1. Descriptive statistics (Part I)
Lecture 2
Descriptive statistics (Part I)

2. Lecture 2: Descriptive statistics
Data in raw form are usually not easy to use for decision making
Some type of organization is needed
Table
Graph
Techniques reviewed here:
Bar charts and pie charts
Ordered array
Stem-and-leaf display
Frequency distributions, histograms
Cumulative distributions
Contingency tables

3. Tabulating and Graphing Univariate Categorical Data
Graphing Data
Tabulating Data
Pie Charts
Summary Table
Bar Charts

4. Summary Table (for an Investor’s Portfolio)
Investment Category Amount Percentage
(in thousands $)
Stocks
Bonds CD
Savings
Total
Variables are Categorical

5. Bar Chart (for an Investor’s Portfolio)

6. Pie Chart (for an Investor’s Portfolio)
Amount Invested in K$
Savings
15%
Stocks
42%
CD 14%
Percentages are rounded to the nearest percent
Bonds 29%

7. Organizing Numerical Data
41, 24, 32, 26, 27, 27, 30, 24, 38, 21
Frequency Distributions & Cumulative Distributions
Ordered Array
Stem and Leaf
Display
3 028
4 1
21, 24, 24, 26, 27,
27, 30, 32, 38, 41
Histograms
Tables

8. The Ordered Array Shows range (min to max)
Data in raw form (as collected):
24, 26, 24, 21, 27, 27, 30, 41, 32, 38
Data in ordered array from smallest to largest: , 24, 24, 26, 27, 27, 30, 32, 38, 41
Shows range (min to max)
May help identify outliers (unusual observations)
If the data set is large, the ordered array is less useful

9. Stem-and-Leaf Display
A simple way to see distribution details in a data set
METHOD: Separate the sorted data series into leading digits (the stem) and the trailing digits (the leaves)

10. Example
Data in Raw Form (as Collected): 24, 26, 24, 21, 27, 27, 30, 41, 32, 38
Data in Ordered Array from Smallest to Largest: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41
Stem-and-Leaf Display:
4 1

11. Tabulating Numerical Data: Frequency Distributions
What is a Frequency Distribution?
A frequency distribution is a list or a table …
containing class groupings (ranges within which the data fall) ...
and the corresponding frequencies with which data fall within each grouping or category
It allows for a quick visual interpretation of the data

12. Tabulating Numerical Data: Frequency Distributions
Example: A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature
24, 35, 17, 21, 24, 37, 26, 46, 58, 30,
32, 13, 12, 38, 41, 43, 44, 27, 53, 27

13. Sort Raw Data on days in Ascending Order 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Find Range: = 46
Select Number of Classes: 5 (usually between 5 and 15)
Compute Class Interval (Width): 10 (46/5 then round up)
Determine Class Boundaries (Limits):10, 20, 30, 40, 50, 60
Count Observations & Assign to Classes

14. Frequency Distributions, Relative Frequency Distributions and Percentage Distributions
Data in Ordered Array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Relative
Frequency
Percentage
Class Frequency
[10, 20)
[20, 30)
[30, 40)
[40, 50)
[50, 60)
Total

15. Graphing Numerical Data: The Histogram
A graph of the data in a frequency distribution is called a histogram
The class boundaries (or class midpoints) are shown on the horizontal axis
the vertical axis is either frequency, relative frequency, or percentage
Bars of the appropriate heights are used to represent the number of observations within each class

16. Histogram Example (No gaps between bars) Class Midpoints
Frequency
[10, 20)
[20, 30)
[30, 40)
[40, 50)
[50, 60)
(No gaps between bars)
Class Midpoints

17. Tabulating Numerical Data: Cumulative Frequency
Data in Ordered Array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Upper Cumulative Cumulative
Limit Frequency % Frequency

18. Two categorical variables (contingency table)
The following data represent the responses to a question asked in a survey of 20 college students majoring in business –
What is your gender? (Male = M; Female = F)
What is your major? (Accountancy = A; Information System = I; Market = M)
Gender: M M M F M F F M F M F M M M M F F M F F
Major: A I I M A I A A I I A A A M I M A A A I

19. Contingency table (cont’d)
Raw data set:
Gender: M M M F M F F M F M F M M M M F F M F F
Major: A I I M A I A A I I A A A M I M A A A I A I M
Total
Male 6 4 1 11
Female 3 2 9 10 7 20

20. Graphical methods are:
Good in presenting data
Not easy for comparison
Difficult to use for statistical inference

21. Numerical description
Summary Measures
Central Tendency (location measures)
Quartiles
Variation
Range
Mean
Median
Mode
Variance
Interquartile range
Standard Deviation

22. Mean Mean (Arithmetic Mean) of Data Values Sample mean Population mean
Sample Size
Population Size

23. An example TV watching hours/week: 5, 7, 3, 38, 7
Mean = ( )/5 = 60/5 = 12
If the correct time for 4th subject is 8 (not 38)
Mean = ( )/5 = 30/5 = 6
Mean = 6
Mean = 12

24. Mean (Cont’d)
The Most Common Measure of Central Tendency especially when n is large due to its good theoretical properties
Affected by Extreme Values (Outliers)

25. Median Robust measure of central tendency
Not affected by extreme values
In an ordered array, the median is the ‘middle’ number
If n is odd, the median is the middle number (i.e,(n+1)/2 th measurement)
If n is even, the median is the average of the n/2 th and (n/2 +1) th measurement
Median = 7
Median = 7

26. Mode A Measure of Central Tendency Value that Occurs Most Often
Not Affected by Extreme Values
There May Not Be a Mode
There May Be Several Modes
Used for Either Numerical or Categorical Data
No Mode
Mode = 9

27. Quartiles 25% 25% 25% 25% Split ordered data into 4 quarters
Position of i-th quartile
(1st quartile) and (3rd quartile) are measures of Noncentral Location
are called 25th, 50th, and 75th percentile respectively. A pth percentile is the value of X such that p% of the measurements are less than X and (100-p)% are greater than X.
25%
25%
25%
25%

28. Quartiles (example) Data in Ordered Array: 3 6 6 12 12 12 15 15 18 21
Position of first quartile is
Position of third quartile is

29. 5-number summary Median( ) X X 3 6 12 15.75 21 Box-and-Whisker Plot
Graphical display of data using 5-numbers
Data in Ordered Array:
Median( ) X X
largest
smallest 3 6 12
15.75 21