1. Chapter 2 – Descriptive Statistics
Tabular
and
Graphical Presentations

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

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. Summarize Qualitative Data
Check out the following data. What pattern can you detect from the raw data?

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. Frequency Distribution for Data in Table 2.1

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.
15/50=0.3

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.
Both charts indicate that the most popular network evening news is on NBC.

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?

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. 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. 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= We can round it up to 10, which is a much more convenient value to work with for class width.

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 =10.

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.

15. Relative Frequency
Example: Monthly Sales Volume of 50 Starbucks Stores
3/50=0.06

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 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. 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. Monthly Sales Volume of 50 Starbucks Stores in NYC
Histogram
Monthly Sales Volume of 50 Starbucks Stores in NYC

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

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

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

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. Cumulative Distributions
Monthly sales volume of 50 Starbucks stores     
2+6+11=19
19/50=0.38 

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. 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. Colonial Log Split A-Frame
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.
quantitative
variable
categorical
variable
Home Style
Price
Range
Colonial Log Split A-Frame
Total 55 45
< $200,000
> $200,000
Total
100

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

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

29. Crosstabulations Example: Finger Lakes Homes Home Style Price Range
Frequency
distribution
for the
price range
variable
Example: Finger Lakes Homes
Home Style
Price
Range
Colonial Log Split A-Frame
Total 55 45
< $200,000
> $200,000
Total
100
Frequency distribution for
the home style variable

30. Crosstabulations: Simpson’s Paradox
Data in two or more crosstabulations are often
aggregated to produce a summary crosstabulation.
We must be careful in drawing conclusions about the
relationship between the two variables in the
aggregated crosstabulation.
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.

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. Scatter Diagrams
A Positive Relationship y x

33. Scatter Diagrams
A Negative Relationship y x

34. Scatter Diagrams
No Relationship y x

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. Data of U.S. Retail Gas Price and S&P 500 Proxy Price (SPY)

37. Scatter Diagrams
The relationship between gas prices and stock prices

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.