Unformatted text preview: kewness is positive (often above 1.0).
Mean will usually be more than the median.
.35 Relative Frequency .30
.25
.20
.15
.10 .05
0 Skewness = 1.25 Distribution Shape: Skewness
Example: Apartment Rents
Seventy efficiency apartments were randomly sampled in
a small college town. The monthly rent prices for these
apartments are listed in ascending order as below. 4 25
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
600 435
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 Distribution Shape: Skewness
Relative Frequency .35
.30
.25 .20
.15 .10
.05
0 Skewness = .92 ZScores
The zscore 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 ZScores
An observation’s zscore is a measure of the relative
location of the observation in a data set. 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 zscore greater than zero. A data value equal to the sample mean will have a
zscore of zero. ZScores zScore 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 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 Empirical Rule
When the data are believed to approximate a
bellshaped distribution …
The empirical rule can be used to determine the
percentage of data values that must be within a
specified number of standard deviations of the
mean.
The empirical rule is based on the normal
distribution, which is covered in Chapter 6. Empirical Rule
For data having a bellshaped
d68.26% of the values of a normal random variable
istribution:
are within +/ 1 standard deviation of its mean. 95.44% of the values of a normal random variable
are within +/ 2 standard deviations of its mean.
99.72% of the values of a normal random variable
are within +/ 3 standard deviations of its mean. Empirical Rule
99.72%
95.44%
68.26% m – 3
m – 1
m – 2 m m + 3
m + 1
m + 2 x Detecting Outliers An outlier is an unusually small or unusually large value
in a data set. A data value with a zscore 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 Detecting Outliers The most extreme zscores are 1.20 and 2.27 Using z > 3 as the criterion for an outlier, there are
no outliers in this data set.
Standardized Va...
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This note was uploaded on 09/14/2013 for the course STAT 1001 taught by Professor Kim during the Fall '11 term at Yonsei University.
 Fall '11
 Kim
 Statistics

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