1. Chapter 3, Part 1 Descriptive Statistics II: Numerical Methods
Measures of Location
Measures of Variability x s s m

2. Measures of Location
Mean
Median
Mode
Percentiles

3. Example: Apartment Rents
Given below is a sample of monthly rent values ($) for one-bedroom apartments. The data is a sample of 70 apartments in a particular city. The data are presented in ascending order.
425
430
430
435
435
435
435
435
440
440
440
440
440
445
445
445
445
445
450
450
450
450
450
450
450
460
460
460
465
465
465
470
470
472
475
475
475
480
480
480
480
485
490
490
490
500
500
500
500
510
510
515
525
525
525
535
549
550
570
570
575
575
580
590
600
600
600
600
615
615

4. Mean
The mean of a data set is the average of all the data values.
If the data are from a sample, the mean is denoted by (x-bar)
If the data are from a population, the mean is denoted by (mu). x å x x = i n m å x m = i N

5. Example: Apartment Rents
Mean å x 34 , ,
356 = i = = x = = =
490 . . 80 n 70

6. Example: Apartment Rents
Trimmed Mean
With n = 70, a 5% trimmed mean removes
.05(70) = 3.5 = 4 values from each end of the set.
5% trimmed mean = 30 , ,
206 = =
487 . . 19 62

7. Median
The median of a data set is the value in the middle when the data items are arranged in ascending order.
If there is an odd number of items, the median is the value of the middle item.
If there is an even number of items, the median is the average of the values for the middle two items.

8. Example: Apartment Rents
Median
Median = 50th percentile
i = (p/100)n = (50/100)70 = Averaging the 35th and 36th data values:
Median = ( )/2 = 475

9. Mode
The mode of a data set is the value that occurs with greatest frequency.

10. Example: Apartment Rents
Mode
450 occurred most frequently (7 times)
Mode = 450

11. Percentiles
The p th percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100-p) percent of the items take on this value or more.
Arrange the data in ascending order.
Compute index i, the position of the p th percentile.
i = (p/100)n
If i is not an integer, round up. The p th percentile is the value in the i th position.
If i is an integer, the p th percentile is the average of the values in positions i and i+1.

12. Example: Apartment Rents
90th Percentile
i = (p/100)n = (90/100)70 = 63
Averaging the 63rd and 64th data values:
90th Percentile = ( )/2 = 585

13. Quartiles Quartiles are specific percentiles.
First Quartile = 25th Percentile
Second Quartile = 50th Percentile = Median
Third Quartile = 75th Percentile

14. Example: Apartment Rents
Third Quartile
Third quartile = 75th percentile
i = (p/100)n = (75/100)70 = 52.5 = 53
Third quartile = 525

15. Measures of Variability
Range
Variance
Standard Deviation
Coefficient of Variation

16. Range
The range of a data set is the difference between the largest and smallest data values.
It is the simplest measure of dispersion.
It is very sensitive to the smallest and largest data values.

17. Example: Apartment Rents
Range
Range = largest value - smallest value
Range = = 190

18. Variance
The variance is the average of the squared differences between each data value and the mean.
If the data set is a sample, the variance is denoted by s2.
If the data set is a population, the variance is denoted by  2. - - 2 å 2 ( x x ) 2 i i s = = - n - 1 x - m ) 2 å ( x ) 2 s = i N

19. Standard Deviation
The standard deviation of a data set is the positive square root of the variance.
It is measured in the same units as the data, making it more easily comparable to the mean.
If the data set is a sample, the standard deviation is denoted s.
If the data set is a population, the standard deviation is denoted  (sigma). = s = s 2 s = = s 2

20. Coefficient of Variation
The coefficient of variation indicates how large the standard deviation is in relation to the mean.
If the data set is a sample, the coefficient of variation is computed as follows:
If the data set is a population, the coefficient of variation is computed as follows: s ( (
100 ) ) x s (
100 ) m

21. Example: Apartment Rents
Variance
Standard Deviation
Coefficient of Variation å 2 ( x x ) 2 i s = = = 2 , ,
996 . . 16 n - - 1 = = s = s 2 =
2996 . . 47 54 . . 74 s 54 . . 74 =
100 = =
100 = 11 . . 15 x
490 . 80

22. Chapter 3, Part 2 Descriptive Statistics II: Numerical Methods
Measures of Relative Location and Locating Outliers
z -Scores
Chebyshev’s Theorem
The Empirical Rule
Detecting Outliers

23. 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. x - x = i z i s

24. Example: Apartment Rents
z -Score of Smallest Value (425)
Standardized Values for Apartment Rents x - - x
425 - -
490 . . 80 = i = = = = - z - 1 . . 20 s 54 . . 74
-1.20
-1.11
-1.02
-0.93
-0.84
-0.75
-0.56
-0.47
-0.38
-0.34
-0.29
-0.20
-0.11
-0.01
0.17
0.35
0.44
0.62
0.81
1.06
1.08
1.45
1.54
1.63
1.81
1.99
2.27

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

26. Example: Apartment Rents
Chebyshev’s Theorem
Let k = 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
- k(s) = (54.74) = 409
and
+ k(s) = (54.74) = 573 x x x

27. Example: Apartment Rents
Chebyshev’s Theorem (continued)
Actually, 86% of the rent values
are between 409 and 573.

28. Empirical Rule For data having a bell-shaped distribution:
Approximately 68% of the data values will be within one standard deviation of the mean.
Approximately 95% of the data values will be within two standard deviations of the mean.
Almost all of the items (99%) will be within three standard deviations of the mean.

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

30. 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.
It might be a data value that was incorrectly included in the data set.
It might be a correctly recorded data value that belongs in the data set!

31. 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.

32. Chapter 3, Part 3 Descriptive Statistics II: Numerical Methods
Measures of Association Between Two Variables
Working with Grouped Data

33. Measures of Association Between Two Variables
Covariance
Correlation Coefficient

34. Covariance Positive values indicate a positive relationship.
Negative values indicate a negative relationship.
If the data sets are samples, the covariance is denoted by sxy.
If the data sets are populations, the covariance is denoted by

35. 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 denoted by rxy.
If the data sets are populations, the coefficient is denoted by
Where Sx and Sy are the standard
deviations for each variable!

36. Mean for Grouped Data Sample Data Population Data
where fi = frequency of class i
Mi = midpoint of class i å f M x = i i n å f M m = i i N

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

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

39. Variance for Grouped Data
Sample Data
Population Data å f ( M - x ) 2 2 = i i s n - 1 å f ( M - m ) 2 s 2 = i i N

40. Example: Apartment Rents
Sample Variance for Grouped Data
Sample Standard Deviation for Grouped Data
This approximation differs by only $.20 from the
actual standard deviation of $54.74. s 2 = 3 ,
017 . 89 s = 3 ,
017 . 89 = 54 . 94