Descriptive Statistics presentation

1. 1
Chapter 2 – Descriptive Statistics
Tabular
and
Graphical Presentations

2. 2
Chapter Outline
 Summarize Qualitative Data
 Frequency Distribution
 Bar Charts and Pie Charts
 Summarize Quantitative Data
 Frequency Distribution
 Histogram
 Cumulative Distributions
 Crosstabulations
 Scatter Diagrams

3. 3
A Note
 An important aspect of statistics is to present the
data in an informative way so as to reveal any
patterns in the data (no pattern is a pattern!).
 Different types of data require different
summarization methods and statistical analyses.

4. 4
Summarize Qualitative Data
 Check out the following data. What pattern can you detect from the
raw data?
NBC CBS NBC ABC FOX
NBC NBC CNN NBC CBS
CBS FOX NBC CNN ABC
FOX NBC CBS FOX ABC
CNN FOX CBS CBS CNN
NBC NBC CBS FOX ABC
CBS NBC FOX NBC FOX
NBC CNN NBC CBS CBS
ABC NBC CNN FOX CBS
FOX CBS ABC NBC CNN
Table 2.1 Data from a sample of 50 individual responses to the question
'Which network's evening news do you prefer to watch?'

5. 5
Summarize Qualitative Data
Frequency Distribution
 The raw data in the previous table does not provide any
meaningful information ( like any pattern) directly. For
qualitative data, we can summarize and present the raw
data with ‘Frequency Distribution’.
 A frequency distribution is a tabular summary of data
showing the number (frequency) of items in each
nonoverlapping class.
• Please refer to the Excel demonstration ( Chapter 2) on how to
construct the frequency distribution for the data in table 2.1.
• The outcome is shown on the next slide.

6. 6
Frequency Distribution for Data in Table 2.1
Network Frequency
ABC 6
CBS 12
CNN 7
FOX 10
NBC 15

7. 7
Relative Frequency
 To obtain relative frequency, simply divide the frequency of each class
by the total number of observations (n). For the data in Table 2.1, n
equals 50.
Network Frequency Relative Frequency Percent Frequency
ABC 6 0.12 12
CBS 12 0.24 24
CNN 7 0.14 14
FOX 10 0.2 20
NBC 15 0.3 30
15/50=0.3

8. 8
Bar Charts and Pie Charts
 A frequency distribution is often presented in a graph (a bar chart or a pie chart)
to communicate information visually.
 Please refer to the Excel demonstration ( Chapter 2) on how to create a bar
chart and a pie chart for the frequency distribution from previous slide.
ABC
12%
CBS
24%
CNN
14%
FOX
20%
NBC
30%
Both charts indicate that the most popular
network evening news is on NBC.

9. 9
Summarize Quantitative Data
 Check out the following data. Can you quickly decide how many
classes there should be in the construction of a frequency distribution?
95 77 97 99 89
108 120 78 79 88
67 97 97 79 93
99 103 106 82 93
93 97 95 61 109
77 88 100 109 90
86 89 97 93 88
93 105 87 82 98
119 104 93 104 101
118 105 82 73 101
Table 2.2 Data of average monthly sales volume
($1000) of a sample of 50 Starbucks stores in New
York City in 2012

10. 10
Summarize Quantitative Data
Frequency Distribution
 Different from the qualitative data in Table 2.1, the
quantitative data in Table 2.2 do not indicate the
number of classes straightforwardly.
 Apply the following procedure to construct a
frequency distribution for quantitative data.
• Determine the number of non-overlapping classes;
• Determine the class width;
• Determine the class limits;
• Count the item numbers in each class.

11. 11
Summarize Quantitative Data
Frequency Distribution
 Step one – Determine the number of non-
overlapping classes
• As a guidance, you can use the ‘2 to the power of k’
rule. That is, to find the smallest integer (k) such that 2k
 n ( n is the sample size). Applying the rule to the data
in Table 2.2, we find k = 6 since 26=64 ( n=50). Thus,
we set the # of classes as 6. (Note that it is only a
suggestion, not an absolute rule.)
• Empirically speaking, the # of classes is between 5 and
20.

12. 12
Summarize Quantitative Data
Frequency Distribution
 Step two – Determine the class width
• Use equal class width to avoid misinterpretation
• Approximately, class width =
• For the data in Table 2.2, class width = (120-61)/6=
9.96. We can round it up to 10, which is a much more
convenient value to work with for class width.
classes
of
#
alue
Smallest v
-
lue
Largest va

13. 13
Summarize Quantitative Data
Frequency Distribution
 Step three – Determine the class limits
• Class limits should be set so that each data point
belongs to one and only one class, and no data point is
left out.
• Similar to class width, class limits can use values that
are convenient to work with.
- In our example, the smallest value is 61 and the class width is
set as 10. So, the lowest class can be set as 61 – 70. Note that
the class width is calculated as 70-61+1=10.

14. 14
Summarize Quantitative Data
Frequency Distribution
 Step four – count the # of items in each class
• For the data in Table 2.2, the frequency distribution is
constructed as follows:
• Please refer to the Excel demonstration ( Chapter 2) on how to construct the
frequency distribution for the data in table 2.2.
Sales Volume ($1000) Frequency
61-70 2
71-80 6
81-90 11
91-100 17
101-110 11
111-120 3
Total 50

15. 15
Relative Frequency
Sales Volume ($1000) Frequency Relative Frequency Percent Freqency
61-70 2 0.04 4
71-80 6 0.12 12
81-90 11 0.22 22
91-100 17 0.34 34
101-110 11 0.22 22
111-120 3 0.06 6
3/50=0.06
Example: Monthly Sales Volume of 50 Starbucks Stores

16. 16
Interpretation of Frequency Distribution
The frequency distribution of monthly sales volume of 50
Starbucks stores in NYC reveals that
 39 stores generated an average monthly sales in 2012
between $81,000 and $110,000.
 4% of the sample stores had an average monthly sales no
more than $70,000.
 6% of the sample stores had an average monthly sales
$111,000 or more.

17. 17
Histogram
 Like a bar chart, a histogram is a graphical presentation of
frequency distribution.
 The height of a rectangle ( a bar) drawn above each class
interval corresponds to that class’ frequency or relative
frequency.
 Unlike a bar chart, a histogram has no gap between
rectangles of adjacent classes.
• Please refer to the Excel demonstration ( Chapter 2) on how to create a
histogram for the frequency distribution of Sales volume of Starbucks
stores.

18. 18
Histogram
Monthly Sales Volume of 50 Starbucks Stores in NYC
Average MonthlySales Volume of ASample of 50 Starbucks Stores
in NYC in 2012
2
6
11
17
11
3
0
5
10
15
20
61-70 71-80 81-90 91-100 101-110 111-120
Sales Volume ($1000)
Frequency

19. 19
Histogram
 Skewness – the lack of symmetry.
 Symmetric distribution, such as height or weight of human
population.
Relative
Frequency
.05
.10
.15
.20
.25
.30
.35
0

20. 20
Histogram
 Negative Skewness – a longer tail to the left.
 An example: exam scores
Relative
Frequency
.05
.10
.15
.20
.25
.30
.35
0

21. 21
Relative
Frequency
.05
.10
.15
.20
.25
.30
.35
0
Histogram
 Positive Skewness – a longer tail to the right.
 An example: home values

22. 22
Cumulative Distributions
 Cumulative frequency distribution – shows the # of items
with values less than or equal to the upper limit of each
class.
 Cumulative relative frequency distribution – shows the
proportion (percentage) of items with values less than or
equal to the upper limit of each class.

23. 23
Cumulative Distributions
Monthly sales volume of 50 Starbucks stores

Sales Volume
($1000)
Cumulative
Frequency
Cumulative
Relative Frequency
70 2 0.04
80 8 0.16
90 19 0.38
100 36 0.72
110 47 0.94
120 50 1





2+6+11=19 19/50=0.38

24. 24
Crosstabulations and Scatter Diagrams
 So far, we have studies the methods of summarizing the
data of one variable at a time.
 In business, it is important to understand the relationships
among different variables. For instance, the relationship
between sales volume and expenditure on advertisement.
 Crosstabulations and scatter diagrams are two
methods of descriptive statistics, which are used to
summarize the data to reveal the relationship of two
variables.

25. 25
Crosstabulations
 A crosstabulation is a tabular summary of data for two
variables.
 The two variables can be either qualitative or quantitative
or one of each.
 The left and top margin labels show the classes for
the two variables.

26. 26
Crosstabulations
 Example: Finger Lakes Homes
The number of Finger Lakes homes sold for each
style and price for the past two years is shown below.
Price
Range Colonial Log Split A-Frame Total
< $200,000
> $200,000
18 6 19 12 55
45
30 20 35 15
Total 100
12 14 16 3
Home Style
quantitative
variable
categorical
variable

27. 27
Crosstabulations
 Example: Finger Lakes Homes
Insights Gained from Preceding Crosstabulation
• Only three homes in the sample are an A-Frame
style and priced at $200,000 or more.
• The greatest number of homes (19) in the sample
are a split-level style and priced at less than
$200,000.

28. 28
Crosstabulation
Insights Gained from Preceding Crosstabulation
 Only three homes in the sample are an A-Frame
style and priced at $200,000 or more.
 The greatest number of homes (19) in the sample
are a split-level style and priced at less than
$200,000.
Example: Finger Lakes Homes

29. 29
Price
Range Colonial Log Split A-Frame Total
< $200,000
> $200,000
18 6 19 12 55
45
30 20 35 15
Total 100
12 14 16 3
Home Style
Crosstabulations
Frequency
distribution
for the
price range
variable
Frequency distribution for
the home style variable
Example: Finger Lakes Homes

30. 30
Crosstabulations: Simpson’s Paradox
 In some cases the conclusions based upon an
aggregated crosstabulation can be completely
reversed if we look at the unaggregated data. The
reversal of conclusions based on aggregate and
unaggregated data is called Simpson’s paradox.
 We must be careful in drawing conclusions about the
relationship between the two variables in the
aggregated crosstabulation.
 Data in two or more crosstabulations are often
aggregated to produce a summary crosstabulation.

31. 31
Scatter Diagrams
 A scatter diagram is a graphical presentation of the
relationship between two quantitative variables.
 One variable is shown on the horizontal axis and the other
variable is shown on the vertical axis.
 The general pattern of the plotted points suggests the
overall relationship between the variables.
 A trendline provides a linear approximation of the
relationship.

32. 32
Scatter Diagrams
 A Positive Relationship
x
y

33. 33
Scatter Diagrams
 A Negative Relationship
y
x

34. 34
Scatter Diagrams
 No Relationship
y
x

35. 35
Scatter Diagrams
 An example
Is there a relationship between gas prices and stock prices?
• For the variable – gas price, let us use the data of the U.S. retail gas
price;
• For the variable – stock prices, let us use the data of the S&P 500
Index ( ticker symbol – SPY);
• Weekly data for both variables.
The data are shown in the next slide.

36. 36
Data of U.S. Retail Gas Price and S&P
500 Proxy Price (SPY)
Date
U.S. Retail
Gas Price
SPY
Jan 28, 2013 3.296 151.24
Feb 04, 2013 3.471 151.8
Feb 11, 2013 3.537 152.11
Feb 18, 2013 3.69 151.89
Feb 25, 2013 3.722 152.11
Mar 04, 2013 3.698 155.44
Mar 11, 2013 3.644 155.83
Mar 18, 2013 3.633 155.6
Mar 25, 2013 3.616 156.67
Apr 01, 2013 3.572 155.86

37. 37
Scatter Diagrams
 The relationship between gas prices and stock prices
Scatter Diagram
150
151
152
153
154
155
156
157
3.25 3.3 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75
U.S. Retail Gas Price ($/gallon)
SPY

38. 38
Scatter Diagrams
The relationship between gas prices and stock prices
 The plots in the previous scatter diagram indicate a positive
relationship between U.S. retail gas price and the value of
SPY.
 The relationship is sketchy. When gas price is high, the
S&P 500 Index tend to be high.
 We need to be cautious in drawing conclusion from a
scatter diagram. In the example, there are only 10 data
points. Much more data are required to rigorously examine
the relationship between gas price and stock prices.