Chapter 2 Descriptive Statistics II: Numerical Methods Part B

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Chapter 2 Descriptive Statistics II 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

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Measures of Relative Locationand Detecting
Outliers

• z-Scores
• The 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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The Empirical Rule

• For data having a bell-shaped distribution
• Approximately 68 of the data values will be
  within one standard deviation of the mean.
• Approximately 95 of the data values will be
  within two standard deviations of the mean.
• Almost all of the items (99.7) will be
  within three standard deviations of the mean.

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Example Apartment Rents

• The 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.
• It might be a data value that was incorrectly
  included in the data set.
• It might be 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
• Smallest Value
• First Quartile
• Median
• Third Quartile
• Largest Value

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Example Apartment Rents

• Five-Number Summary
• Lowest Value 425 First Quartile 445
• Median 475
• Third Quartile 525 Largest Value 615

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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.
• The formula is complex, and well use Excel to do
  the mathematics for us.

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Using Excel to Compute theCovariance and
Correlation Coefficient

• Formula Worksheet

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Using Excel to Compute theCovariance and
Correlation Coefficient

• Value Worksheet

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Excel, covariance and correlation

• Excel calculates a population covariance
• Excel calculates a sample correlation
• It is usually necessary to correct the covariance
  to be a sample covariance (n/(n-1))
  COVAR(array1,array2) n / (n-1) is the
  correction factor

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The Weighted Mean andWorking with Grouped Data

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

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The 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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The Weighted Mean

• xwt ? 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 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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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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End of Chapter 2, Part B