1. Chapter 4
Describing Data

2. What is Average? Central tendency Effect of Outliners Mean Median Mode
A value that is much higher or much lower than almost all other values

3. Mean, Median, Mode Measure Common? Exist?
Takes every values into account?
Affect by outliners?
Advantages
Mean
Most familiar “average”
Always
Yes
Commonly understood;
Statistically used
Median
Common
No (aside from counting) No
Good if there is outliner
Mode
Sometimes used
0, 1 or more no
For nominal level

4. Confusion about “Average”
Example 5 – Which mean?
All 100 first-year students at a small college take three courses. 2 courses are taught in large lectures, with 100 students in a single class. The third course is taught in 10 classes of 10 students each. Students and administrators get into an argument about whether classes are too large. The students claim that the mean size of their core courses is 70. The administrators claim that the mean class size is only 25. ???

5. Example 5 – Which mean? (cont.)
Students
Mean size of the classes in which each student is personally enrolled
Each student is taking 2 classes with enrolment of 100 each and one class with an enrolment of 10, so the mean size of each student’s classes is:

6. Example 5 – Which mean? (cont.)
Administrators
Mean enrollment in all classes
There are 2 classes with 100 students each and 10 classes with 10 students each, thus a total of 300 students in 12 classes:

7. Weighted Mean
Accounts for variations in the relative importance of data values.
Each data value is assigned a weight, and the weighted mean is:

8. Weighted mean (cont.) Example 6 GPA Example 7 Stock Voting Stockholder
Shared owned
Vote A
225 Y B
170 C
275 D
500 N E 90

9. Exercise
Ex29-40 pp

10. Shapes of Distribution
Number of modes
Single-peaked (unimodal), bimodal, trimodal
Symmetry or skewness
Symmetric – mirror image
Left-skew
Mean<Median
Right-skew
Mean>Median
Variation
Spread of data

11. Exercise
Ex15-23 p163

12. Measures of Variation Variation, spread, deviation How data differ?
Quantitative measurement

13. Measures of Variation (cont.)
Range
Difference between its highest and lowest data values
Affect by extreme values
Use 2 data values only and ignore other variation within the data set

14. Measures of Variation (cont.)
Quartiles and the Five-Number Summary
Quartiles
First(25%), second(50%), third(75%) quartiles
Five-Number Summary
Lowest value
Lower quartile (first quartile)
Median (second quartile)
Upper quartile (third quartile)
Highest value

15. Measures of Variation (cont.)
Percentile
Dividing a data set into 100 parts
The nth percentile of a data set has at least n% data lies below it

16. Measures of Variation (cont.)
Standard Deviation
An indicator for spread(dispersion, variation)
SD=0, all data are the same, i.e. no variation
SD>0, the further SD is away from 0, indicating the higher the spread of the data values

17. Measures of Variation (cont.)
Advantage of SD
Using one single value to describe the spread
All data are involved to calculate SD, i.e. representative
Easy to understand and make comparison
Use in statistical analysis
Rough estimate of SD : range/4

18. Statistical Paradoxes
Better in each case, but worse Overall
Basketball Shots
First half
Second half
Baskets
Attempts
Percent
Henry 4 10
40% 3
75%
Michael 1
25% 7
70%
Overall
Baskets
Attempts
Percent
Henry 7 14
50%
Michael 8
57.1%

19. Exercise
Ex 5-8 p174
Chapter Review Exercises
3 a-f p186

20. Focus on Economics
Are the Rich Getting Richer? pp