PETROLEUM ENGINEERING 689 Special Topics in Unconventional Resource Reserves Lecture 1B Descriptive

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PETROLEUM ENGINEERING 689Special Topics
inUnconventional Resource ReservesLecture
1BDescriptive StatisticsTexas AM
University - Spring 2007
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Descriptive Statistics

• Part A
• Measures of Central Tendency

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Learning Objectives

• Determine the mean, median, and mode of a data
  set
• Determine geometric, harmonic, and quadratic
  means of a data set
• Determine weighted averages of data sets
• Determine the range, standard deviation,
  variance, mean absolute deviation, and
  coefficient of variation for a data set

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Definitions

• Population data set that contains all possible
  items of interest
• Sample data set that contains only a few random
  or otherwise representative elements of a data
  set
• Measure of central tendency most likely value of
  a data set
• Parameters measures of central tendency and
  other statistical characteristics that describe a
  population
• Statistics corresponding measures and
  statistical characteristics that describe a sample

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Mean

• Also called arithmetic mean
• Symbol µ represents population mean
• Symbol represents sample mean
• Commonly called average
• Calculated by adding values of all items in data
  set and dividing by total number of items in set

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Properties of Mean

• Sum of deviations from mean is zero

• Sum of squared deviations minimized when
  deviations are measured from mean

Mean may be influenced by extreme values.
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Median

• Central value in array of a data set
• Odd number of elements ? actual data element in
  middle of set
• Even number of elements ? arithmetic average of
  the two data elements in middle of array
• Found by arranging elements of data set in
  ascending or descending order and identifying
  midpoint

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Properties of the Median

• Not influenced by extreme values
• For perfectly symmetrical data set, median equals
  mean

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Mode

• Most frequently occurring data element in data
  set
• Poor measure of central tendency in most
  casesdoes not take into account values of other
  data elements

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Properties of the Mode

• Data set may have more than one mode e.g.,
  bimodal (two modes)
• Unaffected by extreme values

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Calculate Footage Drilled

• 20 bits drilled 2,013 ft
• Determine mean, median, mode

Mode (most frequent)
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Calculate Footage Drilled

• 20 bits drilled 2,013 ft
• Determine mean, median, mode

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Geometric Mean (Gm)

• Nth root of product of individual data elements
  of data set with N elements

• Calculation simplified using logarithms

• In terms of natural logarithms

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Properties of the Geometric Mean

• Biased toward smaller values appropriate for
  skewed data sets (asymmetrical distributions)
• Not affected as much as arithmetic mean by
  extreme values
• Undefined for data sets with negative or zero
  values

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Harmonic Mean, Hm

• Reciprocal of arithmetic mean of reciprocals of
  data elements in data set

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Quadratic Mean, Qm (Root Mean Square)
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Weighted Average

• Averages in which data elements are weighted by
  frequency of occurrence

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Weighted Average

• Weighted geometric mean (Gwm)

• Weighted harmonic mean (Hwm)

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Calculate Footage Drilled

• Determine geometric, harmonic, quadratic means of
  bit record

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Calculate Footage Drilled

• Determine geometric, harmonic, quadratic means of
  bit record

• Geometric mean

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Calculate Footage Drilled

• Determine geometric, harmonic, quadratic means of
  bit record

• Harmonic mean

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Calculate Footage Drilled

• Determine geometric, harmonic, quadratic means of
  bit record

• Quadratic mean

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Calculate Weighted-Average Porosity

• 22-ft pay zone
• Calculate weighted-average porosity

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Calculate Weighted-Average Cost of Capital

• Company will invest in 500,000 project
• 150,000 equity at cost of 8
• 350,000 long-term debt at 18
• Calculate weighted-average cost of capital

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Measures of Variability

• Range, R
• Difference between highest and lowest values in
  data set

• Not particularly useful measure of dispersion,
  since it uses only two values from data set

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Standard Deviation, s

• Measure of dispersion of data elements about mean
• Coupled with mean, provides more information
  about data set than any other measure

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Variance, s 2

• Simply square of standard deviation
• Not used directly in descriptive statistics

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Mean Absolute Deviation, dm

• Average deviation of data from the mean over all
  observations

• For symmetric (bell-shaped) distributions

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Coefficient of Variation, ?

• Ratio of standard deviation to mean of data set
• Expresses standard deviation as fraction of
  percentage of mean

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Calculate Statistical Values for Drilling

• Determine range, standard deviation, for bit
  record

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Calculate Statistical Values for Drilling

• Determine range, standard deviation, for bit
  record

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Calculate Statistical Values for Drilling

• Determine range, standard deviation for bit record

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• Range

• Standard deviation

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Calculate Statistical Values for Drilling

• Determine range, standard deviation for bit record

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• Range

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Calculate Statistical Values for Drilling

• Determine range, standard deviation for bit record

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• Standard deviation

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Calculate Statistical Values for Drilling

• Determine range, standard deviation for bit record

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• Standard deviation (alternate method)

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Calculate Statistical Values for Drilling

• Determine variance, mean absolute deviation for
  bit record

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• Variance (square of standard deviation)

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Calculate Statistical Values for Drilling

• Determine variance, mean absolute deviation for
  bit record

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• Mean absolute deviation

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Calculate Footage Drilled

• Determine coefficient of variation for bit record

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• Coefficient of variation

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What Weve Learned

• Determine the mean, median, and mode of a data
  set
• Determine geometric, harmonic, and quadratic
  means of a data set
• Determine weighted averages of data sets
• Determine the range, standard deviation,
  variance, mean absolute deviation, and
  coefficient of variation for a data set

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Descriptive Statistics

• End Part A

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Descriptive Statistics

• Part B
• Working With Grouped Data

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Learning Objective

• Determine measures of central tendency and
  variability for grouped data sets

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Grouping Data

• Condensing large data sets into groups simplifies
  calculations of parameters
• Steps in grouping
• Define classes for data set to be analyzed
• Determine frequency of data element appearances
  in each class
• Calculate absolute and relative frequency
• Calculate class mark (CM), or midpoint, for each
  class
• Proceed with calculation of parameters

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Guidelines for Defining Classes

• Number of classes should be between 5 and 20
• Define classes so that every element in data set
  falls into one and only one class
• No class should be empty

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Guidelines for Defining Classes

• Approximate number of classes (Nc) for data set
  with N elements

• Class interval (CI) should be same for entire
  data set

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Determining Mean
Frequency of individual class
Class mark (mid value of individual class)
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Determining Median, Mode

• Mode taken as CM of class with highest frequency
  of data elements

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Determining Geometric, Harmonic Mean

• Geometric mean

• Harmonic mean

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Determining Standard Deviation, Variance

• Standard deviation

• Variance s 2

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Calculate Parameters From Grouped Data

• Calculate measures of central tendency for
  grouped drill-bit data

• Group data

• Calculate class interval

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Calculate Parameters From Grouped Data

• Calculate measures of central tendency for
  grouped drill-bit data

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Calculate Parameters From Grouped Data

• Calculate measures of central tendency for
  grouped drill-bit data

• Mean

(compared to 100.65 ft)
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Calculate Parameters From Grouped Data

• Calculate measures of central tendency for
  grouped drill-bit data

• Median

(compared to 103.5 ft)
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Calculate Parameters From Grouped Data

• Calculate measures of central tendency for
  grouped drill-bit data

• Geometric mean

(compared to 97.97 ft)
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Calculate Parameters From Grouped Data

• Calculate measures of central tendency for
  grouped drill-bit data

• Harmonic mean

(compared to 97.97 ft)
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Calculate Parameters From Grouped Data

• Calculate measures of central tendency for
  grouped drill-bit data

• Standard deviation

(compared to 22.74 ft)
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Calculate Parameters From Grouped Data

• Calculate measures of central tendency for
  grouped drill-bit data

• Variance (s 2)

(compared to 517.19)
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What Weve Learned

• Determine measures of central tendency and
  variability for grouped data sets

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Descriptive Statistics

• End Part B