Chapter 3 - Part B Descriptive Statistics: Numerical Methods Measures of Relative Location and...
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Transcript of Chapter 3 - Part B Descriptive Statistics: Numerical Methods Measures of Relative Location and...
Chapter 3 - Part B Descriptive Statistics: Numerical Methods
Measures of Relative Location and Detecting OutliersExploratory Data AnalysisMeasures of Association Between Two VariablesThe Weighted Mean and Working with Grouped Data
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Measures of Relative Locationand Detecting Outliers
z-ScoresChebyshev’s TheoremEmpirical RuleDetecting Outliers
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.
zx xsii
zx xsii
z-Score of Smallest Value (425)
Standardized Values for Apartment Rents
74.54
80.490425
s
xxz i
74.54
80.490425
s
xxz i
-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27
-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27
Example: Apartment Rents
Chebyshev’s Theorem
At least (1 - 1/z2) of the items in any data set will bewithin z standard deviations of the mean, where z isany value greater than 1.
– At least 75% of the items must be withinz = 2 standard deviations of the mean.
– At least 89% of the items must be withinz = 3 standard deviations of the mean.
– At least 94% of the items must be withinz = 4 standard deviations of the mean.
Example: Apartment Rents
Chebyshev’s Theorem
Let z = 1.5 with = 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 - z(s) = 490.80 - 1.5(54.74) =
_______ and
+ z(s) = 490.80 + 1.5(54.74) =_______
xx
xx
xx
Chebyshev’s Theorem (continued) Actually, 86% of the rent values
are between ____ and _____.
425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615
425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615
Example: Apartment Rents
Empirical Rule
For data having a bell-shaped distribution:
– Approximately 68% of the data values will be within one standard deviation of the mean.
Empirical Rule
For data having a bell-shaped distribution:
– Approximately 95% of the data values will be within two standard deviations of the mean.
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.
Example: Apartment Rents
Empirical Rule Interval % in
IntervalWithin +/- 1s 436.06 to 545.54 48/70 = 69%Within +/- 2s 381.32 to 600.28 68/70 = 97%Within +/- 3s 326.58 to 655.02 70/70 = 100%
425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615
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
Example: Apartment Rents
Detecting OutliersThe 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 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27
-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27
Exploratory Data Analysis
Five-Number SummaryBox Plot
Five-Number Summary
Smallest ValueFirst QuartileMedianThird QuartileLargest Value
Example: Apartment Rents
Five-Number SummaryLowest Value = 425 First Quartile
= 445 Median = 475
Third Quartile = 525 Largest Value = 615
425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615
425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615
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
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 * .
Example: Apartment Rents
Box Plot
Lower Limit: Q1 - 1.5(IQR) = 445 - 1.5(75) = 332.5
Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5
There are no outliers.
375375
400400
425425
450450
475475
500500
525525
550550 575575 600600 625625
Measures of Association Between Two Variables
CovarianceCorrelation Coefficient
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.
If the data sets are samples, the covariance is denoted by sxy.
If the data sets are populations, the covariance is denoted by .
Covariance
sx x y ynxy
i i
( )( )
1s
x x y ynxy
i i
( )( )
1
xyi x i yx y
N
( )( )
xy
i x i yx y
N
( )( )
xyxy
x = Number of y = Number ofInterceptions Points Scored
1 14 3 24 2 18 1 17 3 27
----------------- --------------------
Example: Panthers Football Team
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.
rs
s sxyxy
x yrs
s sxyxy
x y
xyxy
x y
xyxy
x y
The Weighted Mean andWorking with Grouped Data
Weighted MeanMean for Grouped DataVariance for Grouped DataStandard Deviation for Grouped Data
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.
Weighted Mean
wi xi
x = ___________ wi
where: xi = value of observation i
wi = weight for observation i
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.
Sample Data
Population Data
where: fi = frequency of class i
Mi = midpoint of class i
Mean for Grouped Data
i
ii
f
Mfx
i
ii
f
Mfx
N
Mf iiN
Mf ii
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.
Rent ($) Frequency420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6
Rent ($) Frequency420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6
Example: Apartment Rents
Mean for Grouped Data
This approximation differs by $2.41 from
the actual sample mean of $_______.
Rent ($) f i M i f iM i
420-439 8 429.5 3436.0440-459 17 449.5 7641.5460-479 12 469.5 5634.0480-499 8 489.5 3916.0500-519 7 509.5 3566.5520-539 4 529.5 2118.0540-559 2 549.5 1099.0560-579 4 569.5 2278.0580-599 2 589.5 1179.0600-619 6 609.5 3657.0
Total 70 34525.0
Rent ($) f i M i f iM i
420-439 8 429.5 3436.0440-459 17 449.5 7641.5460-479 12 469.5 5634.0480-499 8 489.5 3916.0500-519 7 509.5 3566.5520-539 4 529.5 2118.0540-559 2 549.5 1099.0560-579 4 569.5 2278.0580-599 2 589.5 1179.0600-619 6 609.5 3657.0
Total 70 34525.0
21.49370
525,34_
x 21.49370
525,34_
x
Sample Data
Population Data
Variance for Grouped Data
sf M xn
i i22
1
( )s
f M xn
i i22
1
( )
22
f M
Ni i( ) 2
2
f M
Ni i( )
Example: Apartment Rents
Variance for Grouped Data
Standard Deviation for Grouped Data
This approximation differs by only $_____ from the actual standard deviation of $______.
s2 3 017 89 , .s2 3 017 89 , .
s 3 017 89 54 94, . .s 3 017 89 54 94, . .
A 5-Minute In-Class Exercise
With = 490.80 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 =
3.16228, then What Percentage of the data set values must be between what Lower Limit and what Upper Limit?
Percentage = Lower Limit =Upper Limit =
xx
End of Chapter 3, Part B