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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slides by
JohnLoucks
St. Edward’sUniversity
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 3, Part B Descriptive Statistics: Numerical
Measures Measures of Distribution Shape, Relative
Location, and Detecting Outliers Exploratory Data Analysis Measures of Association Between Two
Variables The Weighted Mean and Working with Grouped Data
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Measures of Distribution Shape,Relative Location, and Detecting Outliers
Distribution Shape z-Scores Chebyshev’s
Theorem Empirical Rule Detecting Outliers
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Distribution Shape: Skewness An important measure of the shape of a
distribution is called skewness. The formula for the skewness of sample data is
Skewness can be easily computed using statistical software.
3
)2)(1(Skewness
sxx
nnn i
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Distribution Shape: Skewness Symmetric (not skewed)
Rela
tive
Freq
uenc
y
.05
.10
.15
.20
.25
.30
.35
0
Skewness = 0
• Skewness is zero.• Mean and median are equal.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Rela
tive
Freq
uenc
y
.05
.10
.15
.20
.25
.30
.35
0
Distribution Shape: Skewness Moderately Skewed Left
Skewness = .31
• Skewness is negative.• Mean will usually be less than the median.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Distribution Shape: Skewness Moderately Skewed Right
Rela
tive
Freq
uenc
y
.05
.10
.15
.20
.25
.30
.35
0
Skewness = .31
• Skewness is positive.• Mean will usually be more than the median.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Distribution Shape: Skewness Highly Skewed Right
Rela
tive
Freq
uenc
y
.05
.10
.15
.20
.25
.30
.35
0
Skewness = 1.25
• Skewness is positive (often above 1.0).• Mean will usually be more than the median.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Seventy efficiency apartments were randomly
sampled in a college town. The monthly rent prices
for the apartments are listed below in ascending order.
Distribution Shape: Skewness Example: Apartment Rents
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
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Rela
tive
Freq
uenc
y
.05
.10
.15
.20
.25
.30
.35
0
Skewness = .92
Distribution Shape: Skewness Example: Apartment Rents
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or duplicated, or posted to a publicly accessible website, in whole or in part.
The z-score is often called the standardized value.
It denotes the number of standard deviations a data value xi is from the mean.
z-Scores
z x xsii
Excel’s STANDARDIZE function can be used to compute the z-score.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
z-Scores
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.
An observation’s z-score is a measure of the relative location of the observation in a data set.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
• z-Score of Smallest Value (425)425 490.80 1.2054.74
ix xzs
z-Scores
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
Example: Apartment Rents
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or duplicated, or posted to a publicly accessible website, in whole or in part.
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.
Chebyshev’s theorem requires z > 1, but z need not be an integer.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
At least of the data values must be within of the mean.
75% z = 2 standard deviations
Chebyshev’s Theorem
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
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Chebyshev’s Theorem
Let z = 1.5 with = 490.80 and s = 54.74x
At least (1 1/(1.5)2) = 1 0.44 = 0.56 or 56%of the rent values must be betweenx - z(s) = 490.80 1.5(54.74) = 409
andx + z(s) = 490.80 + 1.5(54.74) = 573
(Actually, 86% of the rent values are between 409 and 573.)
Example: Apartment Rents
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Empirical Rule
When the data are believed to approximate a bell-shaped distribution …
The empirical rule is based on the normal distribution, which is covered in Chapter 6.
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.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Empirical Rule
For data having 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
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Empirical Rule
xm – 3s m – 1s
m – 2sm + 1s
m + 2sm + 3sm
68.26%95.44%99.72%
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or duplicated, or posted to a publicly accessible website, in whole or in part.
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
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or duplicated, or posted to a publicly accessible website, in whole or in part.
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.
-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
Standardized Values for Apartment Rents
Example: Apartment Rents
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Exploratory Data Analysis
Exploratory data analysis procedures enable us to use simple arithmetic and easy-to-draw pictures to summarize data.
We simply sort the data values into ascending order and identify the five-number summary and then construct a box plot.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Five-Number Summary
1 Smallest Value
First Quartile Median Third Quartile Largest Value
2345
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Five-Number Summary
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
Lowest Value = 425 First Quartile = 445Median = 475
Third Quartile = 525Largest Value = 615
Example: Apartment Rents
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Box Plot
A box plot is a graphical summary of data that is based on a five-number summary.
A key to the development of a box plot is the computation of the median and the quartiles Q1 and Q3.
Box plots provide another way to identify outliers.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
400
425
450
475
500
525
550
575
600
625
• A box is drawn with its ends located at the first and third quartiles.
Box Plot
• A vertical line is drawn in the box at the location of the median (second quartile).
Q1 = 445 Q3 = 525Q2 = 475
Example: Apartment Rents
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Box Plot 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 * .
continued
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Box Plot
Lower Limit: Q1 - 1.5(IQR) = 445 - 1.5(80) = 325
Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(80) = 645
• The lower limit is located 1.5(IQR) below Q1.
• The upper limit is located 1.5(IQR) above Q3.
• There are no outliers (values less than 325 or greater than 645) in the apartment rent data.
Example: Apartment Rents
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Box Plot
• Whiskers (dashed lines) are drawn from the ends
of the box to the smallest and largest data values
inside the limits.
400
425
450
475
500
525
550
575
600
625
Smallest valueinside limits = 425
Largest valueinside limits = 615
Example: Apartment Rents
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Measures of Association Between Two Variables
Thus far we have examined numerical methods used to summarize the data for one variable at a time.
Often a manager or decision maker is interested in the relationship between two variables.
Two descriptive measures of the relationship between two variables are covariance and correlation coefficient.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Covariance
Positive values indicate a positive relationship.
Negative values indicate a negative relationship.
The covariance is a measure of the linear association between two variables.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Covariance
The covariance is computed as follows:
forsamples
forpopulations
s x x y ynxy
i i
( )( )
1
sm m
xyi x i yx y
N
( )( )
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Correlation Coefficient
Just because two variables are highly correlated, it does not mean that one variable is the cause of the other.
Correlation is a measure of linear association and not necessarily causation.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
The correlation coefficient is computed as follows:
forsamples
forpopulations
rss sxyxy
x y
ss sxyxy
x y
Correlation Coefficient
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Correlation Coefficient
Values near +1 indicate a strong positive linear relationship.
Values near -1 indicate a strong negative linear relationship.
The coefficient can take on values between -1 and +1.
The closer the correlation is to zero, the weaker the relationship.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
A golfer is interested in investigating therelationship, if any, between driving distance
and 18-hole score.
277.6259.5269.1267.0255.6272.9
697170707169
Average DrivingDistance (yds.)
Average18-Hole Score
Covariance and Correlation Coefficient Example: Golfing Study
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Covariance and Correlation Coefficient
277.6259.5269.1267.0255.6272.9
697170707169
x y
10.65 -7.45 2.15 0.05-11.35 5.95
-1.0 1.0 0 0 1.0-1.0
-10.65 -7.45 0 0-11.35 -5.95
( )ix x ( )( )i ix x y y ( )iy y
AverageStd. Dev.
267.0 70.0 -35.408.2192.8944
Total
Example: Golfing Study
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or duplicated, or posted to a publicly accessible website, in whole or in part.
• Sample Covariance
• Sample Correlation Coefficient
Covariance and Correlation Coefficient
7.08 -.9631(8.2192)(.8944)xy
xyx y
sr
s s
( )( ) 35.40 7.081 6 1i i
xyx x y y
sn
Example: Golfing Study
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Using Excel to Compute theCovariance and Correlation Coefficient
• Excel Formula Worksheet Example: Golfing Study
A B C D
1Average
Drive18-Hole Score
2 277.6 69 Pop. Covariance =COVARIANCE.S(A2:A7,B2:B7)3 259.5 71 Samp. Correlation =CORREL(A2:A7,B2:B7)4 269.1 705 267.0 706 255.6 717 272.9 698
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Using Excel to Compute theCovariance and Correlation Coefficient
• Excel Value Worksheet Example: Golfing Study
A B C D
1Average
Drive18-Hole Score
2 277.6 69 Pop. Covariance -5.93 259.5 71 Samp. Correlation -0.96314 269.1 705 267.0 706 255.6 717 272.9 698
Sample Covariance = sxy = n/(n – 1)sxy = 6/(6 – 1)(-5.9) = -7.08
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or duplicated, or posted to a publicly accessible website, in whole or in part.
The Weighted Mean andWorking with Grouped Data
Weighted Mean Mean for Grouped Data Variance for Grouped Data Standard Deviation for Grouped Data
42 Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
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.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Weighted Mean
i i
i
wxx
w
where: xi = value of observation i wi = weight for observation i
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or duplicated, or posted to a publicly accessible website, in whole or in part.
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.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Mean for Grouped Data
i if Mx
n
NMf iim
where: fi = frequency of class i Mi = midpoint of class i
Sample Data
Population Data
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or duplicated, or posted to a publicly accessible website, in whole or in part.
The previously presented sample of apartment
rents is shown here as grouped data in the form of
a frequency distribution.
Sample Mean for Grouped Data
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
47 Slide© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Sample Mean for Grouped Data
This approximationdiffers by $2.41 fromthe actual samplemean of $490.80.
34,525 493.2170x
Rent ($) f i
420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6
Total 70
M i
429.5449.5469.5489.5509.5529.5549.5569.5589.5609.5
f iM i
3436.07641.55634.03916.03566.52118.01099.02278.01179.03657.034525.0
Example: Apartment Rents
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Variance for Grouped Data
s f M xn
i i22
1
( )
s m22
f M
Ni i( )
For sample data
For population data
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Sample Variance for Grouped Data
continued
Rent ($) f i
420-439 8440-459 17460-479 12480-499 8500-519 7520-539 4540-559 2560-579 4580-599 2600-619 6
Total 70
M i
429.5449.5469.5489.5509.5529.5549.5569.5589.5609.5
M i - x-63.7-43.7-23.7-3.716.336.356.376.396.3116.3
(M i - x )2
4058.961910.56562.1613.76
265.361316.963168.565820.169271.76
13523.36
f i(M i - x )2
32471.7132479.596745.97110.11
1857.555267.866337.13
23280.6618543.5381140.18
208234.29
Example: Apartment Rents
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or duplicated, or posted to a publicly accessible website, in whole or in part.
3,017.89 54.94s
s2 = 208,234.29/(70 – 1) = 3,017.89
This approximation differs by only $.20 from the actual standard deviation of $54.74.
• Sample Variance
• Sample Standard Deviation
Example: Apartment Rents
Sample Variance for Grouped Data
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or duplicated, or posted to a publicly accessible website, in whole or in part.
End of Chapter 3, Part B