
of the two middle scores (17 + 18)/2 = 17.5

Interquartile Range Solution*
Interquartile Range
Raw Data: 17 16 21 18 13 16 12
11
Ordered: 11 12 13 16 16 17 18
21
Position: 1 2 3 4 5 6 7 8
Interquartile Range = Q3 – Q1 = 17.5 – 12.5 = 5

Box Plot
1. Graphical display of data using 5-number
summary
Median
4
4 6
6 8
8 10
10 12
12
Q3
Q1 Xlargest
Xsmallest

Box Plot
1. Draw a rectangle (box) with the ends
(hinges) drawn at the lower and upper
quartiles (QL and QU). The median data is
shown by a line or symbol (such as “+”).
2. The points at distances 1.5(IQR) from each
hinge define the inner fences of the data set.
Line (whiskers) are drawn from each hinge
to the most extreme measurements inside the
inner fence.

Box Plot
3. A second pair of fences, the outer fences, are
defined at a distance of 3(IQR) from the hinges.
One symbol (*) represents measurements falling
between the inner and outer fences, and another (0)
represents measurements beyond the outer fences.
4. Symbols that represent the median and extreme data
points vary depending on software used. You may
use your own symbols if you are constructing a box
plot by hand.

Shape & Box Plot
Right-Skewed
Left-Skewed Symmetric
Q
Q1
1 Median
Median Q
Q3
3
Q
Q1
1 Median
Median Q
Q3
3 Q
Q1
1 Median
Median Q
Q3
3

Detecting Outliers
Box Plots: Observations falling between the
inner and outer fences are deemed suspect
outliers. Observations falling beyond the
outer fence are deemed highly suspect
outliers.
z-scores: Observations with z-scores greater than
3 in absolute value are considered outliers.
(For some highly skewed data sets,
observations with z-scores greater than 2 in
absolute value may be outliers.)

2.9
Graphing Bivariate Relationships

Graphing Bivariate
Relationships
• Describes a relationship between two
quantitative variables
• Plot the data in a scattergram (or scatterplot)
Positive
relationship
Negative
relationship
No
relationship
x x
x
y
y y

Scattergram Example
• You’re a marketing analyst for Hasbro Toys.
You gather the following data:
Ad $ (x) Sales (Units) (y)
1 1
2 1
3 2
4 2
5 4
• Draw a scattergram of the data

Scattergram Example
0
1
2
3
4
0 1 2 3 4 5
Sales
Advertising

2.10
The Time Series Plot

Time Series Plot
• Used to graphically display data produced over
time
• Shows trends and changes in the data over
time
• Time recorded on the horizontal axis
• Measurements recorded on the vertical axis
• Points connected by straight lines

Time Series Plot Example
• The following data shows
the average retail price of
regular gasoline in New
York City for 8 weeks in
2006.
• Draw a time series plot
for this data.
Date
Average
Price
Oct 16, 2006 $2.219
Oct 23, 2006 $2.173
Oct 30, 2006 $2.177
Nov 6, 2006 $2.158
Nov 13, 2006 $2.185
Nov 20, 2006 $2.208
Nov 27, 2006 $2.236
Dec 4, 2006 $2.298

Time Series Plot Example
2.05
2.1
2.15
2.2
2.25
2.3
2.35
10/16 10/23 10/30 11/6 11/13 11/20 11/27 12/4
Date
Price

2.11
Distorting the Truth with
Descriptive Statistics

Errors in Presenting Data
1. Use area to equate to value
2. No relative basis in
comparing data batches
3. Compress the vertical axis
4. No zero point on the vertical
axis
5. Gap in the vertical axis
6. Use of misleading wording
7. Knowing central tendency
without knowing variability

Reader Equates Area to Value
Bad Presentation
Bad Presentation Good 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 Presentation
Good Presentation
A’s by Class A’s by Class
Bad Presentation
Bad Presentation
0
100
200
300
FR SO JR SR
Freq.
0%
10%
20%
30%
FR SO JR SR
%

Compressing
Vertical Axis
Good Presentation
Good Presentation
Quarterly Sales Quarterly Sales
Bad Presentation
Bad Presentation
0
25
50
Q1 Q2 Q3 Q4
$
0
100
200
Q1 Q2 Q3 Q4
$

No Zero Point
on Vertical Axis
Good Presentation
Good Presentation
Monthly Sales Monthly Sales
Bad Presentation
Bad Presentation
0
20
40
60
J M M J S N
$
36
39
42
45
J M M J S N
$

Gap in the Vertical Axis
Bad Presentation
Bad Presentation

Changing the Wording
Changing the title of the graph can influence the reader.
We’re not doing so well. Still in prime years!

Knowing only central tendency
Knowing ONLY the central tendency might lead one
to purchase Model A. Knowing the variability as
well may change one’s decision!

Key Ideas
Describing Qualitative Data
1. Identify category classes
2. Determine class frequencies
3. Class relative frequency = (class freq)/n
4. Graph relative frequencies

Key Ideas
Graphing Quantitative Data
1 Variable
1. Identify class intervals
2. Determine class interval frequencies
3. Class relative relative frequency =
(class interval frequencies)/n
4. Graph class interval relative frequencies

Key Ideas
Graphing Quantitative Data
2 Variables
Scatterplot

Key Ideas
Numerical Description of Quantitative Date
Central Tendency
Mean
Median
Mode

Key Ideas
Variation
Range
Variance
Standard Deviation
Interquartile range

Key Ideas
Relative standing
Percentile score
z-score

Key Ideas
Rules for Detecting Quantitative Outliers
Interval Chebyshev’s Rule Empirical Rule
At least 0%
At least 57%
At least 89%
≈ 68%
≈ 95%
All
x ± s
x ± 2s
x ± 3s

Key Ideas
Rules for Detecting Quantitative Outliers
Method Suspect Highly Suspect
Values
between inner
and outer
fences
2 < |z| < 3
Box plot:
z-score
Values
beyond outer
fences
2 < |z| < 3

Sampling of statistics and its techniques

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