© 2003 Thomson/South-Western
Slide 1
Chapter 3
Descriptive Statistics: Numerical Methods
Part B




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
© 2003 Thomson/South-Western
Slide 2
Measures of Relative Location
and Detecting Outliers




z-Scores
Chebyshev’s Theorem
Empirical Rule
Detecting Outliers
© 2003 Thomson/South-Western
Slide 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.
xi  x
zi 
s



A data value less than the sample mean will have a zscore 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 zscore of zero.
© 2003 Thomson/South-Western
Slide 4
Example: Apartment Rents

z-Score of Smallest Value (425)
xi  x 425  490.80
z

 1. 20
s
54. 74
Standardized Values for Apartment Rents
-1.20
-0.93
-0.75
-0.47
-0.20
0.35
1.54
-1.11
-0.93
-0.75
-0.38
-0.11
0.44
1.54
-1.11
-0.93
-0.75
-0.38
-0.01
0.62
1.63
-1.02
-0.84
-0.75
-0.34
-0.01
0.62
1.81
-1.02
-0.84
-0.75
-0.29
-0.01
0.62
1.99
© 2003 Thomson/South-Western
-1.02
-0.84
-0.56
-0.29
0.17
0.81
1.99
-1.02
-0.84
-0.56
-0.29
0.17
1.06
1.99
-1.02
-0.84
-0.56
-0.20
0.17
1.08
1.99
-0.93
-0.75
-0.47
-0.20
0.17
1.45
2.27
-0.93
-0.75
-0.47
-0.20
0.35
1.45
2.27
Slide 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.
© 2003 Thomson/South-Western
Slide 6
Example: Apartment Rents

Chebyshev’s Theorem
Let z = 1.5 with
x = 490.80 and s = 54.74
At least (1 - 1/(1.5)2) = 1 - 0.44 = 0.56 or 56%
of the rent values must be between
x - z(s) = 490.80 - 1.5(54.74) = 409
and
x + z(s) = 490.80 + 1.5(54.74) = 573
© 2003 Thomson/South-Western
Slide 7
Example: Apartment Rents

Chebyshev’s Theorem (continued)
Actually, 86% of the rent values
are between 409 and 573.
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
© 2003 Thomson/South-Western
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 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.
© 2003 Thomson/South-Western
Slide 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.
© 2003 Thomson/South-Western
Slide 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.
© 2003 Thomson/South-Western
Slide 11
Example: Apartment Rents

Empirical Rule
Within +/- 1s
Within +/- 2s
Within +/- 3s
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
Interval
436.06 to 545.54
381.32 to 600.28
326.58 to 655.02
435
445
450
472
490
525
590
435
445
450
475
490
525
600
© 2003 Thomson/South-Western
435
445
460
475
500
535
600
435
445
460
475
500
549
600
% in Interval
48/70 = 69%
68/70 = 97%
70/70 = 100%
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 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
© 2003 Thomson/South-Western
Slide 13
Example: Apartment Rents

Detecting Outliers
The most extreme z-scores are -1.20 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
-1.20
-0.93
-0.75
-0.47
-0.20
0.35
1.54
-1.11
-0.93
-0.75
-0.38
-0.11
0.44
1.54
-1.11
-0.93
-0.75
-0.38
-0.01
0.62
1.63
-1.02
-0.84
-0.75
-0.34
-0.01
0.62
1.81
-1.02
-0.84
-0.75
-0.29
-0.01
0.62
1.99
© 2003 Thomson/South-Western
-1.02
-0.84
-0.56
-0.29
0.17
0.81
1.99
-1.02
-0.84
-0.56
-0.29
0.17
1.06
1.99
-1.02
-0.84
-0.56
-0.20
0.17
1.08
1.99
-0.93
-0.75
-0.47
-0.20
0.17
1.45
2.27
-0.93
-0.75
-0.47
-0.20
0.35
1.45
2.27
Slide 14
End of Chapter 3, Part B
© 2003 Thomson/South-Western
Slide 15