1. Five-Number Summary 1 Smallest Value 2 First Quartile 3 Median 4
Third Quartile 5
Largest Value

2. Five-Number Summary Lowest Value = 425 First Quartile = 445
Median = 475
Third Quartile = 525
Largest Value = 615

3. Box Plot A box is drawn with its ends located at the first and
third quartiles.
A vertical line is drawn in the box at the location of
the median (second quartile).
375
400
425
450
475
500
525
550
575
600
625
Q1 = 445
Q3 = 525
Q2 = 475

4. Box Plot
Limits are located (not drawn) using the interquartile range (IQR).
Data outside these limits are considered outliers.
The locations of each outlier is shown with the symbol * .
… continued

5. Box Plot The lower limit is located 1.5(IQR) below Q1.
Lower Limit: Q (IQR) = (75) = 332.5
The upper limit is located 1.5(IQR) above Q3.
Upper Limit: Q (IQR) = (75) = 637.5
There are no outliers (values less than or
greater than 637.5) in the apartment rent data.

6. Box Plot
Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits.
375
400
425
450
475
500
525
550
575
600
625
Smallest value
inside limits = 425
Largest value
inside limits = 615

7. Measures of Association Between two Variables
Covariance
Correlation coefficient

8. Covariance
Covariance is a measure of linear association between variables.
Positive values indicate a positive correlation between variables.
Negative values indicate a negative correlation between variables.

9. To compute a covariance for variables x and y
For populations
For samples

10. n = 299 II I IV
III

11. If the majority of the sample points are located in quadrants II and IV, you have a negative correlation between the variables—as we do in this case.
Thus the covariance will have a negative sign.

12. The (Pearson) Correlation Coefficient
A covariance will tell you if 2 variables are positively or negatively correlated—but it will not tell you the degree of correlation. Moreover, the covariance is sensitive to the unit of measurement. The correlation coefficient does not suffer from these defects

13. The (Pearson) Correlation Coefficient
For populations
For samples
Note that:

15. I have 7 hours per week for exercise

16. Example: Golf Stats A golfer is interested in
investigating the relationship, if any,
between driving distance and 18-hole
score.
Average Driving
Distance (yds.)
Average
18-Hole Score
277.6
259.5
269.1
267.0
255.6
272.9 69 71 70

17. Using Excel to Compute the Covariance and Correlation Coefficient
Formula Worksheet

18. Using Excel to Compute the Covariance and Correlation Coefficient
Value Worksheet

19. The Weighted Mean and Working with Grouped Data
Mean for grouped data
Variance for grouped data
Standard deviation for grouped data.

20. GPA Example
A grade point average is a weighted-mean. That is, 4- hour courses are weighted more than 3- hour courses when computing a GPA

21. The Weighted Mean
Where wi is the weight attached to observation i

22. Example: Raw Materials Purchase
Cost per Pound($)
Number of Pounds 1
3.00
1200 2
3.40
500 3
2.80
2750 4
2.90
1000 5
3.25
800
Let x1 = 3.00, x2 = 3.40, x3 = 2.80, x4 =2.90, and x5 = 3.25
Let w1 = 1200, w2 = 500, w3 = 2750, w4 =1000, and w5 =800
Thus:

23. Grouped Data The weighted mean computation can be used to
obtain approximations of the mean, variance, and
standard deviation for the grouped data.
To compute the weighted mean, we treat the
midpoint of each class as though it were the mean
of all items in the class.
We compute a weighted mean of the class midpoints
using the class frequencies as weights.
Similarly, in computing the variance and standard
deviation, the class frequencies are used as weights.

24. Sample Mean for Grouped Data
For populations
For samples
Where fi is the frequency of class i and Mi is the midpoint of class i

25. Example: Apartment Rents
Given below is the previous sample of monthly rents
for 70 studio apartments, presented here as grouped
data in the form of a frequency distribution.

26. Sample Mean for Grouped Data
This approximation
differs by $2.41 from
the actual sample
mean of $

27. Variance for Grouped Data
For populations
For samples

28. Sample Variance for Grouped Data
continued

29. Sample Standard Deviation
Sample Variance for Grouped Data
Sample Variance
Sample Standard Deviation
s2 = 208,234.29/(70 – 1) = 3,017.89
This approximation differs by only $.20
from the actual standard deviation of $54.74.