1. Chapter 3 - Part B Descriptive Statistics: Numerical Methods
Measures of Relative Location and Detecting Outliers
Exploratory Data Analysis
Measures of Association Between Two Variables
The Weighted Mean and
Working with Grouped Data   % x

2. Measures of Relative Location and Detecting Outliers
z-Scores
Chebyshev’s Theorem
Empirical Rule
Detecting Outliers

3. z-Scores The z-score is often called the standardized value.
It denotes the number of standard deviations a data value xi is from the mean.
A data value less than the sample mean will have a z-score less than zero.
A data value greater than the sample mean will have a z-score greater than zero.
A data value equal to the sample mean will have a z-score of zero.

4. Example: Apartment Rents
z-Score of Smallest Value (425)
Standardized Values for Apartment Rents

5. Chebyshev’s Theorem
At least (1 - 1/z2) of the items in any data set will be
within z standard deviations of the mean, where z is
any value greater than 1.
At least 75% of the items must be within
z = 2 standard deviations of the mean.
At least 89% of the items must be within
z = 3 standard deviations of the mean.
At least 94% of the items must be within
z = 4 standard deviations of the mean.

6. Example: Apartment Rents
Chebyshev’s Theorem
Let z = 1.5 with = and s = 54.74
At least (1 - 1/(1.5)2) = = 0.56 or 56%
of the rent values must be between
- z(s) = (54.74) = _______
and
+ z(s) = (54.74) =_______

7. Example: Apartment Rents
Chebyshev’s Theorem (continued)
Actually, 86% of the rent values
are between ____ and _____.

8. Empirical Rule For data having a bell-shaped distribution:
Approximately 68% of the data values will be within one standard deviation of the mean.

9. Empirical Rule For data having a bell-shaped distribution:
Approximately 95% of the data values will be within two standard deviations of the mean.

10. Empirical Rule For data having a bell-shaped distribution:
Almost all (99.7%) of the items will be within
three standard deviations of the mean.

11. Example: Apartment Rents
Empirical Rule
Interval % in Interval
Within +/- 1s to /70 = 69%
Within +/- 2s to /70 = 97%
Within +/- 3s to /70 = 100%

12. Detecting Outliers
An outlier is an unusually small or unusually large value in a data set.
A data value with a z-score less than -3 or greater than +3 might be considered an outlier.
It might be:
an incorrectly recorded data value
a data value that was incorrectly included in the data set
a correctly recorded data value that belongs in the data set

13. Example: Apartment Rents
Detecting Outliers
The most extreme z-scores are and 2.27.
Using |z| > 3 as the criterion for an outlier,
there are no outliers in this data set.
Standardized Values for Apartment Rents

14. Exploratory Data Analysis
Five-Number Summary
Box Plot

15. Five-Number Summary Smallest Value First Quartile Median
Third Quartile
Largest Value

16. Example: Apartment Rents
Five-Number Summary
Lowest Value = First Quartile = 445
Median = 475
Third Quartile = Largest Value = 615

17. 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.
Limits are located (not drawn) using the interquartile range (IQR).
The lower limit is located 1.5(IQR) below Q1.
The upper limit is located 1.5(IQR) above Q3.
Data outside these limits are considered outliers.
… continued

18. Box Plot (Continued)
Whiskers (dashed lines ) are drawn from the ends of the box to the smallest and largest data values inside the limits.
The locations of each outlier is shown with the symbol * .

19. Example: Apartment Rents
Box Plot
Lower Limit: Q (IQR) = (75) = 332.5
Upper Limit: Q (IQR) = (75) = 637.5
There are no outliers.
375
400
425
450
475
500
525
550
575
600
625

20. Measures of Association Between Two Variables
Covariance
Correlation Coefficient

21. Covariance
The covariance is a measure of the linear association between two variables.
Positive values indicate a positive relationship.
Negative values indicate a negative relationship.

22. Covariance
If the data sets are samples, the covariance is denoted by sxy.
If the data sets are populations, the covariance is denoted by

23. Example: Panthers Football Team
x = Number of y = Number of
Interceptions Points Scored

24. Correlation Coefficient
The coefficient can take on values between -1 and +1.
Values near -1 indicate a strong negative linear relationship.
Values near +1 indicate a strong positive linear relationship.
If the data sets are samples, the coefficient is rxy.
If the data sets are populations, the coefficient is xy.

25. The Weighted Mean and Working with Grouped Data
Mean for Grouped Data
Variance for Grouped Data
Standard Deviation for Grouped Data

26. Weighted Mean
When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean.
In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade.
When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value.

27. Weighted Mean  wi xi x = ___________  wi where:
xi = value of observation i
wi = weight for observation i

28. 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.

29. Mean for Grouped Data Sample Data Population Data where:
fi = frequency of class i
Mi = midpoint of class i

30. Example: Apartment Rents
Given below is the previous sample of monthly rents
for one-bedroom apartments presented here as grouped
data in the form of a frequency distribution.

31. Example: Apartment Rents
Mean for Grouped Data
This approximation differs by $2.41 from
the actual sample mean of $_______.

32. Variance for Grouped Data
Sample Data
Population Data

33. Example: Apartment Rents
Variance for Grouped Data
Standard Deviation for Grouped Data
This approximation differs by only $_____
from the actual standard deviation of $______.

34. A 5-Minute In-Class Exercise
With = and s = 54.74:
1. What is the z-score for an observation value Xi= 600?
Z =
2. According to Chebyshev’s Theorem, if z = , then What Percentage of the data set values must be between what Lower Limit and what Upper Limit?
Percentage =
Lower Limit =
Upper Limit =

35. End of Chapter 3, Part B