Chapter 3 Descriptive Statistics: Numerical Methods Part B

1
Chapter 3 Descriptive Statistics Numerical
MethodsPart 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

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Measures of Relative Locationand Detecting
Outliers

• z-Scores
• Chebyshevs Theorem
• Empirical Rule
• Detecting Outliers

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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.

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Example Apartment Rents

• z-Score of Smallest Value (425)
• Standardized Values for Apartment Rents

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Chebyshevs 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.

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Example Apartment Rents

• Chebyshevs 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) 409
• and
• z(s) 490.80 1.5(54.74) 573

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Example Apartment Rents

• Chebyshevs Theorem (continued)
• Actually, 86 of the rent values
• are between 409 and 573.

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Empirical Rule

• For data having a bell-shaped
  distribution
• Approximately 68 of the data values will be
  within one standard deviation of the mean.

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Empirical Rule

• For data having a bell-shaped distribution
• Approximately 95 of the data values will be
  within two standard deviations of the mean.

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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.

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Example Apartment Rents

• Empirical Rule
• Interval in Interval
• Within /- 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

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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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Example Apartment Rents

• Detecting Outliers
• The most extreme z-scores are -1.20 and 2.27.
• Using z gt 3 as the criterion for an outlier,
• there are no outliers in this data set.
• Standardized Values for Apartment Rents

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Exploratory Data Analysis

• Five-Number Summary
• Box Plot

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Five-Number Summary

• Smallest Value
• First Quartile
• Median
• Third Quartile
• Largest Value

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Example Apartment Rents

• Five-Number Summary
• Lowest Value 425 First Quartile 450
• Median 475
• Third Quartile 525 Largest Value 615

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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

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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 .

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Example Apartment Rents

• Box Plot
• Lower Limit Q1 - 1.5(IQR) 450 - 1.5(75)
  337.5
• Upper Limit Q3 1.5(IQR) 525 1.5(75)
  637.5
• There are no outliers.

575
600
625
375
400
450
500
525
425
475
550
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Measures of Association Between Two Variables

• Covariance
• Correlation Coefficient

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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.

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Covariance

• If the data sets are samples, the covariance is
  denoted by sxy.
• If the data sets are populations, the covariance
  is denoted by .

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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 .

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The Weighted Mean andWorking with Grouped Data

• Weighted Mean
• Mean for Grouped Data
• Variance for Grouped Data
• Standard Deviation for Grouped Data

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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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Weighted Mean

• x ? wi xi
• ? wi
• where
• xi value of observation i
• wi weight for observation i

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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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Mean for Grouped Data

• Sample Data
• Population Data
• where
• fi frequency of class i
• Mi midpoint of class i

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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.

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Example Apartment Rents

• Mean for Grouped Data
• This approximation
  differs by 2.41 from
• the actual sample mean of
  490.80.

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Variance for Grouped Data

• Sample Data
• Population Data

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Example Apartment Rents

• Variance for Grouped Data
• Standard Deviation for Grouped Data
• This approximation differs by only .20
• from the actual standard deviation of 54.74.