1. Descriptive Statistics: Numerical Measures
Measures of Shape of a Distribution, Relative Location and Outliers
Measures of Association between Two Variables
Weighted Mean and Grouped Data

2. Shape of A Distribution Depends On the Relative Location, and Outliers
z-Scores (Standardized Values)
Chebyshev’s Theorem
Empirical Rule
Detecting Outliers

3. Distribution Shape: Skewness
An important measure of the shape of a distribution is called skewness.
The formula for computing the skewness of a data set is somewhat complex.

4. Distribution Shape: Skewness
Skeweness (S)
Is a measure of the asymmetry of a probability distribution
S=0: Symmetrical
S>0: the distribution is right (positively) skewed
S<0: the distribution is left (negatively) skewed

5. Distribution Shape: Skewness
Symmetric (not skewed)
Skewness is zero.
Mean and median are equal.
.05
.10
.15
.20
.25
.30
.35
Skewness = 0
Relative Frequency

6. Distribution Shape: Skewness
Moderately Skewed Left
Skewness is negative.
Mean will usually be less than the median.
.05
.10
.15
.20
.25
.30
.35
Skewness =
Relative Frequency

7. Distribution Shape: Skewness
Highly Skewed Right
Skewness is positive (often above 1.0).
Mean will usually be more than the median.
.05
.10
.15
.20
.25
.30
.35
Skewness = 1.25
Relative Frequency

8. Distribution Shape: Skewness
Example: Apartment Rents
The monthly rents for randomly selected 70 efficiency apartments are listed in ascending order on the next slide.

9. Distribution Shape: Skewness

10. Distribution Shape: Skewness
.05
.10
.15
.20
.25
.30
.35
Skewness = .92
Relative Frequency

11. z-Score (Standardized Value)
Is a measure of the relative location of the observation
in a data set.
Z-Score denotes the number of standard units (deviations)
a given data value xi is located from the mean.
As a result, z-score is also called a standardized value.

12. z-Score (Standardized Value)
A data value less than the sample mean
will always have a z-score less than zero.
A data value greater than the sample
mean will always have a z-score greater
than zero.
A data value equal to the sample mean
will always have a z-score of zero.

13. Recall: Rent for 70 Efficiency Apartments
To convert this data into z-scores, first we compute the mean
and the standard deviation of the data. Then we use the formula
for computing the z-score to convert each value into a standardized value;
Mean=490.80; S=54.74

14. Standardized Values for Apartment Rents
z-Scores
z-Score of Smallest Value (425)
Standardized Values for Apartment Rents

15. Chebyshev’s Theorem
A theorem that shows the position of a certain proportion
of observation in any data set relative to the mean of the
data when the values are standardized.

16. The theorem states that, for any data set---
At least of the data values must be
within of the mean.
75%
z = 2 standard deviations
At least of the data values must be
within of the mean.
89%
z = 3 standard deviations
At least of the data values must be
within of the mean.
94%
z = 4 standard deviations

17. For a data with a bell-shaped distribution:
Empirical Rule
For a data with a bell-shaped distribution:
of the values of a normal random variable
are within of its mean.
68.26%
+/- 1 standard deviation
of the values of a normal random variable
are within of its mean.
95.44%
+/- 2 standard deviations
of the values of a normal random variable
are within of its mean.
99.72%
+/- 3 standard deviations

18. Empirical Rule x 99.72% 95.44% 68.26% m m – 3s m – 1s m + 1s m + 3s

19. Z-Scores allow us to Detect Outliers
An outlier is an unusually small or unusually large
value in a data set.
It might be the result of:
an incorrect recording
an incorrectly included value in the data set
a correctly recorded data value but an unusual
occurrence
A data value with a z-score less than -3 or greater
than +3 might be considered an outlier.

20. Standardized Values for 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

21. Exploratory Data Analysis
Five-Number Summary and Box Plot

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

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

24. 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

25. Box Plot
Lower and upper 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 *

26. 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.

27. 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

28. Measures of Association between Two Variables
Covariance
Correlation Coefficient

29. Covariance

30. Covariance Covariance between two random variables ( X and Y)
is computed as follows:
for
samples
for
populations

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

32. Covariance
Use the following data on Age and Weight of five randomly selected UMD students and compute the covariance between age and weight.
Age
Weight 17
160 25
150 20
130 19
145 21
155

33. Correlation Coefficient

34. Correlation Coefficient
Correlation is a measure of the degree of linear association
between two variables.
However, it doesn’t indicate the causation. That is, just because
two variables are highly correlated, it does not mean that
one variable is the cause of the other.

35. Correlation Coefficient
The correlation coefficient is computed as follows:
for
samples
for
populations

36. 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.

37. Correlation Age Weight 17 160 25 150 20 130 19 145 21 155
Use the following data on age and weight of five randomly selected UMD students, compute the correlation between age and weight, and comment on the direction and extent of the association observed

38. Correlation Coefficient
The correlation coefficient is computed as follows:
for
samples

39. Covariance and Correlation Coefficient
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

40. Covariance and Correlation Coefficientx y
277.6
259.5
269.1
267.0
255.6
272.9 69 71 70
10.65
-7.45
2.15
0.05
-11.35
5.95
-1.0
1.0
-10.65
-7.45
-11.35
-5.95
Average
267.0
70.0
Total
-35.40
Std. Dev.
8.2192
.8944

41. Covariance and Correlation Coefficient
Sample Covariance
Sample Correlation Coefficient

42. Mean, Variance, and Standard Deviation of Grouped Data
Weighted Mean
Mean, Variance, and Standard Deviation of Grouped Data

43. Courses Credit Hours Grade Calculus 4 B Psychology 3 A Marketing C
You are taking five courses. The following table depicts the credit hours associated with each course and your grades. Compute your GPA for the semester?
Courses
Credit Hours
Grade
Calculus 4 B
Psychology 3 A
Marketing C
Economics D
Stat 2

44. Weighted Mean where: xi = value of observation i
wi = weight for observation i

45. Courses Credit Hours (Wi) Grade WiX G Calculus 4 B(3) 12 Psychology 3
You are taking five courses. The following table depicts the credit hours associated with each course and your grades. Compute your GPA for the semester?
Courses
Credit Hours (Wi)
Grade
WiX G
Calculus 4
B(3) 12
Psychology 3
A(4)
Marketing
C(2) 6
Economics
D(1)
Stat 2 8 13 41

46. Weighted Mean where: xi = value of observation i
wi = weight for observation i

47. 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.
When data values vary in importance, the analyst
must choose the weight that best reflects the
importance of each value.

48. Working with Grouped Data

49. Given below is a sample of monthly rents for 70 efficiency apartments.

50. Grouped Data

51. Computing the mean and variance of a grouped data
To compute the weighted mean from a grouped data
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 data using
class midpoints and class frequencies as weights.
Similarly, in computing the variance and standard
deviation, the class frequencies are used as weights.

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

53. Sample Mean for Grouped Data
Given below is a sample of monthly rents for 70 efficiency apartments as grouped data--- in the form of a frequency distribution.

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

55. Variance for Grouped Data
For sample data
For population data

56. Sample Variance for Grouped Data

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